On some invariants for algebraic cobordism cycles International Workshop on... · On some invariants for algebraic cobordism cycles International Workshop on Motives, Tokyo February

On some invariants for algebraic cobordism

cycles

International Workshop on Motives, Tokyo February 15, 2016

Gereon QuickNTNU

Point of departure: Poincaré, Lefschetz, Hodge…

Let X⊂PN be a smooth projective complex variety.



Let zj be a local coordinate on X.



Let zj be a local coordinate on X. Write zj = xj + iyj with real coordinates xj and yj.




There are associated differential forms

dzj = dxj + idyj and dzj = dxj - idyj.-




There are associated differential forms

dzj = dxj + idyj and dzj = dxj - idyj.-

Every k-form 𝛼 on X can then be written as

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzjp+1∧…∧dzjk.- -

Integrals over cycles:

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -

Let’s write 𝛼 ∈ Ap,q(X) if 𝛼 is of the form


exactly p dzj’s for all j




exactly p dzj’s for all j exactly q dzj’s for all j-







Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k.





Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k. We can integrate 𝛼 over 𝛤: ∫ 𝜄*𝛼. 𝛤





Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k. We can integrate 𝛼 over 𝛤: ∫ 𝜄*𝛼. 𝛤

If 𝛤 = Z happens to be an algebraic subvariety of X, say of complex dimension n, then ∫ 𝜄*𝛼 = 0 unless 𝛼 lies in An,n(X). Z

Hodge’s question:

Hodge’s question:This imposes a necessary condition on a topological cycle 𝜄: 𝛤 ⊂ X to be algebraic (homologous to an algebraic subvariety Z dimension n):


𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤


𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤

Hodge wondered: Is this condition also sufficient? ? ⇐ ?


𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤

Hodge wondered: Is this condition also sufficient? ? ⇐ ?

Usually, we ask this question in cohomological terms…

Hodge’s question in terms of cohomology:

Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Z

Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Zits Poincaré dual complex cohomology class is represented by a form in Ap,p(X) with p = dimX - n.

Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Zits Poincaré dual complex cohomology class is represented by a form in Ap,p(X) with p = dimX - n.Such classes are said to be “Hodge-type (p,p)”. The subgroup of such classes is denoted by

Hp,p(X) ⊂ H2p(X;C).


Hp,p(X) ⊂ H2p(X;C).The Hodge Conjecture: The map

clH: Zp(X) → Hp,p(X) ∩ H2p(X;Z) is surjective.


Hp,p(X) ⊂ H2p(X;C).The Hodge Conjecture: The map

clH: Zp(X) → Hp,p(X) ∩ H2p(X;Z) is surjective.⊗Q

switch to Q-coefficients

×

A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g.

A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C.


⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,

Then every pair of points p,q∈C defines a g-tuple of complex numbers


µ: Div0(C) → ℂg

⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,




⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,


group of formal sums ∑i(pi-qi)



⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,


group of formal sums ∑i(pi-qi) lattice of integrals

of ωj’s over loops

/Λ



⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,


group of formal sums ∑i(pi-qi) lattice of integrals

of ωj’s over loops

/Λ

Jacobian variety of C

=: J(C)



⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,


group of formal sums ∑i(pi-qi)

Jacobi Inversion Theorem: The map (Abel-Jacobi map)

is surjective.

lattice of integrals of ωj’s over loops

/Λ

Jacobian variety of C

=: J(C)

More canonically:

More canonically:

µ: Div0(C) → J(C) = H0(C; Ω1hol)∨/H1(C;Z)

⌠⌡qiω

pi

∑(pi-qi) ⟼ mod H1(C;Z).∑i⎝⎛ ⟼ ⎞

⎠ω

More canonically:

Abel’s Theorem: The kernel of this map is the subgroup of principal divisors.

µ: Div0(C) → J(C) = H0(C; Ω1hol)∨/H1(C;Z)

⌠⌡qiω

pi

∑(pi-qi) ⟼ mod H1(C;Z).∑i⎝⎛ ⟼ ⎞

⎠ω

Lefschetz’s proof for (1,1)-classes:

For simplicity, let X⊂PN be a surface.


Let {Ct}t be a family of curves on X (parametrized over the projective line P1).




Associated to {Ct}t is the family of Jacobians

J := ⋃t J(Ct)





J := ⋃t J(Ct)

and a fibre space π: J → P1 (of complex Lie groups).





J := ⋃t J(Ct)

and a fibre space π: J → P1 (of complex Lie groups).

A “normal function” 𝜈 is a holomorphic section of π.


