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1
Derived Categories in Algebraic GeometryA first glance
Paolo Stellari
2
Outline
3
Outline
4
The geometric setting
Let X be a smooth projectivevariety defined over analgebraically closed field K
Example
Consider for example the zerolocus of
xd0 + + xd
n
in the projective space PnK for
an integer d ge 1
(Vague) Question
How do we categorize X
Find a categoryassociated to X which encodesimportant bits
of its geometry
5
The geometric setting
In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X
They have locally a finitepresentation
The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence
0 rarr E1 rarr E2 rarr E3 rarr 0
Example
The local descriptionmentioned before can bethought as follows
Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence
Roplusk1 rarr Roplusk2 rarr M rarr 0
for some non-negative integersk1 k2
6
Coherent sheaves are too much
We then have the following (simplified version of a) classicalresult
Theorem (Gabriel)
Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)
It has been generalized invarious contexts by AntieauCanonaco-S Perego
Coh(X ) encodes too muchof the geometry of X
7
Looking for alternatives
Aim
We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)
For example the categories associated to X1 and to X2 must beclose if
X1 and X2 are nicely related by birational transformations
One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)
8
Looking for alternatives
Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves
Db(X ) = Db(Coh(X ))
on X
Objects bounded complexes of coherent sheaves
rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr
Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)
9
Basic operations
Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])
We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )
Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles
E1 rarr E2 rarr E3 rarr E1[1]
We say that E2 is an extension of E1 and E3
Important construction
For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
2
Outline
3
Outline
4
The geometric setting
Let X be a smooth projectivevariety defined over analgebraically closed field K
Example
Consider for example the zerolocus of
xd0 + + xd
n
in the projective space PnK for
an integer d ge 1
(Vague) Question
How do we categorize X
Find a categoryassociated to X which encodesimportant bits
of its geometry
5
The geometric setting
In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X
They have locally a finitepresentation
The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence
0 rarr E1 rarr E2 rarr E3 rarr 0
Example
The local descriptionmentioned before can bethought as follows
Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence
Roplusk1 rarr Roplusk2 rarr M rarr 0
for some non-negative integersk1 k2
6
Coherent sheaves are too much
We then have the following (simplified version of a) classicalresult
Theorem (Gabriel)
Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)
It has been generalized invarious contexts by AntieauCanonaco-S Perego
Coh(X ) encodes too muchof the geometry of X
7
Looking for alternatives
Aim
We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)
For example the categories associated to X1 and to X2 must beclose if
X1 and X2 are nicely related by birational transformations
One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)
8
Looking for alternatives
Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves
Db(X ) = Db(Coh(X ))
on X
Objects bounded complexes of coherent sheaves
rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr
Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)
9
Basic operations
Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])
We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )
Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles
E1 rarr E2 rarr E3 rarr E1[1]
We say that E2 is an extension of E1 and E3
Important construction
For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
3
Outline
4
The geometric setting
Let X be a smooth projectivevariety defined over analgebraically closed field K
Example
Consider for example the zerolocus of
xd0 + + xd
n
in the projective space PnK for
an integer d ge 1
(Vague) Question
How do we categorize X
Find a categoryassociated to X which encodesimportant bits
of its geometry
5
The geometric setting
In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X
They have locally a finitepresentation
The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence
0 rarr E1 rarr E2 rarr E3 rarr 0
Example
The local descriptionmentioned before can bethought as follows
Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence
Roplusk1 rarr Roplusk2 rarr M rarr 0
for some non-negative integersk1 k2
6
Coherent sheaves are too much
We then have the following (simplified version of a) classicalresult
Theorem (Gabriel)
Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)
It has been generalized invarious contexts by AntieauCanonaco-S Perego
Coh(X ) encodes too muchof the geometry of X
7
Looking for alternatives
Aim
We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)
For example the categories associated to X1 and to X2 must beclose if
X1 and X2 are nicely related by birational transformations
One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)
8
Looking for alternatives
Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves
Db(X ) = Db(Coh(X ))
on X
Objects bounded complexes of coherent sheaves
rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr
Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)
9
Basic operations
Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])
We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )
Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles
E1 rarr E2 rarr E3 rarr E1[1]
We say that E2 is an extension of E1 and E3
Important construction
For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
4
The geometric setting
Let X be a smooth projectivevariety defined over analgebraically closed field K
Example
Consider for example the zerolocus of
xd0 + + xd
n
in the projective space PnK for
an integer d ge 1
(Vague) Question
How do we categorize X
Find a categoryassociated to X which encodesimportant bits
of its geometry
5
The geometric setting
In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X
They have locally a finitepresentation
The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence
0 rarr E1 rarr E2 rarr E3 rarr 0
Example
The local descriptionmentioned before can bethought as follows
Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence
Roplusk1 rarr Roplusk2 rarr M rarr 0
for some non-negative integersk1 k2
6
Coherent sheaves are too much
We then have the following (simplified version of a) classicalresult
Theorem (Gabriel)
Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)
It has been generalized invarious contexts by AntieauCanonaco-S Perego
Coh(X ) encodes too muchof the geometry of X
