Hyperbolic algebraic varieties and holomorphic differential - - PowerPoint PPT Presentation

Hyperbolic algebraic varieties and holomorphic differential equations

Jean-Pierre Demailly

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

August 26, 2012 / VIASM Yearly Meeting, Hanoi

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Entire curves

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• Definition. By an entire curve we mean a non constant

holomorphic map f : C → X into a complex n-dimensional manifold.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Entire curves

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• Definition. By an entire curve we mean a non constant

holomorphic map f : C → X into a complex n-dimensional manifold. X is said to be (Brody) hyperbolic if ∃ such f : C → X.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Entire curves

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• Definition. By an entire curve we mean a non constant

holomorphic map f : C → X into a complex n-dimensional manifold. X is said to be (Brody) hyperbolic if ∃ such f : C → X. If X is a bounded open subset Ω ⊂ Cn, then there are no entire curves f : C → Ω (Liouville’s theorem), ⇒ every bounded open set Ω ⊂ Cn is hyperbolic

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Entire curves

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• Definition. By an entire curve we mean a non constant

holomorphic map f : C → X into a complex n-dimensional manifold. X is said to be (Brody) hyperbolic if ∃ such f : C → X. If X is a bounded open subset Ω ⊂ Cn, then there are no entire curves f : C → Ω (Liouville’s theorem), ⇒ every bounded open set Ω ⊂ Cn is hyperbolic X = C {0, 1, ∞} = C {0, 1} has no entire curves, so it is hyperbolic (Picard’s theorem)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Entire curves

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• Definition. By an entire curve we mean a non constant

holomorphic map f : C → X into a complex n-dimensional manifold. X is said to be (Brody) hyperbolic if ∃ such f : C → X. If X is a bounded open subset Ω ⊂ Cn, then there are no entire curves f : C → Ω (Liouville’s theorem), ⇒ every bounded open set Ω ⊂ Cn is hyperbolic X = C {0, 1, ∞} = C {0, 1} has no entire curves, so it is hyperbolic (Picard’s theorem) A complex torus X = Cn/Λ (Λ lattice) has a lot of entire

• curves. As C simply connected, every f : C → X = Cn/Λ

lifts as ˜ f : C → Cn, ˜ f (t) = (˜ f1(t), . . . , ˜ fn(t)), and ˜ fj : C → C can be arbitrary entire functions.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Projective algebraic varieties

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Consider now the complex projective n-space Pn = Pn

C = (Cn+1 {0})/C∗,

[z] = [z0 : z1 : . . . : zn].

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Projective algebraic varieties

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Consider now the complex projective n-space Pn = Pn

C = (Cn+1 {0})/C∗,

[z] = [z0 : z1 : . . . : zn]. An entire curve f : C → Pn is given by a map t − → [f0(t) : f1(t) : . . . : fn(t)] where fj : C → C are holomorphic functions without common zeroes (so there are a lot of them).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Projective algebraic varieties

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Consider now the complex projective n-space Pn = Pn

C = (Cn+1 {0})/C∗,

[z] = [z0 : z1 : . . . : zn]. An entire curve f : C → Pn is given by a map t − → [f0(t) : f1(t) : . . . : fn(t)] where fj : C → C are holomorphic functions without common zeroes (so there are a lot of them). More generally, look at a (complex) projective manifold, i.e. X n ⊂ PN, X = {[z] ; P1(z) = ... = Pk(z) = 0} where Pj(z) = Pj(z0, z1, . . . , zN) are homogeneous polynomials (of some degree dj), such that X is non singular.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Complex curves (genus 0 and 1)

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Canonical bundle KX = ΛnT ∗

X (here KX = T ∗ X)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Complex curves (genus 0 and 1)

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Canonical bundle KX = ΛnT ∗

X (here KX = T ∗ X)

g = 0 : X = P1 courbure TX > 0 not hyperbolic

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Complex curves (genus 0 and 1)

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Canonical bundle KX = ΛnT ∗

X (here KX = T ∗ X)

g = 0 : X = P1 courbure TX > 0 not hyperbolic g = 1 : X = C/(Z + Zτ) courbure TX = 0 not hyperbolic

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Complex curves (genus g ≥ 2)

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deg KX = 2g − 2 If g ≥ 2, X ≃ D/Γ (TX < 0) ⇒ X is hyperbolic.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Complex curves (genus g ≥ 2)

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deg KX = 2g − 2 If g ≥ 2, X ≃ D/Γ (TX < 0) ⇒ X is hyperbolic. In fact every curve f : C → X ≃ D/Γ lifts to f : C → D, and so must be constant by Liouville.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Kobayashi metric / hyperbolic manifolds

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For a complex manifold, n = dimC X, one defines the Kobayashi pseudo-metric : x ∈ X, ξ ∈ TX κx(ξ) = inf{λ > 0 ; ∃f : D → X, f (0) = x, λf∗(0) = ξ} On Cn, Pn or complex tori X = Cn/Λ, one has κX ≡ 0.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Kobayashi metric / hyperbolic manifolds

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For a complex manifold, n = dimC X, one defines the Kobayashi pseudo-metric : x ∈ X, ξ ∈ TX κx(ξ) = inf{λ > 0 ; ∃f : D → X, f (0) = x, λf∗(0) = ξ} On Cn, Pn or complex tori X = Cn/Λ, one has κX ≡ 0. X is said to be hyperbolic in the sense of Kobayashi if the associated integrated pseudo-distance is a distance (i.e. it separates points – i.e. has Hausdorff topology).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Kobayashi metric / hyperbolic manifolds

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For a complex manifold, n = dimC X, one defines the Kobayashi pseudo-metric : x ∈ X, ξ ∈ TX κx(ξ) = inf{λ > 0 ; ∃f : D → X, f (0) = x, λf∗(0) = ξ} On Cn, Pn or complex tori X = Cn/Λ, one has κX ≡ 0. X is said to be hyperbolic in the sense of Kobayashi if the associated integrated pseudo-distance is a distance (i.e. it separates points – i.e. has Hausdorff topology).