Normal functions arise naturally:

Lefschetz’s proof continued:

Normal functions arise naturally: Let D be an algebraic curve on X. It intersects Ct in points p1(t),…, pd(t).




Choose a point p0 on all Ct. Then ∑i pi(t) - dp0 is a divisor of degree 0 and defines a point

𝜈D(t) ∈ J(Ct).



Hence D defines a normal function

𝜈D: t ↦ 𝜈D(t) ∈ J.

Choose a point p0 on all Ct. Then ∑i pi(t) - dp0 is a divisor of degree 0 and defines a point

𝜈D(t) ∈ J(Ct).

Poincaré’s Existence Theorem:

Every normal function 𝜈 arises as the normal function 𝜈D associated to an algebraic curve D.




Then Lefschetz proved:



• Every normal function 𝜈 defines a class 𝜂(𝜈)∈H2(X;Z) of Hodge type (1,1) such that 𝜂(𝜈D) = clH(D).




• Every normal function 𝜈 defines a class 𝜂(𝜈)∈H2(X;Z) of Hodge type (1,1) such that 𝜂(𝜈D) = clH(D).

• Every class in H2(X;Z) of Hodge type (1,1) arises as 𝜂(𝜈) for some normal function 𝜈.


X a smooth projective complex variety with dimX=n.

Griffiths: in higher dimensions…


Z⊂X a closed subvariety of codimension p.





We assume that Z is the boundary of a differentiable chain Γ ∈ C2n-2p+1(X;Z).




Let ω be a smooth form of degree 2n-2p+1.




Then ⌠⌡Γω⟼ defines a map A2n-2p+1(X)→C.ω⎛⎝⎞⎠


Let ω be a smooth form of degree 2n-2p+1.


We would like this map to

Deligne, Griffiths, …:



• descend to a map on cohomology,




• and to become independent of the choice of Γ.





Let ω’ be a form such that dω’= ω.





Let ω’ be a form such that dω’= ω.

Then Stokes’ formula tells us

⌠⌡Γω ⌠

⌡Zω’=

⌠⌡Γω ⌠

⌡Zω’=

The intermediate Jacobian of Griffiths and the Abel-Jacobi map:

⌠⌡Γω ⌠

⌡Zω’=

If ω is a form of degree (p’,2n-2p+1-p’) with p’ ≥ n-p+1, this integral vanishes.


⌠⌡Γω ⌠

⌡Zω’=


We obtain a well-defined map⌠⌡Γ

⟼Z

µ: Zp(X)h → J2p-1(X) = Fn-p+1H2n-2p+1(X;C)∨/H2n-2p+1(X;Z)

for some Γ with Z=∂Γ


⌠⌡Γω ⌠

⌡Zω’=


We obtain a well-defined map⌠⌡Γ

⟼Z

µ: Zp(X)h → J2p-1(X) = Fn-p+1H2n-2p+1(X;C)∨/H2n-2p+1(X;Z)

for some Γ with Z=∂Γ

J2p-1(X) is a complex torus and is called Griffiths’ intermediate Jacobian.


The Jacobian and Griffiths’ theorem:


J2p-1(X) is, in general, not an abelian variety.



But it varies homomorphically in families.




Have an induced a map:

Griffp(X):= Zp(X)h/Zp(X)alg → J2p-1(X)/J2p-1(X)alg




Griffith’s theorem: Let X⊂P4 be a general quintic hypersurface. There are lines L and L’ on X such that µ(L-L’) is a non torsion element in J3(X).

Have an induced a map:

Griffp(X):= Zp(X)h/Zp(X)alg → J2p-1(X)/J2p-1(X)alg

An interesting diagram:Let X be a smooth projective complex variety.

An interesting diagram:

Zp(X)Let X be a smooth projective complex variety.


Zp(X) Z⊂XLet X be a smooth projective complex variety.


Zp(X)

clH

Z⊂XLet X be a smooth projective complex variety.


Hdg2p(X)

Zp(X)

clH

Z⊂XLet X be a smooth projective complex variety.


Hdg2p(X)

Zp(X)

clH

Z⊂X

[Zsm]

−

Let X be a smooth projective complex variety.


