Recent results on jet differentials and entire holomorphic curves in algebraic varieties

Recent results on jet differentials and entire holomorphic curves in algebraic varieties

Jean-Pierre Demailly

Institut Fourier, Universit´e Grenoble Alpes & Acad´emie des Sciences de Paris

Hayama Symposium on Complex Analysis in Several Variables XIX

in honor of the 60th birthday of Professor Hajime Tsuji Shonan Village Center, July 15, 2017

Kobayashi hyperbolicity and entire curves

Definition

A complex space X is said to be Kobayashi hyperbolicif the Kobayashi pseudodistance d_Kob :X ×X →R+ is a distance (i.e. everywhere non degenerate).

By an entire curve we mean a non constant holomorphic map f :C→X into a complex n-dimensional manifold.

Theorem (Brody, 1978)

For a compact complex manifold X, dim_CX =n, TFAE: (i) X is Kobayashi hyperbolic

(ii) X is Brody hyperbolic, i.e. 6 ∃ entire curves f :C→X

(iii) The Kobayashi infinitesimal pseudometrick_x is everywhere non degenerate

Our interest is the study of hyperbolicity for projective varieties. In dim n= 1, X is hyperbolic iff genus g ≥2.

Kobayashi hyperbolicity and entire curves

Definition

A complex space X is said to be Kobayashi hyperbolicif the Kobayashi pseudodistance d_Kob :X ×X →R+ is a distance (i.e. everywhere non degenerate).

By an entire curve we mean a non constant holomorphic map f :C→X into a complex n-dimensional manifold.

Theorem (Brody, 1978)

For a compact complex manifold X, dim_CX =n, TFAE: (i) X is Kobayashi hyperbolic

(ii) X is Brody hyperbolic, i.e. 6 ∃ entire curves f :C→X

(iii) The Kobayashi infinitesimal pseudometrick_x is everywhere non degenerate

Our interest is the study of hyperbolicity for projective varieties. In dim n= 1, X is hyperbolic iff genus g ≥2.

Kobayashi hyperbolicity and entire curves

Definition

A complex space X is said to be Kobayashi hyperbolicif the Kobayashi pseudodistance d_Kob :X ×X →R+ is a distance (i.e. everywhere non degenerate).

By an entire curve we mean a non constant holomorphic map f :C→X into a complex n-dimensional manifold.

Theorem (Brody, 1978)

For a compact complex manifold X, dim_CX =n, TFAE:

(i) X is Kobayashi hyperbolic

(ii) X is Brody hyperbolic, i.e. 6 ∃ entire curves f :C→X

(iii) The Kobayashi infinitesimal pseudometrick_x is everywhere non degenerate

Our interest is the study of hyperbolicity for projective varieties. In dim n= 1, X is hyperbolic iff genus g ≥2.

Kobayashi hyperbolicity and entire curves

Definition

A complex space X is said to be Kobayashi hyperbolicif the Kobayashi pseudodistance d_Kob :X ×X →R+ is a distance (i.e. everywhere non degenerate).

By an entire curve we mean a non constant holomorphic map f :C→X into a complex n-dimensional manifold.

Theorem (Brody, 1978)

For a compact complex manifold X, dim_CX =n, TFAE:

(i) X is Kobayashi hyperbolic

(ii) X is Brody hyperbolic, i.e. 6 ∃ entire curves f :C→X

(iii) The Kobayashi infinitesimal pseudometrick_x is everywhere non degenerate

Our interest is the study of hyperbolicity for projective varieties.

In dim n= 1, X is hyperbolic iff genus g ≥2.

Main conjectures

Conjecture of General Type (CGT)

• A compact complex variety X is volume hyperbolic ⇐⇒ X is of general type, i.e. KX is big [implication ⇐= is well known].

• X Kobayashi (or Brody) hyperbolic should implyK_X ample. Green-Griffiths-Lang Conjecture (GGL)

Let X be a projective variety/C of general type. Then ∃Y (X algebraic such that all entire curves f :C→X satisfy f(C)⊂Y. Arithmetic counterpart (Lang 1987) – very optimistic ?

If X is projective and defined over a number field K⁰, the smallest locus Y =GGL(X) in GGL’s conjecture is also thesmallest Y such that X(K)rY is finite ∀K number field⊃K0.

Consequence of CGT + GGL

A compact complex manifold X should be Kobayashi hyperbolic iff it is projective and every subvariety Y of X is ofgeneral type.

Main conjectures

Conjecture of General Type (CGT)

• A compact complex variety X is volume hyperbolic ⇐⇒ X is of general type, i.e. KX is big [implication ⇐= is well known].

• X Kobayashi (or Brody) hyperbolic should implyK_X ample.

Green-Griffiths-Lang Conjecture (GGL)

Let X be a projective variety/C of general type. Then ∃Y (X algebraic such that all entire curves f :C→X satisfy f(C)⊂Y. Arithmetic counterpart (Lang 1987) – very optimistic ?

If X is projective and defined over a number field K⁰, the smallest locus Y =GGL(X) in GGL’s conjecture is also thesmallest Y such that X(K)rY is finite ∀K number field⊃K0.

Consequence of CGT + GGL

A compact complex manifold X should be Kobayashi hyperbolic iff it is projective and every subvariety Y of X is ofgeneral type.

Main conjectures

Conjecture of General Type (CGT)

• A compact complex variety X is volume hyperbolic ⇐⇒ X is of general type, i.e. KX is big [implication ⇐= is well known].

• X Kobayashi (or Brody) hyperbolic should implyK_X ample.

Green-Griffiths-Lang Conjecture (GGL)

Let X be a projective variety/C of general type. Then ∃Y (X algebraic such that all entire curves f :C→X satisfyf(C)⊂Y.

Arithmetic counterpart (Lang 1987) – very optimistic ?

If X is projective and defined over a number field K⁰, the smallest locus Y =GGL(X) in GGL’s conjecture is also thesmallest Y such that X(K)rY is finite ∀K number field⊃K0.

Consequence of CGT + GGL

A compact complex manifold X should be Kobayashi hyperbolic iff it is projective and every subvariety Y of X is ofgeneral type.

Main conjectures

Conjecture of General Type (CGT)

• A compact complex variety X is volume hyperbolic ⇐⇒ X is of general type, i.e. KX is big [implication ⇐= is well known].

