Birational geometry and moduli spaces of varieties of

Birational geometry and moduli

spaces of varieties of general type

James McKernan

UCSD

Birational geometry and moduli spaces of varieties of general type – p. 1

Review

We would like to construct the moduli space of

varieties of general type— by analogy with Mg.

Birational geometry and moduli spaces of varieties of general type – p. 2

Review

We would like to construct the moduli space of

varieties of general type— by analogy with Mg.

Even if we are interested in irreducible normalvarieties with no boundary we are obliged toconsider non-normal, reducible varieties.

Birational geometry and moduli spaces of varieties of general type – p. 2

Review

We would like to construct the moduli space of

varieties of general type— by analogy with Mg.

Even if we are interested in irreducible normalvarieties with no boundary we are obliged toconsider non-normal, reducible varieties.

We have to consider semi log canonical pairs whichmeans we have to deal with log canonical pairs. Thedouble locus appears with coefficient one.

Birational geometry and moduli spaces of varieties of general type – p. 2

Review

We would like to construct the moduli space of

varieties of general type— by analogy with Mg.

Even if we are interested in irreducible normalvarieties with no boundary we are obliged toconsider non-normal, reducible varieties.

We have to consider semi log canonical pairs whichmeans we have to deal with log canonical pairs. Thedouble locus appears with coefficient one.

We will see that the precise definition of the modulifunctor is unclear for pairs.

Birational geometry and moduli spaces of varieties of general type – p. 2

Review

We would like to construct the moduli space of

varieties of general type— by analogy with Mg.

Even if we are interested in irreducible normalvarieties with no boundary we are obliged toconsider non-normal, reducible varieties.

We have to consider semi log canonical pairs whichmeans we have to deal with log canonical pairs. Thedouble locus appears with coefficient one.

We will see that the precise definition of the modulifunctor is unclear for pairs.

We only know how to construct the moduli space ofsemi log canonical models and not slc pairs but wewill still prove result for pairs.

Birational geometry and moduli spaces of varieties of general type – p. 2

Log canonical pairs (X,∆)

A log resolution π : Y −→ X is a projective

birational map s.t. (Y,Γ = ∆ + E) is log smooth

and E =∑

Ei supports an ample divisor over X .

Birational geometry and moduli spaces of varieties of general type – p. 3

Log canonical pairs (X,∆)

A log resolution π : Y −→ X is a projective

birational map s.t. (Y,Γ = ∆ + E) is log smooth

and E =∑

Ei supports an ample divisor over X .

We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.

Birational geometry and moduli spaces of varieties of general type – p. 3

Log canonical pairs (X,∆)

A log resolution π : Y −→ X is a projective

birational map s.t. (Y,Γ = ∆ + E) is log smooth

and E =∑

Ei supports an ample divisor over X .

We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.

ai = a(Ei, X,∆) is the log discrepancy of Ei.

Birational geometry and moduli spaces of varieties of general type – p. 3

Log canonical pairs (X,∆)

A log resolution π : Y −→ X is a projective

birational map s.t. (Y,Γ = ∆ + E) is log smooth

and E =∑

Ei supports an ample divisor over X .

We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.

ai = a(Ei, X,∆) is the log discrepancy of Ei.

a = infi,Y ai is the log discrepancy of (X,∆).(X,∆) is log canonical if a ≥ 0.

Birational geometry and moduli spaces of varieties of general type – p. 3

Log canonical pairs (X,∆)

A log resolution π : Y −→ X is a projective

birational map s.t. (Y,Γ = ∆ + E) is log smooth

and E =∑

Ei supports an ample divisor over X .

We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.

ai = a(Ei, X,∆) is the log discrepancy of Ei.

a = infi,Y ai is the log discrepancy of (X,∆).(X,∆) is log canonical if a ≥ 0.

(X,∆) is klt if ⌊∆⌋ = 0 and a > 0.

Birational geometry and moduli spaces of varieties of general type – p. 3

Klt vs lc

Varieties come in three types: KX is -ve, 0, +ve.

Birational geometry and moduli spaces of varieties of general type – p. 4

Klt vs lc

Varieties come in three types: KX is -ve, 0, +ve.

Singularities come in three types: klt, lc, not lc.

Birational geometry and moduli spaces of varieties of general type – p. 4

Klt vs lc

Varieties come in three types: KX is -ve, 0, +ve.

Singularities come in three types: klt, lc, not lc.

Let S be the cone over a curve C, genus g, degree d.

Birational geometry and moduli spaces of varieties of general type – p. 4

Klt vs lc

Varieties come in three types: KX is -ve, 0, +ve.

Singularities come in three types: klt, lc, not lc.

Let S be the cone over a curve C, genus g, degree d.

Minimal resolution π : T −→ S exceptional divisor

E ≃ C, E2 = −d. Write KT + E = π∗KS + aE.

Birational geometry and moduli spaces of varieties of general type – p. 4

Klt vs lc

Varieties come in three types: KX is -ve, 0, +ve.

Singularities come in three types: klt, lc, not lc.

Let S be the cone over a curve C, genus g, degree d.

Minimal resolution π : T −→ S exceptional divisor

E ≃ C, E2 = −d. Write KT + E = π∗KS + aE.

2g−2 = degKE = (KT +E) ·E = π∗KS ·E+aE2.

Birational geometry and moduli spaces of varieties of general type – p. 4

Klt vs lc

Varieties come in three types: KX is -ve, 0, +ve.

Singularities come in three types: klt, lc, not lc.

Let S be the cone over a curve C, genus g, degree d.

Minimal resolution π : T −→ S exceptional divisor

E ≃ C, E2 = −d. Write KT + E = π∗KS + aE.

2g−2 = degKE = (KT +E) ·E = π∗KS ·E+aE2.

2g − 2 = −ad =⇒ a =2− 2g

d.

Birational geometry and moduli spaces of varieties of general type – p. 4

Klt vs lc

Varieties come in three types: KX is -ve, 0, +ve.

Singularities come in three types: klt, lc, not lc.

Let S be the cone over a curve C, genus g, degree d.