• Examples. ∗ X = Ω/Γ, Ω bounded symmetric domain.

∗ any product X = X1 × . . . × Xs where Xj hyperbolic.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Kobayashi metric / hyperbolic manifolds

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For a complex manifold, n = dimC X, one defines the Kobayashi pseudo-metric : x ∈ X, ξ ∈ TX κx(ξ) = inf{λ > 0 ; ∃f : D → X, f (0) = x, λf∗(0) = ξ} On Cn, Pn or complex tori X = Cn/Λ, one has κX ≡ 0. X is said to be hyperbolic in the sense of Kobayashi if the associated integrated pseudo-distance is a distance (i.e. it separates points – i.e. has Hausdorff topology).

• Examples. ∗ X = Ω/Γ, Ω bounded symmetric domain.

∗ any product X = X1 × . . . × Xs where Xj hyperbolic. Theorem (dimension n arbitrary) (Kobayashi, 1970) TX negatively curved (T ∗

X > 0, i.e. ample) ⇒ X hyperbolic.

Recall that a holomorphic vector bundle E is ample iff its symmetric powers SmE have global sections which generate 1-jets of (germs of) sections at any point x ∈ X.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Ahlfors-Schwarz lemma

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The proof of the above Kobayashi result depends crucially on: Ahlfors-Schwarz lemma. Let γ = i γjkdtj ∧ dtk be an almost everywhere positive hermitian form on the ball B(0, R) ⊂ Cp, such that −Ricci(γ) := i ∂∂ log det γ ≥ Aγ in the sense of currents, for some constant A > 0 (this means in particular that det γ = det(γjk) is such that log det γ is plurisubharmonic). Then the γ-volume form is controlled by the Poincar´ e volume form : det(γ) ≤ p + 1 AR2 p 1 (1 − |t|2/R2)p+1.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Brody theorem

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Brody reparametrization Lemma. Assume that X is compact, let ω be a hermitian metric on X and f : D → X a holomorphic map. For every ε > 0, there exists a radius R ≥ (1 − ε)f ′(0)ω and a homographic transformation ψ of the disk D(0, R) onto (1 − ε)D such that (f ◦ ψ)′(0)ω = 1 and (f ◦ ψ)′(t)ω ≤ (1 − |t|2/R2)−1 for every t ∈ D(0, R). ⇒ if f ′ unbounded, ∃g = lim f ◦ ψν : C → X with g ′ω ≤ 1.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Brody theorem

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Brody reparametrization Lemma. Assume that X is compact, let ω be a hermitian metric on X and f : D → X a holomorphic map. For every ε > 0, there exists a radius R ≥ (1 − ε)f ′(0)ω and a homographic transformation ψ of the disk D(0, R) onto (1 − ε)D such that (f ◦ ψ)′(0)ω = 1 and (f ◦ ψ)′(t)ω ≤ (1 − |t|2/R2)−1 for every t ∈ D(0, R). ⇒ if f ′ unbounded, ∃g = lim f ◦ ψν : C → X with g ′ω ≤ 1. Brody theorem (1978). If X is compact then X is Kobayashi hyperbolic if and only if there are no entire holomorphic curves f : C → X (Brody hyperbolicity).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Brody theorem

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Brody reparametrization Lemma. Assume that X is compact, let ω be a hermitian metric on X and f : D → X a holomorphic map. For every ε > 0, there exists a radius R ≥ (1 − ε)f ′(0)ω and a homographic transformation ψ of the disk D(0, R) onto (1 − ε)D such that (f ◦ ψ)′(0)ω = 1 and (f ◦ ψ)′(t)ω ≤ (1 − |t|2/R2)−1 for every t ∈ D(0, R). ⇒ if f ′ unbounded, ∃g = lim f ◦ ψν : C → X with g ′ω ≤ 1. Brody theorem (1978). If X is compact then X is Kobayashi hyperbolic if and only if there are no entire holomorphic curves f : C → X (Brody hyperbolicity). Hyperbolic varieties are especially interesting for their expected diophantine properties : Conjecture (S. Lang, 1986) An arithmetic projective variety X is hyperbolic iff X(K) is finite for every number field K.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Varieties of general type

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Definition A non singular projective variety X is said to be of general type if the growth of pluricanonical sections dim H0(X, K ⊗m

X

) ∼ cmn, KX = ΛnT ∗

X

is maximal. (sections locally of the form f (z) (dz1 ∧ . . . ∧ dzn)⊗m) Example: A non singular hypersurface X n ⊂ Pn+1 of degree d satisfies KX = O(d − n − 2), X is of general type iff d > n + 2.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Varieties of general type

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Definition A non singular projective variety X is said to be of general type if the growth of pluricanonical sections dim H0(X, K ⊗m

X

) ∼ cmn, KX = ΛnT ∗

X

is maximal. (sections locally of the form f (z) (dz1 ∧ . . . ∧ dzn)⊗m) Example: A non singular hypersurface X n ⊂ Pn+1 of degree d satisfies KX = O(d − n − 2), X is of general type iff d > n + 2. Conjecture CGT. If a compact variety X is hyperbolic, then it should be of general type, and if X is non singular, then KX = ΛnT ∗