• X Kobayashi (or Brody) hyperbolic should implyK_X ample.

Green-Griffiths-Lang Conjecture (GGL)

Let X be a projective variety/C of general type. Then ∃Y (X algebraic such that all entire curves f :C→X satisfyf(C)⊂Y. Arithmetic counterpart (Lang 1987) – very optimistic ?

If X is projective and defined over a number field K⁰, the smallest locus Y =GGL(X) in GGL’s conjecture is also thesmallest Y such that X(K)rY is finite ∀K number field⊃K0.

Consequence of CGT + GGL

A compact complex manifold X should be Kobayashi hyperbolic iff it is projective and every subvariety Y of X is ofgeneral type.

Main conjectures

Conjecture of General Type (CGT)

• A compact complex variety X is volume hyperbolic ⇐⇒ X is of general type, i.e. KX is big [implication ⇐= is well known].

• X Kobayashi (or Brody) hyperbolic should implyK_X ample.

Green-Griffiths-Lang Conjecture (GGL)

Let X be a projective variety/C of general type. Then ∃Y (X algebraic such that all entire curves f :C→X satisfyf(C)⊂Y. Arithmetic counterpart (Lang 1987) – very optimistic ?

If X is projective and defined over a number field K⁰, the smallest locus Y =GGL(X) in GGL’s conjecture is also thesmallest Y such that X(K)rY is finite ∀K number field⊃K0.

Consequence of CGT + GGL

Kobayashi conjecture on generic hyperbolicity

Kobayashi conjecture (1970)

• Let Xⁿ⊂Pⁿ⁺¹ be a (very) generic hypersurface of degree d ≥d_n large enough. Then X is Kobayashi hyperbolic.

• By a result of M. Zaidenberg, the optimal bound must satisfy dn ≥2n+ 1, and one expects dn= 2n+ 1.

Using “jet technology” and deep results of McQuillanfor curve foliations on surfaces, the following has been proved:

Theorem (D., El Goul, 1998)

A very generic surface X² ⊂P³ of degree d ≥21 is hyperbolic. Independently McQuillan got d ≥35.

This has been improved to d ≥18(P˘aun, 2008).

In 2012, Yum-Tong Siu announced a proof of the case of arbitrary dimension n, with a non explicit d_n (and a rather involved proof).

Kobayashi conjecture on generic hyperbolicity

Kobayashi conjecture (1970)

• Let Xⁿ⊂Pⁿ⁺¹ be a (very) generic hypersurface of degree d ≥d_n large enough. Then X is Kobayashi hyperbolic.

• By a result of M. Zaidenberg, the optimal bound must satisfy dn ≥2n+ 1, and one expects dn= 2n+ 1.

Using “jet technology” and deep results of McQuillanfor curve foliations on surfaces, the following has been proved:

Theorem (D., El Goul, 1998)

A very generic surface X² ⊂P³ of degree d ≥21 is hyperbolic. Independently McQuillan got d ≥35.

This has been improved to d ≥18(P˘aun, 2008).

In 2012, Yum-Tong Siu announced a proof of the case of arbitrary dimension n, with a non explicit d_n (and a rather involved proof).

Kobayashi conjecture on generic hyperbolicity

Kobayashi conjecture (1970)

• Let Xⁿ⊂Pⁿ⁺¹ be a (very) generic hypersurface of degree d ≥d_n large enough. Then X is Kobayashi hyperbolic.

• By a result of M. Zaidenberg, the optimal bound must satisfy dn ≥2n+ 1, and one expects dn= 2n+ 1.

Using “jet technology” and deep results of McQuillanfor curve foliations on surfaces, the following has been proved:

Theorem (D., El Goul, 1998)

A very generic surface X² ⊂P³ of degree d ≥21 is hyperbolic.

Independently McQuillan got d ≥35.

This has been improved to d ≥18(P˘aun, 2008).

In 2012, Yum-Tong Siu announced a proof of the case of arbitrary dimension n, with a non explicit d_n (and a rather involved proof).

Kobayashi conjecture on generic hyperbolicity

Kobayashi conjecture (1970)

• Let Xⁿ⊂Pⁿ⁺¹ be a (very) generic hypersurface of degree d ≥d_n large enough. Then X is Kobayashi hyperbolic.

• By a result of M. Zaidenberg, the optimal bound must satisfy dn ≥2n+ 1, and one expects dn= 2n+ 1.

Using “jet technology” and deep results of McQuillanfor curve foliations on surfaces, the following has been proved:

Theorem (D., El Goul, 1998)

A very generic surface X² ⊂P³ of degree d ≥21 is hyperbolic.

Independently McQuillan got d ≥35.

This has been improved to d ≥18(P˘aun, 2008).

In 2012, Yum-Tong Siu announced a proof of the case of arbitrary

Results on the generic Green-Griffiths conjecture

By a combination of an algebraic existence theorem for jet differentials and of Y.T. Siu’s technique of “slanted vector fields”

(itself derived from ideas of H. Clemens, L. Ein and C. Voisin), the following was proved:

Theorem (S. Diverio, J. Merker, E. Rousseau, 2009)

A generic hypersurface Xⁿ⊂Pⁿ⁺¹ of degreed ≥d_n := 2ⁿ⁵ satisfies the GGL conjecture.

Bound then improved to dn =O(exp(n^1+ε)) : d_n =j

n⁴

3 nlog(nlog(24n))nk

(D-, 2012), d_n= (5n)²nⁿ (Darondeau, 2015). Theorem (S. Diverio, S. Trapani, 2009)

Additionally, a generic hypersurface X³ ⊂P⁴ of degree d ≥593 is hyperbolic.

Results on the generic Green-Griffiths conjecture

By a combination of an algebraic existence theorem for jet differentials and of Y.T. Siu’s technique of “slanted vector fields”

(itself derived from ideas of H. Clemens, L. Ein and C. Voisin), the following was proved:

Theorem (S. Diverio, J. Merker, E. Rousseau, 2009)

A generic hypersurface Xⁿ⊂Pⁿ⁺¹ of degreed ≥d_n := 2ⁿ⁵ satisfies the GGL conjecture. Bound then improved to dn =O(exp(n^1+ε)) :

d_n =j

n⁴

3 nlog(nlog(24n))nk

(D-, 2012), d_n= (5n)²nⁿ (Darondeau, 2015).