Minimal resolution π : T −→ S exceptional divisor

E ≃ C, E2 = −d. Write KT + E = π∗KS + aE.

2g−2 = degKE = (KT +E) ·E = π∗KS ·E+aE2.

2g − 2 = −ad =⇒ a =2− 2g

d.

S is klt, lc, not lc if and only if KC is -ve, 0, +ve.

Birational geometry and moduli spaces of varieties of general type – p. 4

Klt vs lc

Varieties come in three types: KX is -ve, 0, +ve.

Singularities come in three types: klt, lc, not lc.

Let S be the cone over a curve C, genus g, degree d.

Minimal resolution π : T −→ S exceptional divisor

E ≃ C, E2 = −d. Write KT + E = π∗KS + aE.

2g−2 = degKE = (KT +E) ·E = π∗KS ·E+aE2.

2g − 2 = −ad =⇒ a =2− 2g

d.

S is klt, lc, not lc if and only if KC is -ve, 0, +ve.

KS is -ve, 0, +ve if and only if a < 2, a = 2, a > 2.

Birational geometry and moduli spaces of varieties of general type – p. 4

Deformation woes

Two families of surfaces in P5, Veronese P2 ⊂ P5

and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.

Birational geometry and moduli spaces of varieties of general type – p. 5

Deformation woes

Two families of surfaces in P5, Veronese P2 ⊂ P5

and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.

Both surfaces degenerate to the cone S0 over a

rational normal curve of degree four, K2S0

= 9.

Birational geometry and moduli spaces of varieties of general type – p. 5

Deformation woes

Two families of surfaces in P5, Veronese P2 ⊂ P5

and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.

Both surfaces degenerate to the cone S0 over a

rational normal curve of degree four, K2S0

= 9.

Both families are flat and the central fibre iskawamata log terminal so semi log canonical.

Birational geometry and moduli spaces of varieties of general type – p. 5

Deformation woes

Two families of surfaces in P5, Veronese P2 ⊂ P5

and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.

Both surfaces degenerate to the cone S0 over a

rational normal curve of degree four, K2S0

= 9.

Both families are flat and the central fibre iskawamata log terminal so semi log canonical.

Take ramified covers to get families of general type.

Birational geometry and moduli spaces of varieties of general type – p. 5

Deformation woes

Two families of surfaces in P5, Veronese P2 ⊂ P5

and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.

Both surfaces degenerate to the cone S0 over a

rational normal curve of degree four, K2S0

= 9.

Both families are flat and the central fibre iskawamata log terminal so semi log canonical.

Take ramified covers to get families of general type.

To get around this we have to fix the degree in themoduli functor.

Birational geometry and moduli spaces of varieties of general type – p. 5

Deformation woes

Two families of surfaces in P5, Veronese P2 ⊂ P5

and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.

Both surfaces degenerate to the cone S0 over a

rational normal curve of degree four, K2S0

= 9.

Both families are flat and the central fibre iskawamata log terminal so semi log canonical.

Take ramified covers to get families of general type.

To get around this we have to fix the degree in themoduli functor.

(Hacking, Abramovich-Hassett) We also requirereflexive powers of ωX to commute with basechange (over a non-reduced base).

Birational geometry and moduli spaces of varieties of general type – p. 5

Moduli functor

Fix the degree d. The moduli functor of slc models

assigns to every scheme S the set Mslcd (S):

Birational geometry and moduli spaces of varieties of general type – p. 6

Moduli functor

Fix the degree d. The moduli functor of slc models

assigns to every scheme S the set Mslcd (S):

flat projective morphisms X −→ S; the fibres are slc, with

ample canonical class of degree d; ωX is flat over S and

all reflexive powers of ωX commute with base change.

Birational geometry and moduli spaces of varieties of general type – p. 6

Moduli functor

Fix the degree d. The moduli functor of slc models

assigns to every scheme S the set Mslcd (S):

flat projective morphisms X −→ S; the fibres are slc, with

ample canonical class of degree d; ωX is flat over S and

all reflexive powers of ωX commute with base change.

Viehweg: only one power commutes with base change.

Birational geometry and moduli spaces of varieties of general type – p. 6

Moduli functor

Fix the degree d. The moduli functor of slc models

assigns to every scheme S the set Mslcd (S):

flat projective morphisms X −→ S; the fibres are slc, with

ample canonical class of degree d; ωX is flat over S and

all reflexive powers of ωX commute with base change.

Viehweg: only one power commutes with base change.

If we allow pairs, for example in the case of the cone overa rational normal curve, we even get embedded points.

Birational geometry and moduli spaces of varieties of general type – p. 6

Moduli functor

Fix the degree d. The moduli functor of slc models

assigns to every scheme S the set Mslcd (S):

flat projective morphisms X −→ S; the fibres are slc, with

ample canonical class of degree d; ωX is flat over S and

all reflexive powers of ωX commute with base change.

Viehweg: only one power commutes with base change.

If we allow pairs, for example in the case of the cone overa rational normal curve, we even get embedded points.

There are many ways around this; does not occur if

coefficients > 1/2 (Kollár); float the coefficients(Hacking); branch varieties (Alexeev-Knutson).

Birational geometry and moduli spaces of varieties of general type – p. 6

Adjunction and the different

If X is smooth and S is a prime divisor then by

adjunction (KX + S)|S = KS.

Birational geometry and moduli spaces of varieties of general type – p. 7

Adjunction and the different

If X is smooth and S is a prime divisor then by

adjunction (KX + S)|S = KS.

If X is singular then (KX +∆)|S = KS +Θ, wheremultS ∆ = 1 and Θ ≥ 0 is called the different.

Birational geometry and moduli spaces of varieties of general type – p. 7

Adjunction and the different

If X is smooth and S is a prime divisor then by

adjunction (KX + S)|S = KS.

If X is singular then (KX +∆)|S = KS +Θ, wheremultS ∆ = 1 and Θ ≥ 0 is called the different.

(X,∆) is divisorially log terminal if there is a logresolution such that ai > 0 all i. plt if x∆y = S.

Birational geometry and moduli spaces of varieties of general type – p. 7

Adjunction and the different

If X is smooth and S is a prime divisor then by

adjunction (KX + S)|S = KS.

