A simple proof with eﬀective bounds of the Kobayashi conjecture on generic hyperbolicity

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A simple proof with effective bounds of the Kobayashi conjecture

on generic hyperbolicity

Jean-Pierre Demailly

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

Algebraic Geometry Seminar Center for Mathematical Sciences University of Cambridge, May 23, 2018

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Kobayashi hyperbolicity and entire curves

Kobayashi-Eisenman infinitesimal pseudometrics

Let X be a complex space, dim_CX =n, Bp = unit ball inC^p, 1≤p ≤n and τ0 =∂/∂t1∧ · · · ∧∂/∂tp∈Λ^pC^p. The Kobayashi- Eisenman infinitesimal pseudometric e^p_X is the pseudometric defined on decomposable p-vectors ξ=ξ₁∧ · · · ∧ξ_p ∈Λ^pT_X_,x, by

e^p_X(ξ) = inf

λ >0 ;∃f :Bp →X, f(0) =x, λf_?(τ₀) =ξ .

We say that X is (infinitesimally) p-measure hyperbolicif e^p_X is everywhere locally uniformly positive definite on the tautological line bundle of the Grassmannian bundle of p-subspaces Gr(T_X,p). Characterization of Kobayashi hyperbolicity (Brody, 1978)

For a compact complex manifold X, dim_CX =n, TFAE: (i) The pseudometric k_X =e¹_x is everywhere non degenerate ; (ii) the integrated pseudodistance dKob of e¹_X is a distance ;

(iii) X Brody hyperbolic, i.e.6 ∃ entire curves f :C→X, f 6= const.

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Kobayashi hyperbolicity and entire curves

e^p_X(ξ) = inf

λ >0 ;∃f :Bp →X, f(0) =x, λf_?(τ₀) =ξ . We say that X is (infinitesimally) p-measure hyperbolicif e^p_X is everywhere locally uniformly positive definite on the tautological line bundle of the Grassmannian bundle of p-subspaces Gr(T_X,p).

Characterization of Kobayashi hyperbolicity (Brody, 1978) For a compact complex manifold X, dim_CX =n, TFAE: (i) The pseudometric k_X =e¹_x is everywhere non degenerate ; (ii) the integrated pseudodistance dKob of e¹_X is a distance ;

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Kobayashi hyperbolicity and entire curves

e^p_X(ξ) = inf

λ >0 ;∃f :Bp →X, f(0) =x, λf_?(τ₀) =ξ . We say that X is (infinitesimally) p-measure hyperbolicif e^p_X is everywhere locally uniformly positive definite on the tautological line bundle of the Grassmannian bundle of p-subspaces Gr(T_X,p).

Characterization of Kobayashi hyperbolicity (Brody, 1978) For a compact complex manifold X, dim_CX =n, TFAE:

(i) The pseudometric k_X =e¹_x is everywhere non degenerate ; (ii) the integrated pseudodistance dKob of e¹_X is a distance ;

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Main conjectures

Conjecture of General Type (CGT)

• A compact variety X/C isvolume hyperbolic (w.r.t. eⁿ_X) ⇔ X is of general type, i.e. K_X big [implication ⇐ is well known].

• X Kobayashi (or Brody) hyperbolic should implyKX 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 ?

For X projective defined over a number fieldK0, the exceptional locus Y =Exc(X) in GGL’s conjecture equals Mordel(X) = smallest 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.

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Main conjectures

• 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 ?

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Main conjectures

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 ?

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Main conjectures

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 ?

For X projective defined over a number fieldK0, the exceptional locus Y =Exc(X) in GGL’s conjecture equals Mordel(X) = smallest Y such that X(K)rY is finite,∀K number field ⊃K⁰.

Consequence of CGT + GGL

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Main conjectures

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 ?

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Kobayashi conjecture on generic hyperbolicity

Kobayashi conjecture (1970)

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

• By results of Riemann, Poincar´e, Zaidenberg, Clemens, Ein, Voisin, Pacienza, the optimal bound is expected to be d₁ = 4, dn = 2n+ 1 for 2≤n≤4 and dn = 2n for n≥5.