Theorem (S. Diverio, S. Trapani, 2009)

Additionally, a generic hypersurface X³ ⊂P⁴ of degree d ≥593 is hyperbolic.

Results on the generic Green-Griffiths conjecture

By a combination of an algebraic existence theorem for jet differentials and of Y.T. Siu’s technique of “slanted vector fields”

(itself derived from ideas of H. Clemens, L. Ein and C. Voisin), the following was proved:

Theorem (S. Diverio, J. Merker, E. Rousseau, 2009)

A generic hypersurface Xⁿ⊂Pⁿ⁺¹ of degreed ≥d_n := 2ⁿ⁵ satisfies the GGL conjecture. Bound then improved to dn =O(exp(n^1+ε)) :

d_n =j

n⁴

3 nlog(nlog(24n))nk

(D-, 2012), d_n= (5n)²nⁿ (Darondeau, 2015).

Theorem (S. Diverio, S. Trapani, 2009)

Additionally, a generic hypersurface X³ ⊂P⁴ of degree d ≥593 is hyperbolic.

Recent proof of the Kobayashi conjecture

In 2016, Brotbek gave a shorter and more geometric proof of Y.T.

Siu’s result on the Kobayashi conjecture, using again jet techniques.

Theorem (Brotbek, April 2016)

Let Z be a projective n+ 1-dimensional projective manifold and A→Z a very ample line bundle. Let σ∈H⁰(Z,dA) be a generic section. Then, for d 1 large, the hypersurfaceXσ =σ⁻¹(0) is hyperbolic.

The initial proof of Brotbek did not provide effective bounds. Through various improvements, Deng Ya showed in his PhD thesis: Theorem (Y. Deng, May 2016)

In the above setting, a generic hypersurface X_σ =σ⁻¹(0) is hyperbolic as soon as

d ≥dn = (n+ 1)ⁿ⁺²(n+ 2)²ⁿ⁺⁷ =O(n³ⁿ⁺⁹).

Recent proof of the Kobayashi conjecture

In 2016, Brotbek gave a shorter and more geometric proof of Y.T.

Siu’s result on the Kobayashi conjecture, using again jet techniques.

Theorem (Brotbek, April 2016)

Let Z be a projective n+ 1-dimensional projective manifold and A→Z a very ample line bundle. Let σ∈H⁰(Z,dA) be a generic section. Then, for d 1 large, the hypersurfaceXσ =σ⁻¹(0) is hyperbolic.

The initial proof of Brotbek did not provide effective bounds.

Through various improvements, Deng Ya showed in his PhD thesis:

Theorem (Y. Deng, May 2016)

In the above setting, a generic hypersurface X_σ =σ⁻¹(0) is hyperbolic as soon as

d ≥dn = (n+ 1)ⁿ⁺²(n+ 2)²ⁿ⁺⁷ =O(n³ⁿ⁺⁹).

Solution of a conjecture of Debarre (2005)

In the same vein, the following results have also been proved.

Solution of Debarre’s conjecture (Brotbek-Darondeau & Xie, 2015) Let Z be a projective n+c-dimensional projective manifold and A→Z a very ample line bundle. Let σ_j ∈H⁰(Z,d_jA) be generic sections, 1≤j ≤c. Then, forc ≥n and d_j 1 large, the n-dimensional complete intersection X_σ =T

σ_j⁻¹(0)⊂Z has an ample cotangent bundle T_X^∗_σ.

In particular, such a generic complete intersection is hyperbolic.

Xie got the sufficient lower bound d_j ≥d_n,c =N^N2,N =n+c. In his PhD thesis, Y. Deng obtained the improved lower bound

dn,c = 4ν(2N −1)^2ν+1+ 6N −3 =O((2N)^N+3), ν=b^N+1₂ c. The proof is obtained by selecting carefully certain special sections σ_j associated with “lacunary” polynomials of high degree.

Solution of a conjecture of Debarre (2005)

In the same vein, the following results have also been proved.

Solution of Debarre’s conjecture (Brotbek-Darondeau & Xie, 2015) Let Z be a projective n+c-dimensional projective manifold and A→Z a very ample line bundle. Let σ_j ∈H⁰(Z,d_jA) be generic sections, 1≤j ≤c. Then, forc ≥n and d_j 1 large, the n-dimensional complete intersection X_σ =T

σ_j⁻¹(0)⊂Z has an ample cotangent bundle T_X^∗_σ.

In particular, such a generic complete intersection is hyperbolic.

Xie got the sufficient lower bound d_j ≥d_n,c =N^N2,N =n+c.

In his PhD thesis, Y. Deng obtained the improved lower bound dn,c = 4ν(2N −1)^2ν+1+ 6N −3 =O((2N)^N+3), ν=b^N+1₂ c. The proof is obtained by selecting carefully certain special sections σ_j associated with “lacunary” polynomials of high degree.

Solution of a conjecture of Debarre (2005)

In the same vein, the following results have also been proved.

Solution of Debarre’s conjecture (Brotbek-Darondeau & Xie, 2015) Let Z be a projective n+c-dimensional projective manifold and A→Z a very ample line bundle. Let σ_j ∈H⁰(Z,d_jA) be generic sections, 1≤j ≤c. Then, forc ≥n and d_j 1 large, the n-dimensional complete intersection X_σ =T

σ_j⁻¹(0)⊂Z has an ample cotangent bundle T_X^∗_σ.

In particular, such a generic complete intersection is hyperbolic.

Xie got the sufficient lower bound d_j ≥d_n,c =N^N2,N =n+c.

In his PhD thesis, Y. Deng obtained the improved lower bound dn,c = 4ν(2N −1)^2ν+1+ 6N −3 =O((2N)^N+3), ν=b^N+1₂ c.

The proof is obtained by selecting carefully certain special sections σ_j associated with “lacunary” polynomials of high degree.

Solution of a conjecture of Debarre (2005)

In the same vein, the following results have also been proved.