X should be ample, i.e. KX > 0 (Kodaira)

(equivalently ∃ K¨ ahler metric ω such that Ricci(ω) < 0).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Conjectural characterizations of hyperbolicity

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• Theorem. Let X be projective algebraic. Consider the

following properties : (GT) Every subvariety Y of X is of general type. (AH) ∃ε > 0, ∀C ⊂ X algebraic curve 2g(¯ C) − 2 ≥ ε deg(C). (X “algebraically hyperbolic”) (HY) X is hyperbolic (JC) X possesses a jet-metric with negative curvature on its k-jet bundle Xk [to be defined later], for k ≥ k0 ≫ 1. Then (JC) ⇒ (GT), (AH), (HY), (HY) ⇒ (AH), and if Conjecture CGT holds, (HY) ⇒ (GT).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Conjectural characterizations of hyperbolicity

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• Theorem. Let X be projective algebraic. Consider the

following properties : (GT) Every subvariety Y of X is of general type. (AH) ∃ε > 0, ∀C ⊂ X algebraic curve 2g(¯ C) − 2 ≥ ε deg(C). (X “algebraically hyperbolic”) (HY) X is hyperbolic (JC) X possesses a jet-metric with negative curvature on its k-jet bundle Xk [to be defined later], for k ≥ k0 ≫ 1. Then (JC) ⇒ (GT), (AH), (HY), (HY) ⇒ (AH), and if Conjecture CGT holds, (HY) ⇒ (GT). It is expected that all 4 properties are in fact equivalent for projective varieties.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Green-Griffiths-Lang conjecture

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Conjecture (Green-Griffiths-Lang = GGL) Let X be a projective variety of general type. Then there exists an algebraic variety Y X such that for all non-constant holomorphic f : C → X one has f (C) ⊂ Y .

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Green-Griffiths-Lang conjecture

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Conjecture (Green-Griffiths-Lang = GGL) Let X be a projective variety of general type. Then there exists an algebraic variety Y X such that for all non-constant holomorphic f : C → X one has f (C) ⊂ Y . Combining the above conjectures, we get : Expected consequence (of CGT + GGL) Properties: (HY) X is hyperbolic (GT) Every subvariety Y of X is of general type are equivalent if CGT + GGL hold.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Green-Griffiths-Lang conjecture

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Conjecture (Green-Griffiths-Lang = GGL) Let X be a projective variety of general type. Then there exists an algebraic variety Y X such that for all non-constant holomorphic f : C → X one has f (C) ⊂ Y . Combining the above conjectures, we get : Expected consequence (of CGT + GGL) Properties: (HY) X is hyperbolic (GT) Every subvariety Y of X is of general type are equivalent if CGT + GGL hold. Arithmetic counterpart (Lang 1987). If X is a variety

• f general type defined over a number field and Y is the

Green-Griffiths locus (Zariski closure of f (C)), then X(K) Y is finite for every number field K.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Results obtained so far

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Using “jet technology” and deep results of McQuillan for curve foliations on surfaces, D. – El Goul proved Theorem (solution of Kobayashi conjecture, 1998). A very generic surface X⊂P3 of degree ≥ 21 is hyperbolic. Independently McQuillan got degree ≥ 35. Recently improved to degree ≥ 18 (P˘ aun, 2008). For X ⊂ Pn+1, the optimal bound should be degree ≥ 2n + 1 for n ≥ 2 (Zaidenberg). Generic GGL conjecture for dimC X = n (S. Diverio, J. Merker, E. Rousseau, 2009). If X ⊂ Pn+1 is a generic n-fold of degree d ≥ dn := 2n5, [also d3 = 593, d4 = 3203, d5 = 35355, d6 = 172925 ] then ∃Y X s.t. ∀ non const. f : C → X satisfies f (C) ⊂ Y

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Results obtained so far

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Using “jet technology” and deep results of McQuillan for curve foliations on surfaces, D. – El Goul proved Theorem (solution of Kobayashi conjecture, 1998). A very generic surface X⊂P3 of degree ≥ 21 is hyperbolic. Independently McQuillan got degree ≥ 35. Recently improved to degree ≥ 18 (P˘ aun, 2008). For X ⊂ Pn+1, the optimal bound should be degree ≥ 2n + 1 for n ≥ 2 (Zaidenberg). Generic GGL conjecture for dimC X = n (S. Diverio, J. Merker, E. Rousseau, 2009). If X ⊂ Pn+1 is a generic n-fold of degree d ≥ dn := 2n5, [also d3 = 593, d4 = 3203, d5 = 35355, d6 = 172925 ] then ∃Y X s.t. ∀ non const. f : C → X satisfies f (C) ⊂ Y Moreover (S. Diverio, S. Trapani, 2009) codimC Y ≥ 2 ⇒ generic hypersurface X ⊂ P4 of degree ≥ 593 is hyperbolic.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Definition of algebraic differential operators

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The main idea in order to attack GGL is to use differential

• equations. Let

C → X, t → f (t) = (f1(t), . . . , fn(t)) be a curve written in some local holomorphic coordinates (z1, . . . , zn) on X.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Definition of algebraic differential operators

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The main idea in order to attack GGL is to use differential

• equations. Let

C → X, t → f (t) = (f1(t), . . . , fn(t)) be a curve written in some local holomorphic coordinates (z1, . . . , zn) on X. Consider algebraic differential operators which can be written locally in multi-index notation P(f[k]) = P(f ′, f ′′, . . . , f (k)) =