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

Theorem (Work of McQuillan + 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).

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Kobayashi conjecture on generic hyperbolicity

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

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Kobayashi conjecture on generic hyperbolicity

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

Independently McQuillan got d ≥35.

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Kobayashi conjecture on generic hyperbolicity

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Kobayashi conjecture on generic hyperbolicity

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Results on the generic Green-Griffiths conjecture

By combining an algebraic existence theorem for jet differentials and 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, & followers) A generic hypersurface Xⁿ⊂Pⁿ⁺¹ of degreed ≥d_n := 2ⁿ⁵ satisfies the GGL conjecture.

Bound then improved to d_n =O(eⁿ^1+ε) : d_n= 9nⁿ (B´erczi, 2010, using residue formulas), d_n= (5n)²nⁿ (Darondeau, 2015, alternative method), d_n=j

n⁴

3 nlog(nlog(24n))nk

(D-, 2012), weaker bound, but special case of general result for arbitrary projective varieties.

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

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

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Results on the generic Green-Griffiths conjecture

Theorem (S. Diverio, J. Merker, E. Rousseau, 2009, & followers) A generic hypersurface Xⁿ⊂Pⁿ⁺¹ of degreed ≥d_n := 2ⁿ⁵ satisfies the GGL conjecture. Bound then improved to d_n =O(eⁿ^1+ε) :

d_n= 9nⁿ (B´erczi, 2010, using residue formulas), d_n= (5n)²nⁿ (Darondeau, 2015, alternative method), d_n=j

n⁴

3 nlog(nlog(24n))nk

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Results on the generic Green-Griffiths conjecture

Theorem (S. Diverio, J. Merker, E. Rousseau, 2009, & followers) A generic hypersurface Xⁿ⊂Pⁿ⁺¹ of degreed ≥d_n := 2ⁿ⁵ satisfies the GGL conjecture. Bound then improved to d_n =O(eⁿ^1+ε) :

d_n= 9nⁿ (B´erczi, 2010, using residue formulas), d_n= (5n)²nⁿ (Darondeau, 2015, alternative method), d_n=j

n⁴

3 nlog(nlog(24n))nk

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Recent proof of the Kobayashi conjecture

Y.T. Siu’s (Abel conf. 2002, Invent. Math. 2015) detailed a strategy for the proof of the Kobayashi conjecture. In 2016, Brotbek gave a more geometric proof, using Wronskian jet differentials.

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 d1the hypersurface X_σ =σ⁻¹(0) is hyperbolic.

The initial proof of Brotbek didn’t provide effective bounds. Through various improvements, Deng Ya got in his PhD thesis (May 2016) the explicit bound d_n= (n+ 1)ⁿ⁺²(n+ 2)²ⁿ⁺⁷ =O(n³ⁿ⁺⁹). Theorem (D-, 2018, with a much simplified proof)

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

d ≥d_n =b(en)²ⁿ⁺²/3c.

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Recent proof of the Kobayashi conjecture

Y.T. Siu’s (Abel conf. 2002, Invent. Math. 2015) detailed a strategy for the proof of the Kobayashi conjecture. In 2016, Brotbek gave a more geometric proof, using Wronskian jet differentials.

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 d1the hypersurface X_σ =σ⁻¹(0) is hyperbolic.

The initial proof of Brotbek didn’t provide effective bounds. Through various improvements, Deng Ya got in his PhD thesis (May 2016) the explicit bound d_n= (n+ 1)ⁿ⁺²(n+ 2)²ⁿ⁺⁷ =O(n³ⁿ⁺⁹).

Theorem (D-, 2018, with a much simplified proof)

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

d ≥d_n=b(en)²ⁿ⁺²/3c.

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

d_n,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.

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Solution of a conjecture of Debarre (2005)

σ.

In particular, such a generic complete intersection is hyperbolic.

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

d_n,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.

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Solution of a conjecture of Debarre (2005)

σ.

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

In his PhD thesis, Ya Deng obtained the improved lower bound d_n,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.

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Solution of a conjecture of Debarre (2005)

σ.