Hdg2p(X)

Zp(X)

clH

Zp(X)h=Kernel of clH ⊂ Z⊂X

[Zsm]

−



Hdg2p(X)

Zp(X)

clH

Zp(X)h=Kernel of clH ⊂

Abel-Jacobimap µ

Z⊂X

[Zsm]

−



Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−



Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


→ HD (X;Z(p)) →2p


Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


→ HD (X;Z(p)) →2p

Deligne cohomology combines topological with Hodge theoretic information


Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


→ 0→ HD (X;Z(p)) →2p



Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


0 → → 0→ HD (X;Z(p)) →2p



Hdg2p(X)

Zp(X)

clHclHD


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


0 → → 0→ HD (X;Z(p)) →2p


Another interesting map for smooth complex varieties:

Φ: Ω*(X) → MU *(X)2


Φ: Ω*(X) → MU *(X)2


algebraic cobordismof Levine and Morel

Φ: Ω*(X) → MU *(X)2



complex cobordism ofthe top. space X(C)

Φ: Ω*(X) → MU *(X)2



complex cobordism ofthe top. space X(C)

Recall: MUi(X(C)) is (roughly speaking) generated by maps f: M→X(C) of codimension i modulo the relation g-1(0) ∼ g-1(1) for maps g: M’→X(C)xR such that g is transversal to e0: X(C)→X(C)xR and e1: X(C)→X(C)xR.

Algebraic cobordism in a nutshell:


Ωp(X) is generated (roughly speaking) by proj. maps f:Y→X of codimension p with Y smooth modulo …



Levine’s and Pandharipande’s “double point relation”:



Levine’s and Pandharipande’s “double point relation”:

π-1(0) ∼ π-1(∞) for projective morphisms π: Y’→XxP1 such that π-1(0) is smooth and π-1(∞)=A∪DB where A and B are smooth and meet transversally in D.

What can we say about the map Φ?

Ω*(X) MU2*(X)Φ


Ω*(X) MU2*(X)Φ[Y→X]


Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼


• The image: Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼


• The image:

CH*(X)

Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼


• The image:

CH*(X)

Ω*(X) MU2*(X)Φ

Hdg2*(X) ⊆ H2*(X;Z)clH

[Y→X] [Y(C)→X(C)]⟼


• The image:

CH*(X)

Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼


• The image:

CH*(X)

Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼


• The image:

CH*(X)There is a “Hodge-theoretic” restriction for ImΦ.

Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼


• The image:

• The kernel:


Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼


• The image:

• The kernel:


Griffiths’ theorem suggests that Φ is not injective.

Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼


• The image:

• The kernel:


Griffiths’ theorem suggests that Φ is not injective.

Question: Is there is an “Abel-Jacobi-invariant” which is able to detect elements in KerΦ?

Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼

What can we say about ImΦ?

Ω*(X) → MU2*(X)


Ω*(X) → MU2*(X)HdgMU2*(X) ⊂


Ω*(X) → MU2*(X)HdgMU2*(X) ⊂

This map is not surjective, …


Ω*(X) → MU2*(X)

… but the classical topological arguments do not work!

HdgMU2*(X) ⊂

This map is not surjective, …

Ω*(X) MU2*(X)Φ

Atiyah-Hirzebruch and Totaro:

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ



Ω*(X) MU2*(X)Φ


CH*(X)


Ω*(X) MU2*(X)Φ


CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH


Ω*(X) MU2*(X)Φ




Ω*(X) MU2*(X)Φ



Totaro


Ω*(X) MU2*(X)Φ



Totaro


Ω*(X) MU2*(X)Φ


CH*(X)

Levine-Morel ≈


Totaro


Ω*(X) MU2*(X)Φ


CH*(X)

Levine-Morel ≈ ≉ in general


Totaro


Ω*(X) MU2*(X)Φ


CH*(X)


Atiyah-Hirzebruch: clH is not surjective.


Totaro


Ω*(X) MU2*(X)Φ


CH*(X)


Atiyah-Hirzebruch: clH is not surjective.


This argument does not work for Φ.

Totaro

Kollar’s examples: (see also Soulé-Voisin)


Let X⊂P4 a very general hypersurface of degree d.



Lefschetz’s hyperplane theorem:




• H2(X;Z) is torsion-free and generated by the class h=c1(𝓞X(1)) of a hyperplane section on X,




• H2(X;Z) is torsion-free and generated by the class h=c1(𝓞X(1)) of a hyperplane section on X,

• H4(X;Z) is torsion-free and generated by a class α such that ∫ α∙h=1, and all classes are Hodge classes.

X

Kollar’s examples: H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X

Kollar’s examples:

Assume now d=p3 for a prime p≥5.

H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X


Kollar: Then any algebraic curve C on X has degree divisible by p.







Consequence:





Consequence: Let C be a curve on X and [C]∈H4(X;Z) be its cohomology class.





Consequence: Let C be a curve on X and [C]∈H4(X;Z) be its cohomology class.