If X is singular then (KX +∆)|S = KS +Θ, wheremultS ∆ = 1 and Θ ≥ 0 is called the different.

(X,∆) is divisorially log terminal if there is a logresolution such that ai > 0 all i. plt if x∆y = S.

(X,∆) (lc dlt plt) implies (S,Θ) (slc dlt klt).

Birational geometry and moduli spaces of varieties of general type – p. 7

Adjunction and the different

If X is smooth and S is a prime divisor then by

adjunction (KX + S)|S = KS.

If X is singular then (KX +∆)|S = KS +Θ, wheremultS ∆ = 1 and Θ ≥ 0 is called the different.

(X,∆) is divisorially log terminal if there is a logresolution such that ai > 0 all i. plt if x∆y = S.

(X,∆) (lc dlt plt) implies (S,Θ) (slc dlt klt).

If ∆ = S then Θ has coefficients (r − 1)/r.

Birational geometry and moduli spaces of varieties of general type – p. 7

Adjunction and the different

If X is smooth and S is a prime divisor then by

adjunction (KX + S)|S = KS.

If X is singular then (KX +∆)|S = KS +Θ, wheremultS ∆ = 1 and Θ ≥ 0 is called the different.

(X,∆) is divisorially log terminal if there is a logresolution such that ai > 0 all i. plt if x∆y = S.

(X,∆) (lc dlt plt) implies (S,Θ) (slc dlt klt).

If ∆ = S then Θ has coefficients (r − 1)/r.

If coefficients of ∆ belong to I then coefficients of

Θ belong to D(I) = { (r − 1 + s)/r | s ∈ I+ }.

Birational geometry and moduli spaces of varieties of general type – p. 7

Adjunction and the different

If X is smooth and S is a prime divisor then by

adjunction (KX + S)|S = KS.

If X is singular then (KX +∆)|S = KS +Θ, wheremultS ∆ = 1 and Θ ≥ 0 is called the different.

(X,∆) is divisorially log terminal if there is a logresolution such that ai > 0 all i. plt if x∆y = S.

(X,∆) (lc dlt plt) implies (S,Θ) (slc dlt klt).

If ∆ = S then Θ has coefficients (r − 1)/r.

If coefficients of ∆ belong to I then coefficients of

Θ belong to D(I) = { (r − 1 + s)/r | s ∈ I+ }.

Hence if I satisfies DCC then D(I) satisfies DCC.

Birational geometry and moduli spaces of varieties of general type – p. 7

From slc to lc

One can view a stable curve as a set of smoothcurves together with a set of pairs of points.

Birational geometry and moduli spaces of varieties of general type – p. 8

From slc to lc

One can view a stable curve as a set of smoothcurves together with a set of pairs of points.

Given a semi log canonical pair (X,∆) by defn one

can associate a log canonical pair (Xν, D +∆ν).Associate to D an involution τ : D −→ D, whichrepresents the gluing data. Fixes different on D.

Birational geometry and moduli spaces of varieties of general type – p. 8

From slc to lc

One can view a stable curve as a set of smoothcurves together with a set of pairs of points.

Given a semi log canonical pair (X,∆) by defn one

can associate a log canonical pair (Xν, D +∆ν).Associate to D an involution τ : D −→ D, whichrepresents the gluing data. Fixes different on D.

Theorem: (Kollár) Natural bijection between sets:

Birational geometry and moduli spaces of varieties of general type – p. 8

From slc to lc

One can view a stable curve as a set of smoothcurves together with a set of pairs of points.

Given a semi log canonical pair (X,∆) by defn one

can associate a log canonical pair (Xν, D +∆ν).Associate to D an involution τ : D −→ D, whichrepresents the gluing data. Fixes different on D.

Theorem: (Kollár) Natural bijection between sets:

(i) slc pairs (X,∆) such that KX +∆ is ample.

Birational geometry and moduli spaces of varieties of general type – p. 8

From slc to lc

One can view a stable curve as a set of smoothcurves together with a set of pairs of points.

Given a semi log canonical pair (X,∆) by defn one

can associate a log canonical pair (Xν, D +∆ν).Associate to D an involution τ : D −→ D, whichrepresents the gluing data. Fixes different on D.

Theorem: (Kollár) Natural bijection between sets:

(i) slc pairs (X,∆) such that KX +∆ is ample.

(ii) lc pairs (Y,Γ +G) plus an involution τ : Gν −→ Gν

fixing the different such that KY + Γ +G is ample.

Birational geometry and moduli spaces of varieties of general type – p. 8

From slc to lc

One can view a stable curve as a set of smoothcurves together with a set of pairs of points.

Given a semi log canonical pair (X,∆) by defn one

can associate a log canonical pair (Xν, D +∆ν).Associate to D an involution τ : D −→ D, whichrepresents the gluing data. Fixes different on D.

Theorem: (Kollár) Natural bijection between sets:

(i) slc pairs (X,∆) such that KX +∆ is ample.

(ii) lc pairs (Y,Γ +G) plus an involution τ : Gν −→ Gν

fixing the different such that KY + Γ +G is ample.

Correspondence no longer holds if we drop theampleness condition.

Birational geometry and moduli spaces of varieties of general type – p. 8

Moduli space of pairs

Focus on one aspect of the construction of themoduli space of semi log canonical varieties.

Birational geometry and moduli spaces of varieties of general type – p. 9

Moduli space of pairs

Focus on one aspect of the construction of themoduli space of semi log canonical varieties.

Most steps go through as in the construction of thesmooth components.

Birational geometry and moduli spaces of varieties of general type – p. 9

Moduli space of pairs

Focus on one aspect of the construction of themoduli space of semi log canonical varieties.

Most steps go through as in the construction of thesmooth components.

New issue is boundedness of the moduli functor.

Birational geometry and moduli spaces of varieties of general type – p. 9

Moduli space of pairs

Focus on one aspect of the construction of themoduli space of semi log canonical varieties.

Most steps go through as in the construction of thesmooth components.

New issue is boundedness of the moduli functor.

For example, if you fix the degree d how to boundthe number of components of an slc pair?