Solution of Debarre’s conjecture (Brotbek-Darondeau & Xie, 2015) Let Z be a projective n+c-dimensional projective manifold and A→Z a very ample line bundle. Let σ_j ∈H⁰(Z,d_jA) be generic sections, 1≤j ≤c. Then, forc ≥n and d_j 1 large, the n-dimensional complete intersection X_σ =T

σ_j⁻¹(0)⊂Z has an ample cotangent bundle T_X^∗_σ.

In particular, such a generic complete intersection is hyperbolic.

Xie got the sufficient lower bound d_j ≥d_n,c =N^N2,N =n+c.

In his PhD thesis, Y. Deng obtained the improved lower bound dn,c = 4ν(2N −1)^2ν+1+ 6N −3 =O((2N)^N+3), ν=b^N+1₂ c.

The proof is obtained by selecting carefully certain special sections σ_j associated with “lacunary” polynomials of high degree.

Category of directed manifolds

Goal. More generally, we are interested in curves f :C→X such that f⁰(C)⊂V whereV is a subbundle of T_X, or possibly a singular linear subspace, i.e. a closed irreducible analytic subspace such that ∀x ∈X, V_x :=V ∩T_X,x is linear.

Definition (Category of directed manifolds)

– Objects : pairs (X,V), X manifold/C and V ⊂T_X – Arrows ψ : (X,V)→(Y,W) holomorphic s.t. ψ∗V ⊂W – “Absolute case”(X,T_X), i.e. V =T_X

– “Relative case” (X,T_X/S) where X →S

– “Integrable case” when [O(V),O(V)]⊂ O(V) (foliations) Canonical sheaf of a directed manifold (X,V)

When V is nonsingular, i.e. a subbundle, one simply sets KV = det(V^∗) (as a line bundle).

Category of directed manifolds

Goal. More generally, we are interested in curves f :C→X such that f⁰(C)⊂V whereV is a subbundle of T_X, or possibly a singular linear subspace, i.e. a closed irreducible analytic subspace such that ∀x ∈X, V_x :=V ∩T_X,x is linear.

Definition (Category of directed manifolds)

– Objects : pairs (X,V), X manifold/C and V ⊂T_X – Arrows ψ : (X,V)→(Y,W) holomorphic s.t. ψ∗V ⊂W

– “Absolute case”(X,T_X), i.e. V =T_X – “Relative case” (X,T_X/S) where X →S

– “Integrable case” when [O(V),O(V)]⊂ O(V) (foliations) Canonical sheaf of a directed manifold (X,V)

When V is nonsingular, i.e. a subbundle, one simply sets KV = det(V^∗) (as a line bundle).

Category of directed manifolds

Goal. More generally, we are interested in curves f :C→X such that f⁰(C)⊂V whereV is a subbundle of T_X, or possibly a singular linear subspace, i.e. a closed irreducible analytic subspace such that ∀x ∈X, V_x :=V ∩T_X,x is linear.

Definition (Category of directed manifolds)

– Objects : pairs (X,V), X manifold/C and V ⊂T_X – Arrows ψ : (X,V)→(Y,W) holomorphic s.t. ψ∗V ⊂W – “Absolute case”(X,TX), i.e. V =TX

– “Relative case” (X,T_X/S) where X →S

– “Integrable case” when [O(V),O(V)]⊂ O(V) (foliations)

Canonical sheaf of a directed manifold (X,V)

When V is nonsingular, i.e. a subbundle, one simply sets KV = det(V^∗) (as a line bundle).

Category of directed manifolds

Goal. More generally, we are interested in curves f :C→X such that f⁰(C)⊂V whereV is a subbundle of T_X, or possibly a singular linear subspace, i.e. a closed irreducible analytic subspace such that ∀x ∈X, V_x :=V ∩T_X,x is linear.

Definition (Category of directed manifolds)

– Objects : pairs (X,V), X manifold/C and V ⊂T_X – Arrows ψ : (X,V)→(Y,W) holomorphic s.t. ψ∗V ⊂W – “Absolute case”(X,TX), i.e. V =TX

– “Relative case” (X,T_X/S) where X →S

– “Integrable case” when [O(V),O(V)]⊂ O(V) (foliations) Canonical sheaf of a directed manifold (X,V)

When V is nonsingular, i.e. a subbundle, one simply sets KV = det(V^∗) (as a line bundle).

Canonical sheaf of a singular pair (X,V)

When V is singular, we first introduce the rank 1 sheaf ^bK_V of sections of detV^∗ that are locally boundedwith respect to a smooth ambient metric on T_X. One can show that^bK_V is equal to the integral closure of the image of the natural morphism

O(Λ^rT_X^∗)→ O(Λ^rV^∗)→ LV :=invert. sheaf O(Λ^rV^∗)^∗∗

that is, if the image is LV ⊗ JV,JV ⊂ OX,

bK_V =L_V ⊗ J_V, J_V = integral closure of J_V.

Consequence

If µ:Xe →X is a modification and Xe is equipped with the pull-back directed structure Ve = ˜µ⁻¹(V), then

bKV ⊂µ∗(^bK_Ve)⊂ LV

and µ∗(^bK_Ve) increases with µ.

Canonical sheaf of a singular pair (X,V)

When V is singular, we first introduce the rank 1 sheaf ^bK_V of sections of detV^∗ that are locally boundedwith respect to a smooth ambient metric on T_X. One can show that^bK_V is equal to the integral closure of the image of the natural morphism

O(Λ^rT_X^∗)→ O(Λ^rV^∗)→ LV :=invert. sheaf O(Λ^rV^∗)^∗∗

that is, if the image is LV ⊗ JV,JV ⊂ OX,

bK_V =L_V ⊗ J_V, J_V = integral closure of J_V. Consequence

If µ:Xe →X is a modification and Xe is equipped with the pull-back directed structure Ve = ˜µ⁻¹(V), then

bK_V ⊂µ∗(^bK_Ve)⊂ L_V and µ∗(^bK_Ve) increases with µ.

Canonical sheaf of a singular pair (X,V) [cont.]

By Noetherianity, one can define a sequence of rank 1 sheaves K^[m]_V = lim

µ ↑µ∗(^bK_Ve)^⊗m, µ∗(^bKV)^⊗m ⊂ K^[m]_V ⊂ L^⊗m_V which we call the pluricanonical sheaf sequenceof (X,V).

Remark

The blow-up µ for which the limit is attained may depend on m. We do not know if there is a µ that works for all m.