7
Looking for alternatives
Aim
We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)
For example the categories associated to X1 and to X2 must beclose if
X1 and X2 are nicely related by birational transformations
One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)
8
Looking for alternatives
Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves
Db(X ) = Db(Coh(X ))
on X
Objects bounded complexes of coherent sheaves
rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr
Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)
9
Basic operations
Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])
We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )
Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles
E1 rarr E2 rarr E3 rarr E1[1]
We say that E2 is an extension of E1 and E3
Important construction
For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
5
The geometric setting
In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X
They have locally a finitepresentation
The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence
0 rarr E1 rarr E2 rarr E3 rarr 0
Example
The local descriptionmentioned before can bethought as follows
Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence
Roplusk1 rarr Roplusk2 rarr M rarr 0
for some non-negative integersk1 k2
6
Coherent sheaves are too much
We then have the following (simplified version of a) classicalresult
Theorem (Gabriel)
Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)
It has been generalized invarious contexts by AntieauCanonaco-S Perego
Coh(X ) encodes too muchof the geometry of X
7
Looking for alternatives
Aim
We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)
For example the categories associated to X1 and to X2 must beclose if
X1 and X2 are nicely related by birational transformations
One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)
8
Looking for alternatives
Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves
Db(X ) = Db(Coh(X ))
on X
Objects bounded complexes of coherent sheaves
rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr
Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)
9
Basic operations
Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])
We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )
Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles
E1 rarr E2 rarr E3 rarr E1[1]
We say that E2 is an extension of E1 and E3
Important construction
For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
6
Coherent sheaves are too much
We then have the following (simplified version of a) classicalresult
Theorem (Gabriel)
Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)
It has been generalized invarious contexts by AntieauCanonaco-S Perego
Coh(X ) encodes too muchof the geometry of X
7
Looking for alternatives
Aim
We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)
For example the categories associated to X1 and to X2 must beclose if
X1 and X2 are nicely related by birational transformations
One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)
8
Looking for alternatives
Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves
Db(X ) = Db(Coh(X ))
on X
Objects bounded complexes of coherent sheaves
rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr
Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)
9
Basic operations
Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])
We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )
Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles
E1 rarr E2 rarr E3 rarr E1[1]
We say that E2 is an extension of E1 and E3
Important construction
For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
7
Looking for alternatives
Aim
We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)
For example the categories associated to X1 and to X2 must beclose if
X1 and X2 are nicely related by birational transformations
One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)
8
Looking for alternatives
Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves
Db(X ) = Db(Coh(X ))
on X
Objects bounded complexes of coherent sheaves
rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr
Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)
9
Basic operations
Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])
We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )
Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles
E1 rarr E2 rarr E3 rarr E1[1]
We say that E2 is an extension of E1 and E3
Important construction
For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
8
Looking for alternatives
Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves
Db(X ) = Db(Coh(X ))
on X
Objects bounded complexes of coherent sheaves
rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr
Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)
9
Basic operations
Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])
We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )
Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles
E1 rarr E2 rarr E3 rarr E1[1]
We say that E2 is an extension of E1 and E3
Important construction
For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
9
Basic operations
Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])
We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )
Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles
E1 rarr E2 rarr E3 rarr E1[1]
We say that E2 is an extension of E1 and E3
Important construction
For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
10
Outline
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
11
Summary
Coh(X )is too much
A new hope Db(X )
How do wedescribeobjects of
Db(X )
How do weunderstand
the structureof Db(X )
Stability
Applications
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
12
How do we describe objects of Db(X )
Let us go back to a special instance of the first example
Example (n = 2 d ge 1 K = C)
Consider the planar curve C which is the zero locus of
xd0 + xd
1 + xd2
in the projective space P2
Take E isin Db(C) of the form
rarr E iminus1 d iminus1minusrarr E i d i
minusrarr E i+1 rarr
and define
Hi(E) =ker (d i)
Im(d iminus1)isin Coh(C)
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
13
How do we describe objects of Db(X )
In our example we then have an isomorphism in Db(C)
E sim=983120
i
Hi(E)[minusi]
On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points
In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves
Warning 1
For higher dimensional varieties this is false
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
14
How do we understand the structure of Db(X )
We say that Db(X ) has a semiorthogonal decompisition
Db(X ) = 〈A1 Ak 〉
if
Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak
There are no Homs from right to left between the ksubcategories
A1
1003534983577983577A2
1003534983574983574
1003536983640983640
middot middot middot1003534983576983576
1003536983640983640
Ak
1003536983638983638