Birational geometry and moduli spaces of varieties of general type – p. 9

Moduli space of pairs

Focus on one aspect of the construction of themoduli space of semi log canonical varieties.

Most steps go through as in the construction of thesmooth components.

New issue is boundedness of the moduli functor.

For example, if you fix the degree d how to boundthe number of components of an slc pair?

In the smooth case, simply take the closure in theChow variety.

Birational geometry and moduli spaces of varieties of general type – p. 9

Moduli space of pairs

Focus on one aspect of the construction of themoduli space of semi log canonical varieties.

Most steps go through as in the construction of thesmooth components.

New issue is boundedness of the moduli functor.

For example, if you fix the degree d how to boundthe number of components of an slc pair?

In the smooth case, simply take the closure in theChow variety.

A stable curve of genus g has at most 2g − 2components. Each component contributes ≥ 1.

Birational geometry and moduli spaces of varieties of general type – p. 9

Moduli space of pairs

Focus on one aspect of the construction of themoduli space of semi log canonical varieties.

Most steps go through as in the construction of thesmooth components.

New issue is boundedness of the moduli functor.

For example, if you fix the degree d how to boundthe number of components of an slc pair?

In the smooth case, simply take the closure in theChow variety.

A stable curve of genus g has at most 2g − 2components. Each component contributes ≥ 1.

Canonical models of varieties are singular, KnX ∈ Q.

Birational geometry and moduli spaces of varieties of general type – p. 9

DCC for the volume

Theorem: (Hacon,-,Xu) Fix n, I ⊂ [0, 1] satisfyingthe DCC. Let D be the set of log smooth pairs

(X,∆), where X is a projective variety ofdimension n and the coefficients of ∆ belong to I .Then the set

{ vol(X,KX +∆) | (X,∆) ∈ D }

satisfies the DCC.

Birational geometry and moduli spaces of varieties of general type – p. 10

DCC for the volume

Theorem: (Hacon,-,Xu) Fix n, I ⊂ [0, 1] satisfyingthe DCC. Let D be the set of log smooth pairs

(X,∆), where X is a projective variety ofdimension n and the coefficients of ∆ belong to I .Then the set

{ vol(X,KX +∆) | (X,∆) ∈ D }

satisfies the DCC.

Alexeev n = 2.

Birational geometry and moduli spaces of varieties of general type – p. 10

DCC for the volume

Theorem: (Hacon,-,Xu) Fix n, I ⊂ [0, 1] satisfyingthe DCC. Let D be the set of log smooth pairs

(X,∆), where X is a projective variety ofdimension n and the coefficients of ∆ belong to I .Then the set

{ vol(X,KX +∆) | (X,∆) ∈ D }

satisfies the DCC.

Alexeev n = 2.

If (X,∆) is semi log canonical then let (Xi,∆i) bethe components of the normalisation.

Birational geometry and moduli spaces of varieties of general type – p. 10

DCC for the volume

Theorem: (Hacon,-,Xu) Fix n, I ⊂ [0, 1] satisfyingthe DCC. Let D be the set of log smooth pairs

(X,∆), where X is a projective variety ofdimension n and the coefficients of ∆ belong to I .Then the set

{ vol(X,KX +∆) | (X,∆) ∈ D }

satisfies the DCC.

Alexeev n = 2.

If (X,∆) is semi log canonical then let (Xi,∆i) bethe components of the normalisation.

d = (KX +∆)n =∑

(KXi+∆i)

n =∑

di.

Birational geometry and moduli spaces of varieties of general type – p. 10

Log canonical woes

Why is the log canonical case harder than thekawamata log terminal case?

Birational geometry and moduli spaces of varieties of general type – p. 11

Log canonical woes

Why is the log canonical case harder than thekawamata log terminal case?

Run into a brick wall called abundance:

Birational geometry and moduli spaces of varieties of general type – p. 11

Log canonical woes

Why is the log canonical case harder than thekawamata log terminal case?

Run into a brick wall called abundance:

Conjecture: (Abundance) If (X,∆) is kawamata logterminal and KX +∆ is nef then KX +∆ issemiample (some multiple is base point free).

Birational geometry and moduli spaces of varieties of general type – p. 11

Log canonical woes

Why is the log canonical case harder than thekawamata log terminal case?

Run into a brick wall called abundance:

Conjecture: (Abundance) If (X,∆) is kawamata logterminal and KX +∆ is nef then KX +∆ issemiample (some multiple is base point free).

Key case: X not uniruled implies κ(X,KX) 6= −∞.

Birational geometry and moduli spaces of varieties of general type – p. 11

Log canonical woes

Why is the log canonical case harder than thekawamata log terminal case?

Run into a brick wall called abundance:

Conjecture: (Abundance) If (X,∆) is kawamata logterminal and KX +∆ is nef then KX +∆ issemiample (some multiple is base point free).

Key case: X not uniruled implies κ(X,KX) 6= −∞.

Let Y be the cone over X . Finite generation of the

canonical ring R(Y,KY ) implies κ(X,KX) 6= −∞.

Birational geometry and moduli spaces of varieties of general type – p. 11

Log canonical woes

Why is the log canonical case harder than thekawamata log terminal case?

Run into a brick wall called abundance:

Conjecture: (Abundance) If (X,∆) is kawamata logterminal and KX +∆ is nef then KX +∆ issemiample (some multiple is base point free).

Key case: X not uniruled implies κ(X,KX) 6= −∞.

Let Y be the cone over X . Finite generation of the

canonical ring R(Y,KY ) implies κ(X,KX) 6= −∞.

Cascini-Corti-Lazic: New approach to Mori theory.First prove finite generation and use this to derivestandard results of the MMP.

Birational geometry and moduli spaces of varieties of general type – p. 11

Two key results

If π : X −→ U is a morphism then (X,∆) is log smooth

over U if (X,∆) is log smooth and all strata dominate U .

Birational geometry and moduli spaces of varieties of general type – p. 12

Two key results

If π : X −→ U is a morphism then (X,∆) is log smooth

over U if (X,∆) is log smooth and all strata dominate U .