This generalizes the concept of reduced singularitiesof foliations, which is known to work in that form only for surfaces.

Definition

We say that (X,V) is ofgeneral type if the pluricanonical sheaf sequence K^[•]_V is big, i.e. H⁰(X,K^[m]_V ) provides a generic embedding of X for a suitable m 1.

Canonical sheaf of a singular pair (X,V) [cont.]

By Noetherianity, one can define a sequence of rank 1 sheaves K^[m]_V = lim

µ ↑µ∗(^bK_Ve)^⊗m, µ∗(^bKV)^⊗m ⊂ K^[m]_V ⊂ L^⊗m_V which we call the pluricanonical sheaf sequenceof (X,V).

Remark

The blow-up µfor which the limit is attained may depend on m.

We do not know if there is a µ that works for allm.

This generalizes the concept of reduced singularitiesof foliations, which is known to work in that form only for surfaces.

Definition

We say that (X,V) is ofgeneral type if the pluricanonical sheaf sequence K^[•]_V is big, i.e. H⁰(X,K^[m]_V ) provides a generic embedding of X for a suitable m 1.

Canonical sheaf of a singular pair (X,V) [cont.]

By Noetherianity, one can define a sequence of rank 1 sheaves K^[m]_V = lim

µ ↑µ∗(^bK_Ve)^⊗m, µ∗(^bKV)^⊗m ⊂ K^[m]_V ⊂ L^⊗m_V which we call the pluricanonical sheaf sequenceof (X,V).

Remark

The blow-up µfor which the limit is attained may depend on m.

We do not know if there is a µ that works for allm.

This generalizes the concept of reduced singularities of foliations, which is known to work in that form only for surfaces.

Definition

We say that (X,V) is ofgeneral type if the pluricanonical sheaf sequence K^[•]_V is big, i.e. H⁰(X,K_V^[m]) provides a generic embedding

Definition of algebraic differential operators

Let (C,T_C)→(X,V), t 7→f(t) = (f₁(t), . . . ,f_n(t))be a curve written in some local holomorphic coordinates (z₁, . . . ,z_n) on X. It has a local Taylor expansion

f(t) =x +tξ1+. . .+t^kξk +O(t^k+1), ξs = 1

s!∇^sf(0) for some connection ∇ on V.

One considers the Green-Griffiths bundle E_k,m^GGV^∗ of polynomials of weighted degree m written locally in coordinate charts as

P(x; ξ₁, . . . , ξ_k) = X

a_α1α2...αk(x)ξ₁^α1. . . ξ_k^αk, ξ_s ∈V, also viewed as algebraic differential operators

P(f_[k]) = P(f⁰,f⁰⁰, . . . ,f^(k))

= X

a_α1α2...αk(f(t))f⁰(t)^α1f⁰⁰(t)^α2. . .f^(k)(t)^αk.

Definition of algebraic differential operators

Let (C,T_C)→(X,V), t 7→f(t) = (f₁(t), . . . ,f_n(t))be a curve written in some local holomorphic coordinates (z₁, . . . ,z_n) on X. It has a local Taylor expansion

f(t) =x +tξ1+. . .+t^kξk +O(t^k+1), ξs = 1

s!∇^sf(0) for some connection ∇ on V.

One considers the Green-Griffiths bundle E_k,m^GGV^∗ of polynomials of weighted degree m written locally in coordinate charts as

P(x; ξ₁, . . . , ξ_k) = X

a_α1α2...αk(x)ξ₁^α1. . . ξ_k^αk, ξ_s ∈V, also viewed as algebraic differential operators

P(f[k]) = P(f⁰,f⁰⁰, . . . ,f^(k))

Definition of algebraic differential operators [cont.]

Here t 7→z =f(t) is a curve, f_[k] = (f⁰,f⁰⁰, . . . ,f^(k)) its k-jet, and a_α1α2...αk(z) are supposed to holomorphic functions on X.

The reparametrization action : f 7→f ◦ϕ_λ, ϕ_λ(t) = λt,λ∈C^∗ yields (f ◦ϕ_λ)^(k)(t) =λ^kf^(k)(λt), whence a C^∗-action

λ·(ξ₁, ξ₁, . . . , ξ_k) = (λξ₁, λ²ξ₂, . . . , λ^kξ_k).

E_k,m^GG is precisely the set of polynomials of weighted degreem, corresponding to coefficients a_α1...αk with

m =|α₁|+ 2|α₂|+. . .+k|α_k|. Direct image formula

If J_k^ncV is the set of non constantk-jets, one defines the Green-Griffiths bundle to beX_k^GG =J_k^ncV/C^∗ and O_X^GG

k (1) to be the associated tautological rank 1 sheaf. Then we have

π_k :X_k^GG →X, E_k,m^GGV^∗ = (π_k)∗O_X^GG

k (m)

Definition of algebraic differential operators [cont.]

Here t 7→z =f(t) is a curve, f_[k] = (f⁰,f⁰⁰, . . . ,f^(k)) its k-jet, and a_α1α2...αk(z) are supposed to holomorphic functions on X.

The reparametrization action : f 7→f ◦ϕ_λ, ϕ_λ(t) = λt, λ∈C^∗ yields (f ◦ϕ_λ)^(k)(t) =λ^kf^(k)(λt), whence a C^∗-action

λ·(ξ1, ξ1, . . . , ξk) = (λξ1, λ²ξ2, . . . , λ^kξk).

E_k,m^GG is precisely the set of polynomials of weighted degreem, corresponding to coefficients a_α1...αk with

m =|α₁|+ 2|α₂|+. . .+k|α_k|. Direct image formula

If J_k^ncV is the set of non constantk-jets, one defines the Green-Griffiths bundle to beX_k^GG =J_k^ncV/C^∗ and O_X^GG

k (1) to be the associated tautological rank 1 sheaf. Then we have

π_k :X_k^GG →X, E_k,m^GGV^∗ = (π_k)∗O_X^GG

k (m)

Definition of algebraic differential operators [cont.]

Here t 7→z =f(t) is a curve, f_[k] = (f⁰,f⁰⁰, . . . ,f^(k)) its k-jet, and a_α1α2...αk(z) are supposed to holomorphic functions on X.