Then [C]= kα for some k and ∫ [C]∙h=k.X


Kollar: p divides the degree of any curve on X.





k = ∫ [C]∙hX




k = ∫ [C]∙hX

= number of intersection points of C with h




k = ∫ [C]∙hX


= degree of C




k = ∫ [C]∙hX


= degree of C ⟹ p divides k




k = ∫ [C]∙hX



In particular, α is not algebraic (for k cannot be 1).




k = ∫ [C]∙hX



In particular, α is not algebraic (for k cannot be 1).

But dα is algebraic (for ∫ dα∙h = d = ∫ h2∙h ⇒ dα=h2). XX

Consequences for Φ: Ω*(X) → MU2*(X):


Let X⊂P4 be a very general hypersurface as above.



Then MU4(X) ↠ H4(X;Z) is surjective, and thus Kollar’s argument implies that Φ is not surjective (on Hodge classes).



Then MU4(X) ↠ H4(X;Z) is surjective, and thus Kollar’s argument implies that Φ is not surjective (on Hodge classes).

These examples are “not topological”:there is a dense subset of hypersurfaces Y⊂P4 such that the generator in H4(Y;Z) is algebraic.

What about the kernel?

Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼

Φ(Y)=0 creates a new equivalence relation.


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼


top. cobordismrelation


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼


double point relation



Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼




⊆


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼




⊆ algebraic equivalence


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼





of A. Krishna and J. Park


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼





⊆

of A. Krishna and J. Park

A new diagram:


A new diagram:

Ωp(X)


A new diagram:

Ωp(X) [Y→X]


A new diagram:

Ωp(X) [Y→X]


HdgMU(X)2p

A new diagram:

Ωp(X) [Y→X]

[Y(C)→X(C)]

−


HdgMU(X)2p

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


HdgMU(X)2p

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


MUD (p)(X) →2p HdgMU(X)2p

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


MUD (p)(X) →2p HdgMU(X)2p

combines topol. cobordism with Hodge theoretic information (jt. with M. Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


MUD (p)(X) →2p HdgMU(X)2p → 0


A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p → 0


A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p

complex torus ≈ MU2p-1(X)⊗R/Z

→ 0


A new diagram:

Ωp(X)

Φ

Kernel of Φ ⊂ [Y→X]

[Y(C)→X(C)]

−




→ 0


A new diagram:

Ωp(X)

Φ

Kernel of Φ ⊂

“Abel-Jacobimap” µMU

[Y→X]

[Y(C)→X(C)]

−




→ 0


A new diagram:

Ωp(X)

ΦΦD

Kernel of Φ ⊂

“Abel-Jacobimap” µMU

[Y→X]

[Y(C)→X(C)]

−




→ 0


Examples:

The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes:

Ωp(X)

MU2p(X)CHp(X)

ΦµMU

JMU (X)2p-1

Examples:


Ωp(X)

MU2p(X)CHp(X)

∃ α ∊

ΦµMU

JMU (X)2p-1

Examples:


Ωp(X)

MU2p(X)CHp(X)

∃ α ∊

0ΦµMU

JMU (X)2p-1

Examples:


Ωp(X)

MU2p(X)CHp(X)

∃ α ∊

0 0ΦµMU

JMU (X)2p-1

Examples:


Ωp(X)

MU2p(X)CHp(X)

∃ α ∊

0 0≠0ΦµMU

JMU (X)2p-1

The Abel-Jacobi map:


Given α ∈ Ωp(X), by Rost’s degree formula (due to Levine-Morel)

with Zi⊂X of codim ci≥0, Zi a res. of singularities and 𝛾i ∈ π2p+2 MU and

α = ∑i=1𝛾i[Zi→X] k

~~

ci





~~

ci

Assume Φ(α)=0. Then there are chains (modulo torsion)Γi ∈ C2n-2p+1-2ci(X;Z) such that





~~

ci

Assume Φ(α)=0. Then there are chains (modulo torsion)Γi ∈ C2n-2p+1-2ci(X;Z) such that

“Image of α = ∂(∑i=1𝛾iΓi) ∈ C2n-2p-2∗(X;Z)⊗π2∗MU”k


Taking integrals modulo MU2n-2p+1(X) yields a map

n = dim X

Fn-p+1H2n-2p+1(X;π2∗MU⊗C)∨ MU2n-2p+1(X)/Ker(Φ) →

α ⌠⌡Γi

⟼ mod MU2n-2p+1(X).

≈ JMU2p-1(X)

∑i=1 𝛾i k

Thank you!

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On some invariants for algebraic cobordism cycles International Workshop on... · On some invariants for algebraic cobordism cycles International Workshop on Motives, Tokyo February