Theorem: (Berndtsson-Paun) If π : X −→ D is amorphism, (X,∆) is log smooth over D and x∆y = 0then h0(Xt,m(KXt

+∆t)) is independent of t for allm ≥ 0.

Birational geometry and moduli spaces of varieties of general type – p. 12

Two key results

If π : X −→ U is a morphism then (X,∆) is log smooth

over U if (X,∆) is log smooth and all strata dominate U .

Theorem: (Berndtsson-Paun) If π : X −→ D is amorphism, (X,∆) is log smooth over D and x∆y = 0then h0(Xt,m(KXt

+∆t)) is independent of t for allm ≥ 0.

A good minimal model of a log canonical pair (X,∆) isa minimal model f : X 99K Y such that KY + Γ issemiample, where Γ = f∗∆.

Birational geometry and moduli spaces of varieties of general type – p. 12

Two key results

If π : X −→ U is a morphism then (X,∆) is log smooth

over U if (X,∆) is log smooth and all strata dominate U .

Theorem: (Berndtsson-Paun) If π : X −→ D is amorphism, (X,∆) is log smooth over D and x∆y = 0then h0(Xt,m(KXt

+∆t)) is independent of t for allm ≥ 0.

A good minimal model of a log canonical pair (X,∆) isa minimal model f : X 99K Y such that KY + Γ issemiample, where Γ = f∗∆.

Theorem: (Birkar; Hacon-Xu) If π : X −→ U is amorphism, (X,∆) is dlt, every non klt centre dominates

U and (X,∆) has a good model over an open subset of

U then (X,∆) has a good model over U .Birational geometry and moduli spaces of varieties of general type – p. 12

Boundedness of moduli functor

Theorem: Fix an integer n, a positive rational

number d and a set I ⊂ [0, 1] which satisfies DCC.

Then the set Fslc(n, d, I) of all log pairs (X,∆) s.t.

Birational geometry and moduli spaces of varieties of general type – p. 13

Boundedness of moduli functor

Theorem: Fix an integer n, a positive rational

number d and a set I ⊂ [0, 1] which satisfies DCC.

Then the set Fslc(n, d, I) of all log pairs (X,∆) s.t.

X is projective of dimension n,

Birational geometry and moduli spaces of varieties of general type – p. 13

Boundedness of moduli functor

Theorem: Fix an integer n, a positive rational

number d and a set I ⊂ [0, 1] which satisfies DCC.

Then the set Fslc(n, d, I) of all log pairs (X,∆) s.t.

X is projective of dimension n,

(X,∆) is semi log canonical,

Birational geometry and moduli spaces of varieties of general type – p. 13

Boundedness of moduli functor

Theorem: Fix an integer n, a positive rational

number d and a set I ⊂ [0, 1] which satisfies DCC.

Then the set Fslc(n, d, I) of all log pairs (X,∆) s.t.

X is projective of dimension n,

(X,∆) is semi log canonical,

the coefficients of ∆ belong to I ,

Birational geometry and moduli spaces of varieties of general type – p. 13

Boundedness of moduli functor

Theorem: Fix an integer n, a positive rational

number d and a set I ⊂ [0, 1] which satisfies DCC.

Then the set Fslc(n, d, I) of all log pairs (X,∆) s.t.

X is projective of dimension n,

(X,∆) is semi log canonical,

the coefficients of ∆ belong to I ,

KX +∆ is an ample Q-divisor, and

Birational geometry and moduli spaces of varieties of general type – p. 13

Boundedness of moduli functor

Theorem: Fix an integer n, a positive rational

number d and a set I ⊂ [0, 1] which satisfies DCC.

Then the set Fslc(n, d, I) of all log pairs (X,∆) s.t.

X is projective of dimension n,

(X,∆) is semi log canonical,

the coefficients of ∆ belong to I ,

KX +∆ is an ample Q-divisor, and

(KX +∆)n = d,is bounded. In particular there is a finite set I0 such

that Fslc(n, d, I) = Fslc(n, d, I0).

Birational geometry and moduli spaces of varieties of general type – p. 13

Boundedness of moduli functor

Theorem: Fix an integer n, a positive rational

number d and a set I ⊂ [0, 1] which satisfies DCC.

Then the set Fslc(n, d, I) of all log pairs (X,∆) s.t.

X is projective of dimension n,

(X,∆) is semi log canonical,

the coefficients of ∆ belong to I ,

KX +∆ is an ample Q-divisor, and

(KX +∆)n = d,is bounded. In particular there is a finite set I0 such

that Fslc(n, d, I) = Fslc(n, d, I0).

Alexeev: n = 2.

Birational geometry and moduli spaces of varieties of general type – p. 13

Generic abundance

A key result to prove boundedness is:

Birational geometry and moduli spaces of varieties of general type – p. 14

Generic abundance

A key result to prove boundedness is:

Theorem: (Hacon,-,Xu) If π : X −→ U is morphism

and (X,∆) is log smooth over U then (X,∆) has agood minimal model over U if and only if one fibrehas a good minimal model over U .

Birational geometry and moduli spaces of varieties of general type – p. 14

Generic abundance

A key result to prove boundedness is:

Theorem: (Hacon,-,Xu) If π : X −→ U is morphism

and (X,∆) is log smooth over U then (X,∆) has agood minimal model over U if and only if one fibrehas a good minimal model over U .

In particular to prove abundance it suffices to prove

abundance for some deformation of (X,∆).

Birational geometry and moduli spaces of varieties of general type – p. 14

Generic abundance

A key result to prove boundedness is:

Theorem: (Hacon,-,Xu) If π : X −→ U is morphism

and (X,∆) is log smooth over U then (X,∆) has agood minimal model over U if and only if one fibrehas a good minimal model over U .

In particular to prove abundance it suffices to prove

abundance for some deformation of (X,∆).

We now give some of the ideas behind some resultswhich are a warm up to a proof of boundedness.

Birational geometry and moduli spaces of varieties of general type – p. 14

IOU argument

Theorem: (HM, T, T) Fix n. There is a positiveinteger m0 such that φmKX

is birational for allm ≥ m0, where X is of general type, dim. n.