The reparametrization action : f 7→f ◦ϕ_λ, ϕ_λ(t) = λt, λ∈C^∗ yields (f ◦ϕ_λ)^(k)(t) =λ^kf^(k)(λt), whence a C^∗-action

λ·(ξ1, ξ1, . . . , ξk) = (λξ1, λ²ξ2, . . . , λ^kξk).

E_k,m^GG is precisely the set of polynomials of weighted degree m, corresponding to coefficients a_α1...αk with

m =|α₁|+ 2|α₂|+. . .+k|α_k|.

Direct image formula

If J_k^ncV is the set of non constantk-jets, one defines the Green-Griffiths bundle to beX_k^GG =J_k^ncV/C^∗ and O_X^GG

k (1) to be the associated tautological rank 1 sheaf. Then we have

π_k :X_k^GG →X, E_k,m^GGV^∗ = (π_k)∗O_X^GG

k (m)

Definition of algebraic differential operators [cont.]

Here t 7→z =f(t) is a curve, f_[k] = (f⁰,f⁰⁰, . . . ,f^(k)) its k-jet, and a_α1α2...αk(z) are supposed to holomorphic functions on X.

The reparametrization action : f 7→f ◦ϕ_λ, ϕ_λ(t) = λt, λ∈C^∗ yields (f ◦ϕ_λ)^(k)(t) =λ^kf^(k)(λt), whence a C^∗-action

λ·(ξ1, ξ1, . . . , ξk) = (λξ1, λ²ξ2, . . . , λ^kξk).

E_k,m^GG is precisely the set of polynomials of weighted degree m, corresponding to coefficients a_α1...αk with

m =|α₁|+ 2|α₂|+. . .+k|α_k|.

Direct image formula

If J_k^ncV is the set of non constantk-jets, one defines the Green-Griffiths bundle to be X_k^GG =J_k^ncV/C^∗ and O_X^GG

k (1) to be the associated tautological rank 1 sheaf. Then we have

Generalized GGL conjecture

Generalized GGL conjecture

If (X,V) is directed manifold of general type, i.e.K^[•]_V is big, then

∃Y (X such that ∀f : (C,T_C)→(X,V), one has f(C)⊂Y.

Remark. Elementary by Ahlfors-Schwarz if r = rkV = 1.

t 7→logkf⁰(t)k_V,h is strictly subharmonic ifr = 1 and (V^∗,h^∗) big. Strategy : fundamental vanishing theorem

[Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996]

∀P ∈H⁰(X,E_k,m^GGV^∗⊗ O(−A)): global diff. operator onX (A ample divisor), ∀f : (C,T_C)→(X,V), one has P(f_[k])≡0. Theorem on existence of jet differentials (D-, 2010)

Let (X,V) be of general type, such that ^bK^⊗p_V is a bigrank 1 sheaf. Then ∃ many global sectionsP, mk1⇒ ∃ alg. hypersurface Z (X_k^GG s.t. all entire f : (C,T_C)7→(X,V) satisfy f_[k](C)⊂Z.

Generalized GGL conjecture

Generalized GGL conjecture

If (X,V) is directed manifold of general type, i.e.K^[•]_V is big, then

∃Y (X such that ∀f : (C,T_C)→(X,V), one has f(C)⊂Y. Remark. Elementary by Ahlfors-Schwarz if r = rkV = 1.

t 7→logkf⁰(t)k_V,h is strictly subharmonic ifr = 1 and (V^∗,h^∗) big.

Strategy : fundamental vanishing theorem

[Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996]

∀P ∈H⁰(X,E_k,m^GGV^∗⊗ O(−A)): global diff. operator onX (A ample divisor), ∀f : (C,T_C)→(X,V), one has P(f_[k])≡0. Theorem on existence of jet differentials (D-, 2010)

Let (X,V) be of general type, such that ^bK^⊗p_V is a bigrank 1 sheaf. Then ∃ many global sectionsP, mk1⇒ ∃ alg. hypersurface Z (X_k^GG s.t. all entire f : (C,T_C)7→(X,V) satisfy f_[k](C)⊂Z.

Generalized GGL conjecture

Generalized GGL conjecture

If (X,V) is directed manifold of general type, i.e.K^[•]_V is big, then

∃Y (X such that ∀f : (C,T_C)→(X,V), one has f(C)⊂Y. Remark. Elementary by Ahlfors-Schwarz if r = rkV = 1.

t 7→logkf⁰(t)k_V,h is strictly subharmonic ifr = 1 and (V^∗,h^∗) big.

Strategy : fundamental vanishing theorem

[Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996]

∀P ∈H⁰(X,E_k,m^GGV^∗⊗ O(−A)) : global diff. operator onX (A ample divisor), ∀f : (C,T_C)→(X,V), one has P(f_[k])≡0.

Theorem on existence of jet differentials (D-, 2010)

Let (X,V) be of general type, such that ^bK^⊗p_V is a bigrank 1 sheaf. Then ∃ many global sectionsP, mk1⇒ ∃ alg. hypersurface Z (X_k^GG s.t. all entire f : (C,T_C)7→(X,V) satisfy f_[k](C)⊂Z.

Generalized GGL conjecture

Generalized GGL conjecture

If (X,V) is directed manifold of general type, i.e.K^[•]_V is big, then

∃Y (X such that ∀f : (C,T_C)→(X,V), one has f(C)⊂Y. Remark. Elementary by Ahlfors-Schwarz if r = rkV = 1.

t 7→logkf⁰(t)k_V,h is strictly subharmonic ifr = 1 and (V^∗,h^∗) big.

Strategy : fundamental vanishing theorem

[Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996]

∀P ∈H⁰(X,E_k,m^GGV^∗⊗ O(−A)) : global diff. operator onX (A ample divisor), ∀f : (C,T_C)→(X,V), one has P(f_[k])≡0.

Theorem on existence of jet differentials (D-, 2010)

Let (X,V) be of general type, such that ^bK^⊗p_V is a bigrank 1 sheaf.

Then ∃ many global sectionsP, mk1⇒ ∃ alg. hypersurface

1

step: take a Finsler metric on k -jet bundles

Let J_kV be the bundle of k-jets of curves f : (C,T_C)→(X,V)

Assuming that V is equipped with a hermitian metric h, one defines a ”weighted Finsler metric” on J^kV by takingp =k! and

Ψ_hk(f) := X

1≤s≤k

ε_sk∇^sf(0)k^2p/s_h(x)1/p

, 1 = ε₁ ε₂ · · · ε_k.