• aα1α2...αk(f (t)) f ′(t)α1f ′′(t)α2 . . . f (k)(t)αk

where aα1α2...αk(z) are holomorphic coefficients on X and t → z = f (t) is a curve, f[k] = (f ′, f ′′, . . . , f (k)) its k-jet.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Definition of algebraic differential operators

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The main idea in order to attack GGL is to use differential

• equations. Let

C → X, t → f (t) = (f1(t), . . . , fn(t)) be a curve written in some local holomorphic coordinates (z1, . . . , zn) on X. Consider algebraic differential operators which can be written locally in multi-index notation P(f[k]) = P(f ′, f ′′, . . . , f (k)) =

• aα1α2...αk(f (t)) f ′(t)α1f ′′(t)α2 . . . f (k)(t)αk

where aα1α2...αk(z) are holomorphic coefficients on X and t → z = f (t) is a curve, f[k] = (f ′, f ′′, . . . , f (k)) its k-jet. Obvious C∗-action : λ · f (t) = f (λt), (λ · f )(k)(t) = λkf (k)(λt) ⇒ weighted degree m = |α1| + 2|α2| + . . . + k|αk|.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Vanishing theorem for differential operators

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• Definition. E GG

k,m is the sheaf (bundle) of algebraic

differential operators of order k and weighted degree m.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Vanishing theorem for differential operators

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• Definition. E GG

k,m is the sheaf (bundle) of algebraic

differential operators of order k and weighted degree m. Fundamental vanishing theorem [Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996] Let P ∈ H0(X, E GG

k,m ⊗ O(−A)) be a global algebraic

differential operator whose coefficients vanish on some ample divisor A. Then ∀f : C → X, P(f[k]) ≡ 0.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Vanishing theorem for differential operators

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• Definition. E GG

k,m is the sheaf (bundle) of algebraic

differential operators of order k and weighted degree m. Fundamental vanishing theorem [Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996] Let P ∈ H0(X, E GG

k,m ⊗ O(−A)) be a global algebraic

differential operator whose coefficients vanish on some ample divisor A. Then ∀f : C → X, P(f[k]) ≡ 0.

• Proof. One can assume that A is very ample and

intersects f (C). Also assume f ′ bounded (this is not so restrictive by Brody !). Then all f (k) are bounded by Cauchy inequality. Hence C ∋ t → P(f ′, f ′′, . . . , f (k))(t) is a bounded holomorphic function on C which vanishes at some point. Apply Liouville’s theorem !

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Geometric interpretation of vanishing theorem

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Let X GG

k

= Jk(X)∗/C∗ be the projectivized k-jet bundle

• f X = quotient of non constant k-jets by C∗-action.

Fibers are weighted projective spaces.

• Observation. If πk : X GG

k

→ X is canonical projection and OX GG

k

(1) is the tautological line bundle, then E GG

k,m = (πk)∗OX GG

k

(m)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Geometric interpretation of vanishing theorem

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Let X GG

k

= Jk(X)∗/C∗ be the projectivized k-jet bundle

• f X = quotient of non constant k-jets by C∗-action.

Fibers are weighted projective spaces.

• Observation. If πk : X GG

k

→ X is canonical projection and OX GG

k

(1) is the tautological line bundle, then E GG

k,m = (πk)∗OX GG

k

(m) Saying that f : C → X satisfies the differential equation P(f[k]) = 0 means that f[k](C) ⊂ ZP where ZP is the zero divisor of the section σP ∈ H0(X GG

k

, OX GG

k

(m) ⊗ π∗

kO(−A))

associated with P.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Consequence of fundamental vanishing theorem 40/74

Consequence of fundamental vanishing theorem. If Pj ∈ H0(X, E GG

k,m ⊗ O(−A)) is a basis of sections then

the image f (C) lies in Y = πk( ZPj), hence property asserted by the GGL conjecture holds true if there are “enough independent differential equations” so that Y = πk(

• j

ZPj) X.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Consequence of fundamental vanishing theorem 41/74

Consequence of fundamental vanishing theorem. If Pj ∈ H0(X, E GG

k,m ⊗ O(−A)) is a basis of sections then

the image f (C) lies in Y = πk( ZPj), hence property asserted by the GGL conjecture holds true if there are “enough independent differential equations” so that Y = πk(

• j

ZPj) X. However, some differential equations are not very useful. On a surface with coordinates (z1, z2), a Wronskian equation f ′

1f ′′ 2 − f ′ 2f ′′ 1 = 0 tells us that f (C) sits on a line,

but f ′′

2 (t) = 0 says that the second component is linear

affine in time, an essentially meaningless information which is lost by a change of parameter t → ϕ(t).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Invariant differential operators

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The k-th order Wronskian operator Wk(f ) = f ′ ∧ f ′′ ∧ . . . ∧ f (k) (locally defined in coordinates) has degree m = k(k+1)

2

and Wk(f ◦ ϕ) = ϕ′mWk(f ) ◦ ϕ.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Invariant differential operators

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The k-th order Wronskian operator Wk(f ) = f ′ ∧ f ′′ ∧ . . . ∧ f (k) (locally defined in coordinates) has degree m = k(k+1)

2

and Wk(f ◦ ϕ) = ϕ′mWk(f ) ◦ ϕ.