Birational geometry and moduli spaces of varieties of general type – p. 15

IOU argument

Theorem: (HM, T, T) Fix n. There is a positiveinteger m0 such that φmKX

is birational for allm ≥ m0, where X is of general type, dim. n.

Tsuji’s key observation. It is enough to prove:

Birational geometry and moduli spaces of varieties of general type – p. 15

IOU argument

Theorem: (HM, T, T) Fix n. There is a positiveinteger m0 such that φmKX

is birational for allm ≥ m0, where X is of general type, dim. n.

Tsuji’s key observation. It is enough to prove:

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

Birational geometry and moduli spaces of varieties of general type – p. 15

IOU argument

Theorem: (HM, T, T) Fix n. There is a positiveinteger m0 such that φmKX

is birational for allm ≥ m0, where X is of general type, dim. n.

Tsuji’s key observation. It is enough to prove:

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

Indeed, if vol(X,KX) > 1 there is nothing to prove.

Birational geometry and moduli spaces of varieties of general type – p. 15

IOU argument

Theorem: (HM, T, T) Fix n. There is a positiveinteger m0 such that φmKX

is birational for allm ≥ m0, where X is of general type, dim. n.

Tsuji’s key observation. It is enough to prove:

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

Indeed, if vol(X,KX) > 1 there is nothing to prove.

Otherwise pick r such that 1 < vol(X, rKX) ≤ 2n.

Birational geometry and moduli spaces of varieties of general type – p. 15

IOU argument

Theorem: (HM, T, T) Fix n. There is a positiveinteger m0 such that φmKX

is birational for allm ≥ m0, where X is of general type, dim. n.

Tsuji’s key observation. It is enough to prove:

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

Indeed, if vol(X,KX) > 1 there is nothing to prove.

Otherwise pick r such that 1 < vol(X, rKX) ≤ 2n.

vol(X,mrKX) < (2m)n and so X belongs to abirationally bounded family, using the Chow variety.

Birational geometry and moduli spaces of varieties of general type – p. 15

IOU argument

Theorem: (HM, T, T) Fix n. There is a positiveinteger m0 such that φmKX

is birational for allm ≥ m0, where X is of general type, dim. n.

Tsuji’s key observation. It is enough to prove:

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

Indeed, if vol(X,KX) > 1 there is nothing to prove.

Otherwise pick r such that 1 < vol(X, rKX) ≤ 2n.

vol(X,mrKX) < (2m)n and so X belongs to abirationally bounded family, using the Chow variety.

We may find ǫ > 0 such that vol(X,KX) > ǫ and

vol(X, rKX) > 1, r = ⌈1/ǫ⌉. m0r works.Birational geometry and moduli spaces of varieties of general type – p. 15

Isolated non klt centres

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

Birational geometry and moduli spaces of varieties of general type – p. 16

Isolated non klt centres

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

A non klt centre of the pair (X,∆) is the image of adivisor of log discrepancy at most zero.

Birational geometry and moduli spaces of varieties of general type – p. 16

Isolated non klt centres

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

A non klt centre of the pair (X,∆) is the image of adivisor of log discrepancy at most zero.

Lemma 1: Given x ∈ X general, find D ∈ |krKX |such that x is an isolated non kawamata log terminal

centre of (X,B = D/kr).

Birational geometry and moduli spaces of varieties of general type – p. 16

Isolated non klt centres

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

A non klt centre of the pair (X,∆) is the image of adivisor of log discrepancy at most zero.

Lemma 1: Given x ∈ X general, find D ∈ |krKX |such that x is an isolated non kawamata log terminal

centre of (X,B = D/kr).

Lemma 2: Given x, y ∈ X general, find B ∼Q rKX

such that x is an isolated lc centre of (X,B) and ybelongs to a non klt centre V .

Birational geometry and moduli spaces of varieties of general type – p. 16

Isolated non klt centres

Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.

A non klt centre of the pair (X,∆) is the image of adivisor of log discrepancy at most zero.

Lemma 1: Given x ∈ X general, find D ∈ |krKX |such that x is an isolated non kawamata log terminal

centre of (X,B = D/kr).

Lemma 2: Given x, y ∈ X general, find B ∼Q rKX

such that x is an isolated lc centre of (X,B) and ybelongs to a non klt centre V .

It is enough to prove Lemma 2; lift section (1, 0)∈ Γ(X,Ox ⊕OV ), using the theory of multiplierideal sheaves. Birational geometry and moduli spaces of varieties of general type – p. 16

Lifting sections

Lemma 1: Given x ∈ X general, find B ∼Q rKX

such such that x is an isolated non kawamata log

terminal centre of (X,D).

Birational geometry and moduli spaces of varieties of general type – p. 17

Lifting sections

Lemma 1: Given x ∈ X general, find B ∼Q rKX

such such that x is an isolated non kawamata log

terminal centre of (X,D).

By assumption vol(X,nrKX) > (rn)n.

Birational geometry and moduli spaces of varieties of general type – p. 17

Lifting sections

Lemma 1: Given x ∈ X general, find B ∼Q rKX

such such that x is an isolated non kawamata log

terminal centre of (X,D).

By assumption vol(X,nrKX) > (rn)n.

By asymptotic Riemann-Roch find D ∈ |krKX |such that multX D > kn.

Birational geometry and moduli spaces of varieties of general type – p. 17

Lifting sections

Lemma 1: Given x ∈ X general, find B ∼Q rKX

such such that x is an isolated non kawamata log

terminal centre of (X,D).

By assumption vol(X,nrKX) > (rn)n.

By asymptotic Riemann-Roch find D ∈ |krKX |such that multX D > kn.

(X,B) is not kawamata log terminal at x; let x ∈ Vbe a non kawamata log terminal centre.

Birational geometry and moduli spaces of varieties of general type – p. 17

Lifting sections

Lemma 1: Given x ∈ X general, find B ∼Q rKX

such such that x is an isolated non kawamata log

terminal centre of (X,D).

By assumption vol(X,nrKX) > (rn)n.

By asymptotic Riemann-Roch find D ∈ |krKX |such that multX D > kn.

(X,B) is not kawamata log terminal at x; let x ∈ Vbe a non kawamata log terminal centre.