In his PhD thesis, Ya Deng obtained the improved lower bound d_n,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.

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Solution of a conjecture of Debarre (2005)

σ.

In his PhD thesis, Ya Deng obtained the improved lower bound d_n,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 σj associated with “lacunary” polynomials of high degree.

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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

– Morphisms 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).

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Category of directed manifolds

– Morphisms 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

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Category of directed manifolds

– Morphisms ψ : (X,V)→(Y,W) holomorphic s.t. ψ∗V ⊂W – “Absolute case”(X,TX), i.e. V =TX

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

Canonical sheaf of a directed manifold (X,V)

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Category of directed manifolds

– Morphisms ψ : (X,V)→(Y,W) holomorphic s.t. ψ∗V ⊂W – “Absolute case”(X,TX), i.e. V =TX

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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^∗)→ L_V := 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_V_e) increases with µ.

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

that is, if the image is L_V ⊗ J_V,J_V ⊂ O_X,

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

Consequence

bK_V ⊂µ∗(^bK

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

that is, if the image is L_V ⊗ J_V,J_V ⊂ O_X,

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

bK_V ⊂µ∗(^bK

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

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

µ ↑µ∗(^bK_V_e)^⊗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.

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

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

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

Remark

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

Definition

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

Remark

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 of X for a suitable m 1.

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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_α₁_α₂_...α_k(x)ξ₁^α¹. . . ξ_k^α^k, ξ_s ∈V. One can view them as algebraic differential operators

P(f_[k]) = P(f⁰,f⁰⁰, . . . ,f^(k)), P(f_[k_])(t) = X

a_α₁_α₂_...α_k(f(t)) f⁰(t)^α¹f⁰⁰(t)^α². . .f^(k⁾(t)^α^k.

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Definition of algebraic differential operators

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

P(x; ξ₁, . . . , ξ_k) = X

a_α₁_α₂_...α_k(x)ξ₁^α¹. . . ξ_k^α^k, ξ_s ∈V.

One can view them as algebraic differential operators P(f_[k]) = P(f⁰,f⁰⁰, . . . ,f^(k)),

P(f_[k_])(t) = X

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Definition of algebraic differential operators

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

P(x; ξ₁, . . . , ξ_k) = X

a_α₁_α₂_...α_k(x)ξ₁^α¹. . . ξ_k^α^k, ξ_s ∈V. One can view them as algebraic differential operators

P(f[k]) = P(f⁰,f⁰⁰, . . . ,f^(k)), P(f_[k])(t) = X

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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_α₁_α₂_...α_k(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, ξ₂, . . . , 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_...α_k 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)

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Definition of algebraic differential operators [cont.]

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, ξ2, . . . , ξk) = (λξ1, λ²ξ2, . . . , λ^kξk).

E_k,m^GG is precisely the set of polynomials of weighted degreem, corresponding to coefficients a_α₁_...α_k with

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

k (m)

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Definition of algebraic differential operators [cont.]

λ·(ξ1, ξ2, . . . , ξk) = (λξ1, λ²ξ2, . . . , λ^kξk).

E_k,m^GG is precisely the set of polynomials of weighted degree m, corresponding to coefficients a_α₁_...α_k with

m =|α₁|+ 2|α₂|+. . .+k|α_k|.

Direct image formula

k (m)

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Definition of algebraic differential operators [cont.]

λ·(ξ1, ξ2, . . . , ξk) = (λξ1, λ²ξ2, . . . , λ^kξk).

E_k,m^GG is precisely the set of polynomials of weighted degree m, corresponding to coefficients a_α₁_...α_k 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

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

k (m)

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Generalized GGL conjecture, strategy of attack

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.

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.

⇐⇒ f_[k](C)⊂σ⁻¹(0), ∀σ∈H⁰(X_k^GG,O_X^GG

k (m)⊗π_k^∗O(−A)). Corollary: exploit base locus of algebraic differential equations Exceptional locus: Exc(X,V) =S

f f(C)^Zar, f : (C,T_C)→(X,V), Green-Griffiths locus: GG(X,V) =T

kπ_k(GG_k(X,V)), where GG_k(X,V) = T

σσ⁻¹(0), σ∈H⁰(X_k^GG,O_X^GG

k (m)⊗π_k^∗O(−A)). Then Exc(X,V)⊂GG(X,V).