• Definition. A differential operator P of order k and

degree m is said to be invariant by reparametrization if P(f ◦ ϕ) = ϕ′mP(f ) ◦ ϕ for any parameter change t → ϕ(t). Consider their set Ek,m ⊂ E GG

k,m

(a subbundle) (Any polynomial Q(W1, W2, . . . Wk) is invariant, but for k ≥ 3 there are other invariant operators.)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Category of directed manifolds

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• Goal. We are interested in curves f : C → X such that

f ′(C) ⊂ V where V is a subbundle (or subsheaf) of TX.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Category of directed manifolds

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• Goal. We are interested in curves f : C → X such that

f ′(C) ⊂ V where V is a subbundle (or subsheaf) of TX.

• Definition. Category of directed manifolds :

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

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Category of directed manifolds

46/74

• Goal. We are interested in curves f : C → X such that

f ′(C) ⊂ V where V is a subbundle (or subsheaf) of TX.

• Definition. Category of directed manifolds :

– Objects : pairs (X, V ), X manifold/C and V ⊂ O(TX) – Arrows ψ : (X, V ) → (Y , W ) holomorphic s.t. ψ∗V ⊂ W – “Absolute case” (X, TX) – “Relative case” (X, TX/S) where X → S – “Integrable case” when [V , V ] ⊂ V (foliations)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Category of directed manifolds

47/74

• Goal. We are interested in curves f : C → X such that

f ′(C) ⊂ V where V is a subbundle (or subsheaf) of TX.

• Definition. Category of directed manifolds :

– Objects : pairs (X, V ), X manifold/C and V ⊂ O(TX) – Arrows ψ : (X, V ) → (Y , W ) holomorphic s.t. ψ∗V ⊂ W – “Absolute case” (X, TX) – “Relative case” (X, TX/S) where X → S – “Integrable case” when [V , V ] ⊂ V (foliations) Fonctor “1-jet” : (X, V ) → ( ˜ X, ˜ V ) where : ˜ X = P(V ) = bundle of projective spaces of lines in V π : ˜ X = P(V ) → X, (x, [v]) → x, v ∈ Vx ˜ V(x,[v]) =

• ξ ∈ T ˜

X,(x,[v]) ; π∗ξ ∈ Cv ⊂ TX,x

• Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012

Hyperbolic algebraic varieties & holomorphic differential equations

Semple jet bundles

48/74

For every entire curve f : (C, TC) → (X, V ) tangent to V f[1](t) := (f (t), [f ′(t)]) ∈ P(Vf (t)) ⊂ ˜ X f[1] : (C, TC) → ( ˜ X, ˜ V ) (projectivized 1st-jet)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Semple jet bundles

49/74

For every entire curve f : (C, TC) → (X, V ) tangent to V f[1](t) := (f (t), [f ′(t)]) ∈ P(Vf (t)) ⊂ ˜ X f[1] : (C, TC) → ( ˜ X, ˜ V ) (projectivized 1st-jet)

• Definition. Semple jet bundles :

– (Xk, Vk) = k-th iteration of fonctor (X, V ) → ( ˜ X, ˜ V ) – f[k] : (C, TC) → (Xk, Vk) is the projectivized k-jet of f .

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Semple jet bundles

50/74

For every entire curve f : (C, TC) → (X, V ) tangent to V f[1](t) := (f (t), [f ′(t)]) ∈ P(Vf (t)) ⊂ ˜ X f[1] : (C, TC) → ( ˜ X, ˜ V ) (projectivized 1st-jet)

• Definition. Semple jet bundles :

– (Xk, Vk) = k-th iteration of fonctor (X, V ) → ( ˜ X, ˜ V ) – f[k] : (C, TC) → (Xk, Vk) is the projectivized k-jet of f . Basic exact sequences 0 → T ˜

X/X → ˜

V

π⋆

→ O˜

X(−1) → 0

⇒ rk ˜ V = r = rk V 0 → O˜

X → π⋆V ⊗ O˜ X(1) → T ˜ X/X → 0 (Euler)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Semple jet bundles

51/74

For every entire curve f : (C, TC) → (X, V ) tangent to V f[1](t) := (f (t), [f ′(t)]) ∈ P(Vf (t)) ⊂ ˜ X f[1] : (C, TC) → ( ˜ X, ˜ V ) (projectivized 1st-jet)

• Definition. Semple jet bundles :

– (Xk, Vk) = k-th iteration of fonctor (X, V ) → ( ˜ X, ˜ V ) – f[k] : (C, TC) → (Xk, Vk) is the projectivized k-jet of f . Basic exact sequences 0 → T ˜

X/X → ˜

V

π⋆

→ O˜

X(−1) → 0

⇒ rk ˜ V = r = rk V 0 → O˜

X → π⋆V ⊗ O˜ X(1) → T ˜ X/X → 0 (Euler)

0 → TXk/Xk−1 → Vk

(πk)⋆

→ OXk(−1) → 0 ⇒ rk Vk = r 0 → OXk → π⋆

kVk−1 ⊗ OXk(1) → TXk/Xk−1 → 0 (Euler)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Direct image formula

52/74

For n = dim X and r = rk V , get a tower of Pr−1-bundles πk,0 : Xk

πk

→ Xk−1 → · · · → X1

π1

→ X0 = X with dim Xk = n + k(r − 1), rk Vk = r, and tautological line bundles OXk(1) on Xk = P(Vk−1).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Direct image formula

53/74

For n = dim X and r = rk V , get a tower of Pr−1-bundles πk,0 : Xk

πk

→ Xk−1 → · · · → X1

π1

→ X0 = X with dim Xk = n + k(r − 1), rk Vk = r, and tautological line bundles OXk(1) on Xk = P(Vk−1).