As x ∈ V is general, V is of general type.

Birational geometry and moduli spaces of varieties of general type – p. 17

Lifting sections

Lemma 1: Given x ∈ X general, find B ∼Q rKX

such such that x is an isolated non kawamata log

terminal centre of (X,D).

By assumption vol(X,nrKX) > (rn)n.

By asymptotic Riemann-Roch find D ∈ |krKX |such that multX D > kn.

(X,B) is not kawamata log terminal at x; let x ∈ Vbe a non kawamata log terminal centre.

As x ∈ V is general, V is of general type.

Kawamata subadjunction, (KX +B)|V = KV +Θ.

Birational geometry and moduli spaces of varieties of general type – p. 17

Lifting sections

Lemma 1: Given x ∈ X general, find B ∼Q rKX

such such that x is an isolated non kawamata log

terminal centre of (X,D).

By assumption vol(X,nrKX) > (rn)n.

By asymptotic Riemann-Roch find D ∈ |krKX |such that multX D > kn.

(X,B) is not kawamata log terminal at x; let x ∈ Vbe a non kawamata log terminal centre.

As x ∈ V is general, V is of general type.

Kawamata subadjunction, (KX +B)|V = KV +Θ.

Lift sections by Serre vanishing and proceed byinduction on the dimension.

Birational geometry and moduli spaces of varieties of general type – p. 17

Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).

Birational geometry and moduli spaces of varieties of general type – p. 18

Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).

If X is the canonical model then Aut(X) = Bir(X).

Birational geometry and moduli spaces of varieties of general type – p. 18

Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).

If X is the canonical model then Aut(X) = Bir(X).

Take a G = Aut(X)-equivariant resolution.

Birational geometry and moduli spaces of varieties of general type – p. 18

Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).

If X is the canonical model then Aut(X) = Bir(X).

Take a G = Aut(X)-equivariant resolution.

Let π : X −→ Y = X/G be the natural morphism.

Birational geometry and moduli spaces of varieties of general type – p. 18

Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).

If X is the canonical model then Aut(X) = Bir(X).

Take a G = Aut(X)-equivariant resolution.

Let π : X −→ Y = X/G be the natural morphism.

KX = π∗(KY + Γ), Γ has coefficients (r − 1)/r.

Birational geometry and moduli spaces of varieties of general type – p. 18

Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).

If X is the canonical model then Aut(X) = Bir(X).

Take a G = Aut(X)-equivariant resolution.

Let π : X −→ Y = X/G be the natural morphism.

KX = π∗(KY + Γ), Γ has coefficients (r − 1)/r.

Run IOU argument in orbifold case: volumebounded from below.

Birational geometry and moduli spaces of varieties of general type – p. 18

Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).

If X is the canonical model then Aut(X) = Bir(X).

Take a G = Aut(X)-equivariant resolution.

Let π : X −→ Y = X/G be the natural morphism.

KX = π∗(KY + Γ), Γ has coefficients (r − 1)/r.

Run IOU argument in orbifold case: volumebounded from below.

If vol(Y,KY + Γ) ≥ δ then c = 1

δ.

Birational geometry and moduli spaces of varieties of general type – p. 18

Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).

If X is the canonical model then Aut(X) = Bir(X).

Take a G = Aut(X)-equivariant resolution.

Let π : X −→ Y = X/G be the natural morphism.

KX = π∗(KY + Γ), Γ has coefficients (r − 1)/r.

Run IOU argument in orbifold case: volumebounded from below.

If vol(Y,KY + Γ) ≥ δ then c = 1

δ.

n = 1, (P1, 12p+ 2

3q + 6

7r), δ = 1/42, c = 42.

Birational geometry and moduli spaces of varieties of general type – p. 18

Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).

If X is the canonical model then Aut(X) = Bir(X).

Take a G = Aut(X)-equivariant resolution.

Let π : X −→ Y = X/G be the natural morphism.

KX = π∗(KY + Γ), Γ has coefficients (r − 1)/r.

Run IOU argument in orbifold case: volumebounded from below.

If vol(Y,KY + Γ) ≥ δ then c = 1

δ.

n = 1, (P1, 12p+ 2

3q + 6

7r), δ = 1/42, c = 42.

n = 2 Alexeev, Xiao.Birational geometry and moduli spaces of varieties of general type – p. 18

New directions

What general statements apply to all moduli spaces?

Birational geometry and moduli spaces of varieties of general type – p. 19

New directions

What general statements apply to all moduli spaces?

Let Msm be the open locus of normal varieties.

Birational geometry and moduli spaces of varieties of general type – p. 19

New directions

What general statements apply to all moduli spaces?

Let Msm be the open locus of normal varieties.

Theorem: Every subvariety V of Msm has loggeneral type.

Birational geometry and moduli spaces of varieties of general type – p. 19

New directions

What general statements apply to all moduli spaces?

Let Msm be the open locus of normal varieties.

Theorem: Every subvariety V of Msm has loggeneral type.

Vast generalisation of Shafarevich’s conjecture that

every family of curves over P1, A1, C∗ and everyelliptic curve, is isotrivial.

Birational geometry and moduli spaces of varieties of general type – p. 19

New directions

What general statements apply to all moduli spaces?

Let Msm be the open locus of normal varieties.

Theorem: Every subvariety V of Msm has loggeneral type.

Vast generalisation of Shafarevich’s conjecture that

every family of curves over P1, A1, C∗ and everyelliptic curve, is isotrivial.

Combined work, including Campana, Kebekus,Kovács, Paun, Popa, Schnell, Taji, Viehweg, Zuo.

Birational geometry and moduli spaces of varieties of general type – p. 19

New directions

What general statements apply to all moduli spaces?

Let Msm be the open locus of normal varieties.

Theorem: Every subvariety V of Msm has loggeneral type.

Vast generalisation of Shafarevich’s conjecture that

every family of curves over P1, A1, C∗ and everyelliptic curve, is isotrivial.

Combined work, including Campana, Kebekus,Kovács, Paun, Popa, Schnell, Taji, Viehweg, Zuo.