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Generalized GGL conjecture, strategy of attack

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

∀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.

⇐⇒ f_[k](C)⊂σ⁻¹(0), ∀σ∈H⁰(X_k^GG,O_X^GG

k (m)⊗π_k^∗O(−A)). Corollary: exploit base locus of algebraic differential equations Exceptional locus: Exc(X,V) =S

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Generalized GGL conjecture, strategy of attack

⇐⇒ f_[k](C)⊂σ⁻¹(0), ∀σ∈H⁰(X_k^GG,O_X^GG

k (m)⊗π_k^∗O(−A)).

Corollary: exploit base locus of algebraic differential equations Exceptional locus: Exc(X,V) =S

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Generalized GGL conjecture, strategy of attack

⇐⇒ f_[k](C)⊂σ⁻¹(0), ∀σ∈H⁰(X_k^GG,O_X^GG

k (m)⊗π_k^∗O(−A)).

Corollary: exploit base locus of algebraic differential equations Exceptional locus: Exc(X,V) =S

kπ_k(GG_k(X,V)), where GG_k(X,V) =T

k (m)⊗π_k^∗O(−A)).

Then Exc(X,V)⊂GG(X,V).

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Proof of the fundamental vanishing theorem

Simple case. First assume that f is a Brody curve, i.e. that sup_t∈

Ckf⁰(t)k_ω <+∞ for some hermitian metric ω on X. By raising P to a power, we can assumeA very ample, and view P as a Cvalued differential operator whose coefficients vanish on a very ample divisor A.

The Cauchy inequalities imply that all derivatives f^(s) are bounded in any coordinate chart. Hence u_A(t) :=P(f_[k_])(t) is bounded, and must be constant by Liouville’s theorem.

Since Ais very ample, we can move A∈ |A| such that A hits f(C)⊂X. Bu then u_A vanishes somewhere and so u_A ≡0. General case of a general entire curve f : (C,T_C)→(X,V). Instead, one makes use of Nevanlinna theory arguments

(logarithmic derivative lemma).

Remark. Generalized GGL conjecture is easy if rankV = 1.

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Proof of the fundamental vanishing theorem

Since Ais very ample, we can move A∈ |A| such that A hits f(C)⊂X. Bu then u_A vanishes somewhere and so u_A ≡0. General case of a general entire curve f : (C,T_C)→(X,V). Instead, one makes use of Nevanlinna theory arguments

(logarithmic derivative lemma).

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Proof of the fundamental vanishing theorem

Since Ais very ample, we can move A∈ |A| such that A hits f(C)⊂X. Bu then u_A vanishes somewhere and so u_A ≡0.

General case of a general entire curve f : (C,T_C)→(X,V).

Instead, one makes use of Nevanlinna theory arguments (logarithmic derivative lemma).

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And now ... the Semple jet bundles

Functor “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 curve f : (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 functor (X,V)7→( ˜X,V˜) – f_[k]: (C,T_C)→(X_k,V_k) is theprojectivizedk-jet of f. Basic exact sequences

0→T_X_k_/X_k−1 →V_k ^(π→ O^k⁾^? _X_k(−1)→0 ⇒rankV_k =r 0→ O_X_k →π_k^?Vk−1⊗ O_X_k(1) →T_X_k_/X_k−1 →0 (Euler)

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And now ... the Semple jet bundles

Functor “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 curve f : (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 functor (X,V)7→( ˜X,V˜) – f_[k]: (C,T_C)→(X_k,V_k) is theprojectivizedk-jet of f. Basic exact sequences

0→T_X_k_/X_k−1 →V_k ^(π→ O^k⁾^? _X_k(−1)→0 ⇒rankV_k =r 0→ O_X_k →π_k^?Vk−1⊗ O_X_k(1) →T_X_k_/X_k−1 →0 (Euler)