• Theorem. Xk is a smooth compactification of

X GG,reg

k

/Gk = JGG,reg

k

/Gk where Gk is the group of k-jets of germs of biholomorphisms of (C, 0), acting on the right by reparametrization: (f , ϕ) → f ◦ ϕ, and Jreg

k

is the space

• f k-jets of regular curves.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Direct image formula

54/74

For n = dim X and r = rk V , get a tower of Pr−1-bundles πk,0 : Xk

πk

→ Xk−1 → · · · → X1

π1

→ X0 = X with dim Xk = n + k(r − 1), rk Vk = r, and tautological line bundles OXk(1) on Xk = P(Vk−1).

• Theorem. Xk is a smooth compactification of

X GG,reg

k

/Gk = JGG,reg

k

/Gk where Gk is the group of k-jets of germs of biholomorphisms of (C, 0), acting on the right by reparametrization: (f , ϕ) → f ◦ ϕ, and Jreg

k

is the space

• f k-jets of regular curves.

Direct image formula. (πk,0)∗OXk(m) = Ek,mV ∗ = invariant algebraic differential operators f → P(f[k]) acting on germs of curves f : (C, TC) → (X, V ).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Algebraic structure of differential rings

55/74

Although very interesting, results are currently limited by lack of knowledge on jet bundles and differential operators Theorem (B´ erczi-Kirwan, 2009).The ring of germs of invariant differential operators on (Cn, TCn) at the origin Ak,n =

• m

Ek,mT ∗

Cn

is finitely generated.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Algebraic structure of differential rings

56/74

Although very interesting, results are currently limited by lack of knowledge on jet bundles and differential operators Theorem (B´ erczi-Kirwan, 2009).The ring of germs of invariant differential operators on (Cn, TCn) at the origin Ak,n =

• m

Ek,mT ∗

Cn

is finitely generated. Checked by direct calculations ∀n, k ≤ 2 and n = 2, k ≤ 4 : A1,n = O[f ′

1, . . . , f ′ n]

A2,n = O[f ′

1, . . . , f ′ n, W [ij]],

W [ij] = f ′

i f ′′ j − f ′ j f ′′ i

A3,2 = O[f ′

1, f ′ 2, W1, W2][W ]2,

Wi = f ′

i DW − 3f ′′ i W

A4,2 = O[f ′

1, f ′ 2, W11, W22, S][W ]6,

Wii = f ′

i DWi − 5f ′′ i Wi

where W = f ′

1f ′′ 2 − f ′ 2f ′′ 1 , S = (W1DW2 − W2DW1)/W .

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Generalized GGL conjecture

57/74

Generalized GGL conjecture. If (X, V ) is directed manifold of general type, i.e. det V ∗ big, then ∃Y X such that ∀f : (C, TC) → (X, V ) non const., f (C) ⊂ Y .

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Generalized GGL conjecture

58/74

Generalized GGL conjecture. If (X, V ) is directed manifold of general type, i.e. det V ∗ big, then ∃Y X such that ∀f : (C, TC) → (X, V ) non const., f (C) ⊂ Y .

• Remark. Elementary by Ahlfors-Schwarz if r = rk V = 1.

t → log f ′(t)V ,h is strictly subharmonic if r = 1 and (V ∗, h∗) has > 0 curvature in the sense of currents.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Generalized GGL conjecture

59/74

Generalized GGL conjecture. If (X, V ) is directed manifold of general type, i.e. det V ∗ big, then ∃Y X such that ∀f : (C, TC) → (X, V ) non const., f (C) ⊂ Y .

• Remark. Elementary by Ahlfors-Schwarz if r = rk V = 1.

t → log f ′(t)V ,h is strictly subharmonic if r = 1 and (V ∗, h∗) has > 0 curvature in the sense of currents.

• Strategy. Try some sort of induction on r = rk V .

First try to get differential equations f[k](C) ⊂ Z Xk. Take minimal such k. If k = 0, we are done! Otherwise k ≥ 1 and πk,k−1(Z) = Xk−1, thus V ′ = Vk ∩ TZ has rank < rk Vk = r and should have again det V ′∗ big (unless some unprobable geometry situation occurs ?).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Generalized GGL conjecture

60/74

Generalized GGL conjecture. If (X, V ) is directed manifold of general type, i.e. det V ∗ big, then ∃Y X such that ∀f : (C, TC) → (X, V ) non const., f (C) ⊂ Y .

• Remark. Elementary by Ahlfors-Schwarz if r = rk V = 1.

t → log f ′(t)V ,h is strictly subharmonic if r = 1 and (V ∗, h∗) has > 0 curvature in the sense of currents.

• Strategy. Try some sort of induction on r = rk V .

First try to get differential equations f[k](C) ⊂ Z Xk. Take minimal such k. If k = 0, we are done! Otherwise k ≥ 1 and πk,k−1(Z) = Xk−1, thus V ′ = Vk ∩ TZ has rank < rk Vk = r and should have again det V ′∗ big (unless some unprobable geometry situation occurs ?). Needed induction step. If (X, V ) has det V ∗ big and Z ⊂ Xk irreducible with πk,k−1(Z) = Xk−1, then (Z, V ′), V ′ = Vk ∩ TZ has OZℓ(1) big on (Zℓ, V ′

ℓ), ℓ ≫ 0.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Holomorphic Morse inequalities

61/74

Holomorphic Morse inequalities (D-, 1985) Let L → X be a holomorphic line bundle on a compact complex manifold X, h a smooth hermitian metric on L and ΘL,h = i 2π∇2

L,h = − i

2π∂∂ log h its curvature form. Then ∀q = 0, 1, . . . , n = dimC X

q

• j=0

(−1)q−jhj(X, L⊗k) ≤ kn n!