Letting ξ_s =∇^sf(0), this can actually be viewed as a metrich_k on L_k :=O_X^GG

k (1), with curvature form (x, ξ₁, . . . , ξ_k)7→ Θ_Lk,hk =ω_FS,k(ξ)+ i

2π X

1≤s≤k

1 s

|ξ_s|^2p/s P

t|ξ_t|^2p/t X

i,j,α,β

c_ijαβξ_sαξ_sβ

|ξ_s|² dz_i∧d z_j where (c_ijαβ)are the coefficients of the curvature tensor Θ_V^∗_,h^∗ and ω_FS,k is the vertical Fubini-Study metric on the fibers ofX_k^GG →X. The expression gets simpler by using polar coordinates

x_s =|ξ_s|^2p/s_h , u_s =ξ_s/|ξ_s|_h =∇^sf(0)/|∇^sf(0)|.

1

step: take a Finsler metric on k -jet bundles

Let J_kV be the bundle of k-jets of curves f : (C,T_C)→(X,V) Assuming that V is equipped with a hermitian metric h, one defines a ”weighted Finsler metric” on J^kV by takingp =k! and

Ψ_hk(f) := X

1≤s≤k

ε_sk∇^sf(0)k^2p/s_h(x)1/p

, 1 = ε₁ ε₂ · · · ε_k.

Letting ξ_s =∇^sf(0), this can actually be viewed as a metrich_k on L_k :=O_X^GG

k (1), with curvature form (x, ξ₁, . . . , ξ_k)7→ Θ_Lk,hk =ω_FS,k(ξ)+ i

2π X

1≤s≤k

1 s

|ξ_s|^2p/s P

t|ξ_t|^2p/t X

i,j,α,β

c_ijαβξ_sαξ_sβ

|ξ_s|² dz_i∧d z_j where (c_ijαβ)are the coefficients of the curvature tensor Θ_V^∗_,h^∗ and ω_FS,k is the vertical Fubini-Study metric on the fibers ofX_k^GG →X. The expression gets simpler by using polar coordinates

x_s =|ξ_s|^2p/s_h , u_s =ξ_s/|ξ_s|_h =∇^sf(0)/|∇^sf(0)|.

1

step: take a Finsler metric on k -jet bundles

Let J_kV be the bundle of k-jets of curves f : (C,T_C)→(X,V) Assuming that V is equipped with a hermitian metric h, one defines a ”weighted Finsler metric” on J^kV by takingp =k! and

Ψ_hk(f) := X

1≤s≤k

ε_sk∇^sf(0)k^2p/s_h(x)1/p

, 1 = ε₁ ε₂ · · · ε_k.

Letting ξ_s =∇^sf(0), this can actually be viewed as a metrich_k on L_k :=O_X^GG

k (1), with curvature form (x, ξ₁, . . . , ξ_k)7→

Θ_Lk,hk =ω_FS,k(ξ)+ i 2π

X

1≤s≤k

1 s

|ξ_s|^2p/s P

t|ξ_t|^2p/t X

i,j,α,β

c_ijαβξ_sαξ_sβ

|ξ_s|² dz_i∧d z_j where (c_ijαβ)are the coefficients of the curvature tensor Θ_V^∗_,h^∗ and ω_FS,k is the vertical Fubini-Study metric on the fibers ofX_k^GG →X.

The expression gets simpler by using polar coordinates x_s =|ξ_s|^2p/s_h , u_s =ξ_s/|ξ_s|_h =∇^sf(0)/|∇^sf(0)|.

1

step: take a Finsler metric on k -jet bundles

Let J_kV be the bundle of k-jets of curves f : (C,T_C)→(X,V) Assuming that V is equipped with a hermitian metric h, one defines a ”weighted Finsler metric” on J^kV by takingp =k! and

Ψ_hk(f) := X

1≤s≤k

ε_sk∇^sf(0)k^2p/s_h(x)1/p

, 1 = ε₁ ε₂ · · · ε_k.

Letting ξ_s =∇^sf(0), this can actually be viewed as a metrich_k on L_k :=O_X^GG

k (1), with curvature form (x, ξ₁, . . . , ξ_k)7→

Θ_Lk,hk =ω_FS,k(ξ)+ i 2π

X

1≤s≤k

1 s

|ξ_s|^2p/s P

t|ξ_t|^2p/t X

i,j,α,β

c_ijαβξ_sαξ_sβ

|ξ_s|² dz_i∧d z_j where (c_ijαβ)are the coefficients of the curvature tensor Θ_V^∗_,h^∗ and ω_FS,k is the vertical Fubini-Study metric on the fibers ofX_k^GG →X. The expression gets simpler by using polar coordinates

2

step: probabilistic interpretation of the curvature

In such polar coordinates, one gets the formula ΘL_k,h_k =ωFS,p,k(ξ) + i

2π X

1≤s≤k

1 sxs

X

i,j,α,β

cijαβ(z)usαusβdzi ∧d zj

where ω_FS,k(ξ) is positive definite in ξ. The other terms are a weighted average of the values of the curvature tensor Θ_V,h on vectors u_s in the unit sphere bundle SV ⊂V.

The weighted projective space can be viewed as a circle quotient of the pseudosphere P|ξ_s|^2p/s = 1, so we can take here x_s ≥0, Px_s = 1. This is essentially a sum of the form P₁

sγ(u_s) whereu_s are random points of the sphere, and so as k →+∞ this can be estimated by a “Monte-Carlo” integral

1 + 1

2 +. . .+ 1 k

Z

u∈SV

γ(u)du. As γ is quadratic here, R

u∈SV γ(u)du = ¹_r Tr(γ).

2

step: probabilistic interpretation of the curvature

In such polar coordinates, one gets the formula ΘL_k,h_k =ωFS,p,k(ξ) + i

2π X

1≤s≤k

1 sxs

X

i,j,α,β

cijαβ(z)usαusβdzi ∧d zj

where ω_FS,k(ξ) is positive definite in ξ. The other terms are a weighted average of the values of the curvature tensor Θ_V,h on vectors u_s in the unit sphere bundle SV ⊂V.