Idle speculation: To construct the moduli space ofsemi log canonical pairs we will need anotherinsight, akin to the idea of dropping GIT.

Birational geometry and moduli spaces of varieties of general type – p. 19

Kähler geometry

Mori theory is only supposed to apply to thecategory of projective varieties.

Birational geometry and moduli spaces of varieties of general type – p. 20

Kähler geometry

Mori theory is only supposed to apply to thecategory of projective varieties.

Theorem: (Höring-Peternell) We can run theKM -MMP for terminal Kähler threefolds.

Birational geometry and moduli spaces of varieties of general type – p. 20

Kähler geometry

Mori theory is only supposed to apply to thecategory of projective varieties.

Theorem: (Höring-Peternell) We can run theKM -MMP for terminal Kähler threefolds.

Theorem: (Campana-Höring-Peternell) Abundanceholds for terminal Kähler threefolds.

Birational geometry and moduli spaces of varieties of general type – p. 20

Kähler geometry

Mori theory is only supposed to apply to thecategory of projective varieties.

Theorem: (Höring-Peternell) We can run theKM -MMP for terminal Kähler threefolds.

Theorem: (Campana-Höring-Peternell) Abundanceholds for terminal Kähler threefolds.

Therefore every smooth Kähler manifold has a goodminimal model and we get very strong classificationresults for Kähler manifolds.

Birational geometry and moduli spaces of varieties of general type – p. 20

Characteristic p

Missing Kodaira vanishing and we only haveresolution of singularities for 3-folds.

Birational geometry and moduli spaces of varieties of general type – p. 21

Characteristic p

Missing Kodaira vanishing and we only haveresolution of singularities for 3-folds.

Keel proved a strong base point free theorem, usingFrobenius instead of Kawamata-Viehweg vanishing(inspired by results of Smith, which in turn use thetheory of tight closure, Hochster-Huneke).

Birational geometry and moduli spaces of varieties of general type – p. 21

Characteristic p

Missing Kodaira vanishing and we only haveresolution of singularities for 3-folds.

Keel proved a strong base point free theorem, usingFrobenius instead of Kawamata-Viehweg vanishing(inspired by results of Smith, which in turn use thetheory of tight closure, Hochster-Huneke).

Hacon and Xu proved existence of threefold flipsp > 5 , using Grothendieck duality and trace ofFrobenius (Schwede) and termination for KX-flips.

Birational geometry and moduli spaces of varieties of general type – p. 21

Characteristic p

Missing Kodaira vanishing and we only haveresolution of singularities for 3-folds.

Keel proved a strong base point free theorem, usingFrobenius instead of Kawamata-Viehweg vanishing(inspired by results of Smith, which in turn use thetheory of tight closure, Hochster-Huneke).

Hacon and Xu proved existence of threefold flipsp > 5 , using Grothendieck duality and trace ofFrobenius (Schwede) and termination for KX-flips.

Birkar established MMP for threefolds, p > 5.

Birational geometry and moduli spaces of varieties of general type – p. 21

Characteristic p

Missing Kodaira vanishing and we only haveresolution of singularities for 3-folds.

Keel proved a strong base point free theorem, usingFrobenius instead of Kawamata-Viehweg vanishing(inspired by results of Smith, which in turn use thetheory of tight closure, Hochster-Huneke).

Hacon and Xu proved existence of threefold flipsp > 5 , using Grothendieck duality and trace ofFrobenius (Schwede) and termination for KX-flips.

Birkar established MMP for threefolds, p > 5.

Patakfalvi proved a positivity result for thepushforward of the dualising sheaf.

Birational geometry and moduli spaces of varieties of general type – p. 21

Foliations

A foliation is a subsheaf of F ⊂ TX closed underLie bracket.

Birational geometry and moduli spaces of varieties of general type – p. 22

Foliations

A foliation is a subsheaf of F ⊂ TX closed underLie bracket.

Associate a canonical divisor KF = −c1(F).

Birational geometry and moduli spaces of varieties of general type – p. 22

Foliations

A foliation is a subsheaf of F ⊂ TX closed underLie bracket.

Associate a canonical divisor KF = −c1(F).

Bogomolov-McQuillan: Cone theorem for rank onefoliations.

Birational geometry and moduli spaces of varieties of general type – p. 22

Foliations

A foliation is a subsheaf of F ⊂ TX closed underLie bracket.

Associate a canonical divisor KF = −c1(F).

Bogomolov-McQuillan: Cone theorem for rank onefoliations.

Brunella-McQuillan: Classification of rank onefoliations on surfaces. Abundance fails (for varietieswith universal cover D× D).

Birational geometry and moduli spaces of varieties of general type – p. 22

Foliations

A foliation is a subsheaf of F ⊂ TX closed underLie bracket.

Associate a canonical divisor KF = −c1(F).

Bogomolov-McQuillan: Cone theorem for rank onefoliations.

Brunella-McQuillan: Classification of rank onefoliations on surfaces. Abundance fails (for varietieswith universal cover D× D).

Araujo-Druel: results for Fano foliations.

Birational geometry and moduli spaces of varieties of general type – p. 22

Foliations

A foliation is a subsheaf of F ⊂ TX closed underLie bracket.

Associate a canonical divisor KF = −c1(F).

Bogomolov-McQuillan: Cone theorem for rank onefoliations.

Brunella-McQuillan: Classification of rank onefoliations on surfaces. Abundance fails (for varietieswith universal cover D× D).

Araujo-Druel: results for Fano foliations.

Pereira-Touzet: results for K-trivial foliations.

Birational geometry and moduli spaces of varieties of general type – p. 22

Foliations

A foliation is a subsheaf of F ⊂ TX closed underLie bracket.

Associate a canonical divisor KF = −c1(F).

Bogomolov-McQuillan: Cone theorem for rank onefoliations.

Brunella-McQuillan: Classification of rank onefoliations on surfaces. Abundance fails (for varietieswith universal cover D× D).

Araujo-Druel: results for Fano foliations.

Pereira-Touzet: results for K-trivial foliations.

Spicer: partial results for a cone theorem for ranktwo foliations on threefolds. theorem

Birational geometry and moduli spaces of varieties of general type – p. 22