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
15
How do we understand the structure of Db(X )
Consider again the case of the planar curve C
xd0 + xd
1 + xd2 = 0
d = 1 2Genus g = 0
We have
Db(C) = 〈OC OC(1)〉
where
〈OC(i)〉 sim= Db(pt)
for i = 0 1
d = 3Genus g = 1
Db(C) is indec
But interestingautoequivalecegroup
d ge 4Genus g ge 2
Db(C) is indec
But uninterestingautoequivalecegroup
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
16
How do we understand the structure of Db(X )
Conclusion
In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that
The components are simpler and of lsquosmaller dimensionrsquo
Get a dimension reduction a component encodes much ofthe geometry of X
Warning 2
Semiorthogonal decompositions are in general non-canonical
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
17
Summary
Warning 1Difficult to describe
objects
Warning 2Difficult to describe
the structure of Db(X )
We needmore structure
on Db(X )
Stability
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
18
Outline
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
19
Back to sheaves
Take n = 2 in our example (but this can make it work for anysmooth projective variety)
The idea is that we want to filter any coherent sheaf in acanonical way
More precisely
We have an abelian category Coh(C)
We define a function
microslope(minus) =deg(minus)
rk (minus)
(or +infin when the denominator is 0) defined on Coh(C)
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
20
Back to sheaves
Definition
A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have
microslope(F ) lt (le)microslope(E)
HarderndashNarasimhan filtration
Any sheaf E has a filtration
0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E
such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
21
From sheaves to complexes
Question
Can we have something similar for objects in Db(X )
Mathematical PhysicsHomological Mirror Symmetry
(Kontsevich)
Birational geometryHomological MMP
(Bondal Orlov Kawamata)
Bridgelandstability
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
22
Stability conditions
Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )
Let T be a triangulated category
Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ
Example
T = Db(C) for C asmooth projective curve
Γ = N(C) = H0 oplus H2
with
v = (rk deg)
A Bridgeland stability condition on T is a pair σ = (AZ )
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
23
Stability conditions
A is the heart of abounded t-structure on T
Z Γ rarr C is a grouphomomorphism
Example
A = Coh(C)
Z (v(minus)) = minusdeg +radicminus1rk
such that for any 0 ∕= E isin A
1 Z (v(E)) isin Rgt0e(01]πradicminus1
2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)
Im(Z )(minus) (or +infin)
3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
24
Stability conditions
Warning 3
The example is somehow misleading it only works indimension 1
The following is a remarkable result
Theorem (Bridgeland)
If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
25
Stability conditions
Warning 4
It is a very difficult problem to construct stability conditions
It is solved mainly in these cases
Curves (Bridgeland Macrı)
Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)
Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)
Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)
Fourfolds in general is a big mistery
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
26
Outline
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
27
Main idea
We want to combine all the ideas we discussed so far
Semiorthogonal decompositions
Stability conditions
Geometry ofcubic 4-folds
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
28
The setting
Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section
Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2
Example (d = 3 and n = 5)
Consider for example the zero locus of
x30 + + x3
5 = 0
in the projective space P5
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
29
Homological algebra
Let us now look at the bounded derived category of coherentsheaves on X
Db(X )
=
〈 Ku(X ) OX OX (H)OX (2H) 〉
Ku(X )
=983069E isin Db(X )
Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z
983070
Kuznetsov component of X
Exceptional objects
〈OX (iH)〉 sim= Db(pt)
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
30
The idea
X (and Db(X )) arecomplicated objects
Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )
Stabilityconditions
New appli-cations to
hyperkahlergeometry
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
31
Noncommutative K3s
Remark
For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)
But almost never Ku(X ) sim= Db(S) for S a K3 surface
This is related to the following
Conjecture (Kuznetsov)
A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
32
The results existence of stability conditions
We have seen that if S is a K3 surface then Db(S) carriesstability conditions
Question (Addington-Thomas Huybrechts)
If X is a cubic 4-fold does Ku(X ) carry stability conditions
Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)
For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
33
The results moduli spaces
Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it
We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector
Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)
Denote byMσ(Ku(X ) v)
the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
34
The results moduli spaces
Question
What is the geometry of Mσ(Ku(X ) v)
This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects
Theorem 2 (BLMNPS)
Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
35
The results moduli spaces
Definition
A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2
X ) is generated by an everywherenon-degenerate holomorphic 2-form
There are very few examples (up to deformation)
1 K3 surfaces
2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)
3 Generalized Kummer varieties (from abelian surfaces)
4 Two sporadic examples by OrsquoGrady
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
36
Hyperkahler geometry and cubic 4-folds
The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X
Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold
Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold
Question
Can we recover these classical HK manifolds in our newframework
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
37
Hyperkahler geometry and cubic 4-folds
Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)
The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent
New (BLMNPS)
In these two cases we can explicitely describe the birationalmodels via variation of stability
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family
38
Hyperkahler geometry and cubic 4-folds
But the most striking application is the following
Corollary (BLMNPS)
For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2
This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family