The weighted projective space can be viewed as a circle quotient of the pseudosphere P|ξ_s|^2p/s = 1, so we can take here x_s ≥0, Px_s = 1. This is essentially a sum of the formP₁

sγ(u_s) whereu_s are random points of the sphere, and so as k →+∞ this can be estimated by a “Monte-Carlo” integral

1 + 1

2 +. . .+ 1 k

Z

u∈SV

γ(u)du.

3

step: getting the main cohomology estimates

⇒ the leading term only involves the trace of Θ_V^∗_,h^∗, i.e. the curvature of (detV^∗,deth^∗), that can be taken >0 if detV^∗ is big.

Corollary of holomorphic Morse inequalities (D-, 2010)

Let (X,V) be a directed manifold, F →X a Q-line bundle, (V,h) and (F,h_F) hermitian. Define

Lk =O_X^GG

k (1)⊗π^∗_kO 1 kr

1 + 1

2+. . .+ 1 k

F

, η = Θ_detV^∗_,deth^∗+ Θ_F,hF.

Then for all q ≥0 and allmk 1 such that m is sufficiently divisible, we have upper and lower bounds [q = 0 most useful!]

h^q(X_k^GG,O(L^⊗m_k ))≤ m^n+kr−1 (n+kr−1)!

(logk)ⁿ n! (k!)^r

Z

X(η,q)

(−1)^qηⁿ+ C logk

h^q(X_k^GG,O(L^⊗m_k ))≥ m^n+kr−1 (n+kr−1)!

(logk)ⁿ n! (k!)^r

Z

X(η,q,q±1)

(−1)^qηⁿ− C logk

.

3

step: getting the main cohomology estimates

⇒ the leading term only involves the trace of Θ_V^∗_,h^∗, i.e. the curvature of (detV^∗,deth^∗), that can be taken >0 if detV^∗ is big.

Corollary of holomorphic Morse inequalities (D-, 2010)

Let (X,V) be a directed manifold, F →X a Q-line bundle, (V,h) and (F,h_F) hermitian. Define

Lk =O_X^GG

k (1)⊗π^∗_kO 1 kr

1 + 1

2+. . .+ 1 k

F

, η = Θ_detV^∗_,deth^∗+ Θ_F,hF.

Then for all q ≥0 and allmk 1 such that m is sufficiently divisible, we have upper and lower bounds [q = 0 most useful!]

h^q(X_k^GG,O(L^⊗m_k ))≤ m^n+kr−1 (n+kr−1)!

(logk)ⁿ n! (k!)^r

Z

X(η,q)

(−1)^qηⁿ+ C logk

q GG ⊗m m^n+kr−1 (logk)ⁿ Z

q n C

And now ... the Semple jet bundles

Fonctor “1-jet” : (X,V)7→( ˜X,V˜) where :

X˜ =P(V) = bundle of projective spaces of lines in V π : ˜X =P(V)→X, (x,[v])7→x, v ∈V_x

V˜_(x,[v]) =

ξ∈TX˜,(x,[v]); π∗ξ ∈Cv ⊂T_X,x

For every entire curvef : (C,T_C)→(X,V) tangent to V f lifts as

f_[1](t) := (f(t),[f⁰(t)])∈P(V_f(t))⊂X˜

f_[1]: (C,T_C)→( ˜X,V˜) (projectivized 1st-jet) Definition. Semple jet bundles :

– (X_k,V_k) = k-th iteration of fonctor (X,V)7→( ˜X,V˜) – f_[k]: (C,T_C)→(X_k,V_k) is theprojectivizedk-jet of f. Basic exact sequences

0→T_Xk/Xk−1 →V_k ^(π→ O^k)? _Xk(−1)→0 ⇒rkV_k =r 0→ O_Xk →π^?_kVk−1⊗ O_Xk(1)→T_Xk/Xk−1 →0 (Euler)

And now ... the Semple jet bundles

Fonctor “1-jet” : (X,V)7→( ˜X,V˜) where :

X˜ =P(V) = bundle of projective spaces of lines in V π : ˜X =P(V)→X, (x,[v])7→x, v ∈V_x

V˜_(x,[v]) =

ξ∈TX˜,(x,[v]); π∗ξ ∈Cv ⊂T_X,x

For every entire curvef : (C,T_C)→(X,V) tangent to V f lifts as

f_[1](t) := (f(t),[f⁰(t)])∈P(V_f(t))⊂X˜

f_[1]: (C,T_C)→( ˜X,V˜) (projectivized 1st-jet)

Definition. Semple jet bundles :

– (X_k,V_k) = k-th iteration of fonctor (X,V)7→( ˜X,V˜) – f_[k]: (C,T_C)→(X_k,V_k) is theprojectivizedk-jet of f. Basic exact sequences

0→T_Xk/Xk−1 →V_k ^(π→ O^k)? _Xk(−1)→0 ⇒rkV_k =r 0→ O_Xk →π^?_kVk−1⊗ O_Xk(1)→T_Xk/Xk−1 →0 (Euler)

And now ... the Semple jet bundles

Fonctor “1-jet” : (X,V)7→( ˜X,V˜) where :

X˜ =P(V) = bundle of projective spaces of lines in V π : ˜X =P(V)→X, (x,[v])7→x, v ∈V_x

V˜_(x,[v]) =

ξ∈TX˜,(x,[v]); π∗ξ ∈Cv ⊂T_X,x

For every entire curvef : (C,T_C)→(X,V) tangent to V f lifts as

f_[1](t) := (f(t),[f⁰(t)])∈P(V_f(t))⊂X˜

f_[1]: (C,T_C)→( ˜X,V˜) (projectivized 1st-jet) Definition. Semple jet bundles :

– (X_k,V_k) = k-th iteration of fonctor (X,V)7→( ˜X,V˜) – f_[k]: (C,T_C)→(X_k,V_k) is theprojectivizedk-jet of f.

Basic exact sequences

0→T_Xk/Xk−1 →V_k ^(π→ O^k)? _Xk(−1)→0 ⇒rkV_k =r 0→ O_Xk →π^?_kVk−1⊗ O_Xk(1)→T_Xk/Xk−1 →0 (Euler)