• X(L,h,≤q)

(−1)qΘn

L,h + o(kn).

where X(L, h, q) = {x ∈ X ; ΘL,h(x) has signature (n − q, q)} (q-index set), and X(L, h, ≤ q) =

• 0≤j≤q

X(L, h, ≤ j)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Holomorphic Morse inequalities (continued)

62/74

As a consequence, one gets an upper bound h0(X, L⊗k) ≤ kn n!

• X(L,h,0)

Θn

L,h + o(kn)

and a lower bound h0(X, L⊗k) ≥ h0(X, L⊗k) − h1(X, L⊗k) ≥ ≥ kn n!

X(L,h,0)

Θn

L,h −

• X(L,h,1)

|Θn

L,h|

• − o(kn)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Holomorphic Morse inequalities (continued)

63/74

As a consequence, one gets an upper bound h0(X, L⊗k) ≤ kn n!

• X(L,h,0)

Θn

L,h + o(kn)

and a lower bound h0(X, L⊗k) ≥ h0(X, L⊗k) − h1(X, L⊗k) ≥ ≥ kn n!

X(L,h,0)

Θn

L,h −

• X(L,h,1)

|Θn

L,h|

• − o(kn)

and similar bounds for the higher cohomology groups Hq: hq(X, L⊗k) ≤ kn n!

• X(L,h,q)

|Θn

L,h|+o(kn)

hq(X, L⊗k) ≥ kn n!

X(L,h,q)

−

• X(L,h,q−1)

−

• X(L,h,q+1)

|Θn

L,h|

• −o(kn)

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Finsler metric on the k-jet bundles

64/74

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

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Finsler metric on the k-jet bundles

65/74

Let JkV be the bundle of k-jets of curves f : (C, TC) → (X, V ) Assuming that V is equipped with a hermitian metric h, one defines a ”weighted Finsler metric” on JkV by taking p = k! and Ψhk(f ) :=

1≤s≤k

εs∇sf (0)2p/s

h(x)

1/p , 1 = ε1 ≫ ε2 ≫ · · · ≫ εk.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Finsler metric on the k-jet bundles

66/74

Let JkV be the bundle of k-jets of curves f : (C, TC) → (X, V ) Assuming that V is equipped with a hermitian metric h, one defines a ”weighted Finsler metric” on JkV by taking p = k! and Ψhk(f ) :=

1≤s≤k

εs∇sf (0)2p/s

h(x)

1/p , 1 = ε1 ≫ ε2 ≫ · · · ≫ εk. Letting ξs = ∇sf (0), this can actually be viewed as a metric hk on Lk := OX GG

k

(1), with curvature form (x, ξ1, . . . , ξk) → ΘLk,hk = ωFS,k(ξ)+ i 2π

• 1≤s≤k

1 s |ξs|2p/s

• t |ξt|2p/t
• i,j,α,β

cijαβ ξsαξsβ |ξs|2 dzi∧dzj where (cijαβ) are the coefficients of the curvature tensor ΘV ∗,h∗ and ωFS,k is the vertical Fubini-Study metric on the fibers of X GG

k

→ X. The expression gets simpler by using polar coordinates xs = |ξs|2p/s

h

, us = ξs/|ξs|h = ∇sf (0)/|∇sf (0)|.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Probabilistic interpretation of the curvature

67/74

In such polar coordinates, one gets the formula ΘLk,hk = ωFS,p,k(ξ)+ i 2π

• 1≤s≤k

1 s xs

• i,j,α,β

cijαβ(z) usαusβ dzi ∧dzj where ωFS,k(ξ) is positive definite in ξ. The other terms are a weighted average of the values of the curvature tensor ΘV ,h on vectors us in the unit sphere bundle SV ⊂ V . The weighted projective space can be viewed as a circle quotient of the pseudosphere |ξs|2p/s = 1, so we can take here xs ≥ 0, xs = 1. This is essentially a sum of the form 1

s γ(us)

where us are random points of the sphere, and so as k → +∞ this can be estimated by a “Monte-Carlo” integral

• 1 + 1

2 + . . . + 1 k

u∈SV

γ(u) du. As γ is quadratic here,

• u∈SV γ(u) du = 1

r Tr(γ).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Main cohomological estimate

68/74

It follows that the leading term in the estimate only involves the trace of ΘV ∗,h∗, i.e. the curvature of (det V ∗, det h∗), which can be taken to be > 0 if det V ∗ is big. Corollary (D-, 2010) Let (X, V ) be a directed manifold, F → X a Q-line bundle, (V , h) and (F, hF) hermitian. Define Lk = OX GG

k

(1) ⊗ π∗

kO

1 kr

• 1 + 1

2 + . . . + 1 k

• F
• ,

η = Θdet V ∗,det h∗ + ΘF,hF. Then for all q ≥ 0 and all m ≫ k ≫ 1 such that m is sufficiently divisible, we have hq(X GG

k

, O(L⊗m

k

)) ≤ mn+kr−1 (n+kr−1)! (log k)n n! (k!)r

X(η,q)

(−1)qηn + C log k

• Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012

Hyperbolic algebraic varieties & holomorphic differential equations

Main cohomological estimate

69/74

It follows that the leading term in the estimate only involves the trace of ΘV ∗,h∗, i.e. the curvature of (det V ∗, det h∗), which can be taken to be > 0 if det V ∗ is big. Corollary (D-, 2010) Let (X, V ) be a directed manifold, F → X a Q-line bundle, (V , h) and (F, hF) hermitian. Define Lk = OX GG

k

(1) ⊗ π∗

kO

1 kr

• 1 + 1

2 + . . . + 1 k

• F
• ,

η = Θdet V ∗,det h∗ + ΘF,hF. Then for all q ≥ 0 and all m ≫ k ≫ 1 such that m is sufficiently divisible, we have hq(X GG

k

, O(L⊗m

k

)) ≤ mn+kr−1 (n+kr−1)! (log k)n n! (k!)r

X(η,q)

(−1)qηn + C log k

• h0(X GG

k

, O(L⊗m

k

)) ≥ mn+kr−1 (n+kr−1)! (log k)n n! (k!)r

X(η,≤1)

ηn − C log k

• .

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Partial solution of the GGL conjecture

70/74

Using the above cohomological estimate, we obtain: Theorem (D-, 2010) Let (X, V ) be of general type, i.e. KV = (det V )∗ is a big line bundle. Then there exists k ≥ 1 and an algebraic hypersurface Z Xk such that every entire curve f : (C, TC) → (X, V ) satisfies f[k](C) ⊂ Z (in other words, f satisfies an algebraic differential equation of order k).

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Partial solution of the GGL conjecture

71/74

Using the above cohomological estimate, we obtain: Theorem (D-, 2010) Let (X, V ) be of general type, i.e. KV = (det V )∗ is a big line bundle. Then there exists k ≥ 1 and an algebraic hypersurface Z Xk such that every entire curve f : (C, TC) → (X, V ) satisfies f[k](C) ⊂ Z (in other words, f satisfies an algebraic differential equation of order k). Another important consequence is: Theorem (D-, 2012) A generic hypersurface X ⊂ Pn+1 of degree d ≥ dn with d2 = 286, d3 = 7316, dn = n4 3

• n log(n log(24n))

n

• (for n ≥ 4) satisfies the Green-Griffiths conjecture.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

A differentiation technique by Yum-Tong Siu

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The proof of the last result uses an important idea due to Yum-Tong Siu, itself based on ideas of Claire Voisin and Herb Clemens, and then refined by M. P˘ aun [Pau08], E. Rousseau [Rou06b] and J. Merker [Mer09]. The idea consists of studying vector fields on the relative jet space of the universal family of hypersurfaces of Pn+1.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

A differentiation technique by Yum-Tong Siu

73/74

The proof of the last result uses an important idea due to Yum-Tong Siu, itself based on ideas of Claire Voisin and Herb Clemens, and then refined by M. P˘ aun [Pau08], E. Rousseau [Rou06b] and J. Merker [Mer09]. The idea consists of studying vector fields on the relative jet space of the universal family of hypersurfaces of Pn+1. Let X ⊂ Pn+1 × PNd be the universal hypersurface, i.e. X = {(z, a) ; a = (aα) s.t. Pa(z) =

• aαzα = 0},

let Ω ⊂ PNd be the open subset of a’s for which Xa = {Pa(z) = 0} is smooth, and let p : X → Pn+1, π : X|Ω → Ω ⊂ PNd be the natural projections.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

Meromorphic vector fields on jet spaces

74/74

Let pk : Xk → X → Pn+1, πk : Xk → Ω ⊂ PNd be the relative Green-Griffiths k-jet space of X → Ω. Then

• J. Merker [Mer09] has shown that global sections ηj of

O(TXk) ⊗ p∗

kOPn+1(k2 + 2k) ⊗ π∗ kOPNd (1)

generate the bundle at all points of X reg

k

for k = n = dim Xa. From this, it follows that if P is a non zero global section over Ω of E GG

k,mT ∗ X ⊗ p∗ kOPn+1(−s) for some s, then for a suitable

collection of η = (η1, . . . , ηm), the m-th derivatives Dη1 . . . DηmP yield sections of H0 X , E GG

k,mT ∗ X ⊗ p∗ kOPn+1(m(k2 + 2k) − s)

• whose joint base locus is contained in X sing

k

, whence the result.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

References [Dem85] Demailly, J.-P.: Champs Magn´ etiques et In´ egalit´ es de Morse pour la d”-cohomologie. Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 189–229. [Dem95] Demailly, J.-P.: Algebraic Criteria for Kobayashi Hyperbolic Projective Varieties and Jet Differentials. Algebraic geometry – Santa Cruz 1995, 285–360, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997. [Dem10] Demailly, J.-P.: Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, November 2010, arXiv: math.AG/1011.3636, dedicated to the memory of Eckart Viehweg; Pure and Applied Mathematics Quarterly 7 (2011) 1165–1208 [Dem12] Demailly, J.-P.: Hyperbolic algebraic varieties and holomorphic differential equations, lecture given at the VIASM Annual Meeting 2012, Hanoi 26 August 2012, 63p, to appear

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

in Acta Vietnamica [D-EG00] Demailly, J.-P., El Goul, J.: Hyperbolicity of Generic Surfaces of High Degree in Projective 3-Space. Amer.

• J. Math. 122 (2000), no. 3, 515–546.

[Div09] Diverio, S.: Existence of global invariant jet differentials on projective hypersurfaces of high degree. Math.

• Ann. 344 (2009) 293-315.

[DMR09] Diverio, S., Merker, J., Rousseau, E.: Effective algebraic degeneracy. e-print arXiv:0811.2346v5. [DT9] Diverio, S., Trapani, T.: A remark on the codimension

• f the Green-Griffiths locus of generic projective hypersurfaces
• f high degree. e-print arXiv:0902.3741v2.

[F-H91] Fulton, W., Harris, J.: Representation Theory: A First Course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991, xvi+551 pp.

Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations

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Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations