Holomorphic Morse inequalities, old and new

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Holomorphic Morse inequalities, old and new

Jean-Pierre Demailly

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

35th Annual Geometry Festival Stony Brook University

April 23 – 25, 2021

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Introduction and goals

Let X be acompact complex manifold, and L→X a holomorphic line bundle. Assume L equipped with a Hermitian metrich, written locally as h=e^−ϕ in a trivialization.

The curvature form of (L,h) is θ = Θ_L,h=− i

2π∂∂logh = i 2π∂∂ϕ. Important problems in algebraic or analytic geometry

Find upper and lower bounds for the dimensions of

cohomology groups h^q(X,L^⊗m⊗ F) whereF is a coherent sheaf, asymptotically as m→+∞, e.g. in terms of θ= Θ_L,h. (Harder question ?) In case q = 0 andF is invertible (say), try to analyze the base locus of H⁰(X,L^⊗m⊗ F), i.e. the set of common zeroes of all holomorphic sections.

Holomorphic Morse inequalities (D-, 1985) provide workable answers in terms of the q-index sets of the curvature form.

J.-P. Demailly (Grenoble), 35th Geom. Festival, April 24, 2021 Holomorphic Morse inequalities, old and new 2/24

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Introduction and goals

Let X be acompact complex manifold, and L→X a holomorphic line bundle. Assume L equipped with a Hermitian metrich, written locally as h=e^−ϕ in a trivialization. The curvature form of (L,h) is

θ = Θ_L,h=− i

2π∂∂logh= i 2π∂∂ϕ.

Important problems in algebraic or analytic geometry Find upper and lower bounds for the dimensions of

cohomology groups h^q(X,L^⊗m⊗ F) whereF is a coherent sheaf, asymptotically as m→+∞, e.g. in terms of θ= Θ_L,h. (Harder question ?) In case q = 0 andF is invertible (say), try to analyze the base locus of H⁰(X,L^⊗m⊗ F), i.e. the set of common zeroes of all holomorphic sections.

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Introduction and goals

θ = Θ_L,h=− i

cohomology groups h^q(X,L^⊗m⊗ F) whereF is a coherent sheaf, asymptotically as m →+∞, e.g. in terms of θ= Θ_L,h.

(Harder question ?) In case q = 0 andF is invertible (say), try to analyze the base locus of H⁰(X,L^⊗m⊗ F), i.e. the set of common zeroes of all holomorphic sections.

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Introduction and goals

θ = Θ_L,h=− i

cohomology groups h^q(X,L^⊗m⊗ F) whereF is a coherent sheaf, asymptotically as m →+∞, e.g. in terms of θ= Θ_L,h. (Harder question ?) In case q = 0 andF is invertible (say), try to analyze the base locus ofH⁰(X,L^⊗m⊗ F), i.e. the set of common zeroes of all holomorphic sections.

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Introduction and goals

θ = Θ_L,h=− i

cohomology groups h^q(X,L^⊗m⊗ F) whereF is a coherent sheaf, asymptotically as m →+∞, e.g. in terms of θ= Θ_L,h. (Harder question ?) In case q = 0 andF is invertible (say), try to analyze the base locus ofH⁰(X,L^⊗m⊗ F), i.e. the set of common zeroes of all holomorphic sections.

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Holomorphic Morse inequalities: main statement

The q-index setof a real (1,1)-form θ is defined to be X(θ,q) = {x ∈X|θ(x) has signature (n−q,q)}

(exactly q negative eigenvalues and n−q positive ones)

Set also X(θ,≤q) = [

0≤j≤q

X(θ,j).

X(θ,q) and X(θ,≤q) are open sets. sign(θⁿ) = (−1)^q onX(θ,q).

X

0 1 2

q q

⁺

1 n

=

0

θⁿ Theorem (D-, 1985)

Let θ = ΘL,h and r =rankF. Then, as m→+∞

q

X

j=0

(−1)^q−jh^j(X,L^⊗m⊗ F)≤r mⁿ n!

Z

X(θ,≤q)

(−1)^qθⁿ+o(mⁿ).

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Holomorphic Morse inequalities: main statement

(exactly q negative eigenvalues and n−q positive ones) Set also X(θ,≤q) = [

0≤j≤q

X(θ,j).

X(θ,q) andX(θ,≤q) are open sets.

sign(θⁿ) = (−1)^q on X(θ,q).

X

0 1 2

q q

⁺

1 n

=

0

θⁿ

Theorem (D-, 1985)

Let θ = ΘL,h and r =rankF. Then, as m→+∞

q

X

j=0

(−1)^q−jh^j(X,L^⊗m⊗ F)≤r mⁿ n!

Z

X(θ,≤q)

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Holomorphic Morse inequalities: main statement

(exactly q negative eigenvalues and n−q positive ones) Set also X(θ,≤q) = [

0≤j≤q

X(θ,j).

X(θ,q) andX(θ,≤q) are open sets.

sign(θⁿ) = (−1)^q on X(θ,q).

X

0 1 2

q q

⁺

1 n

=

0

θⁿ Theorem (D-, 1985)

Let θ= ΘL,h and r =rankF. Then, as m→+∞

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Strategy of proof and consequences

The proof proceeds by considering the ∂-complex and looking at the spectral theory of =∂∂^∗+∂^∗∂ acting on sections of L^⊗m⊗ F.

The curvature form of L^⊗m is mθ =imP

j,kθjkdzj ∧d zk =iP

j,kθjkdζj ∧dζ_k in rescaled coordinates ζ_j =√

m z_j. The “wavelength” of eigenfunctions is ∼1/√

m and the estimates localize at this scale. Various formulations of holomorphic Morse inequalities

h^q(X,L^⊗m⊗ F)≤r mⁿ n!

Z

X(θ,q)

h^q(X,L^⊗m⊗ F)≥r mⁿ n!

Z

S

q−1≤j≤q+1X(θ,j)

(−1)^qθⁿ−o(mⁿ).

For q = 0, h⁰(X,L^⊗m⊗ F)≥r mⁿ n!

Z

X(θ,≤1)

θⁿ−o(mⁿ).

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Strategy of proof and consequences

The proof proceeds by considering the ∂-complex and looking at the spectral theory of =∂∂^∗+∂^∗∂ acting on sections of L^⊗m⊗ F. The curvature form of L^⊗m is

mθ =imP

m z_j.

The “wavelength” of eigenfunctions is ∼1/√

m and the estimates localize at this scale. Various formulations of holomorphic Morse inequalities

h^q(X,L^⊗m⊗ F)≤r mⁿ n!

Z

X(θ,q)

Z

S

Z

X(θ,≤1)

θⁿ−o(mⁿ).

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Strategy of proof and consequences

mθ =imP

m and the estimates localize at this scale.

Various formulations of holomorphic Morse inequalities h^q(X,L^⊗m⊗ F)≤r mⁿ

n! Z

X(θ,q)

Z

S

Z

X(θ,≤1)

θⁿ−o(mⁿ).

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Strategy of proof and consequences

mθ =imP

n!

Z

X(θ,q)

Z

S

Z

X(θ,≤1)

θⁿ−o(mⁿ).

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Strategy of proof and consequences

mθ =imP

n!

Z

X(θ,q)

Z

S

Z

X(θ,≤1)

θⁿ−o(mⁿ).

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Strategy of proof and consequences

mθ =imP

n!

Z

X(θ,q)

Z

S X(θ,j)

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Singular version of holomorphic Morse inequalities

We assume here that L is equipped with a possibly singular metric h =e^−ϕ were ϕis quasi-psh with analytic singularities, i.e. locally ϕ(z) = clogX

j

|g_j(z)|²+u(z), g_j holomorphic, u ∈C^∞, c >0.

Then L² estimates involve multiplier ideal sheaves I(mϕ)⊂ O_X I(mϕ)_x =

f ∈ O_X_,x; ∃U 3x s.t. Z

U

|f|²e^−mϕdV <+∞ .

Theorem (L. Bonavero 1996 – proof based on blowing up) The same estimates as above are still valid, when one considers instead the twisted cohomology groups

H^q(X,L^⊗m⊗ I(mϕ)⊗ F)

and Morse integrals in the complement of Σ =ϕ⁻¹(−∞) = singular set of θ= Θ_L,h : Z

X(θ,q)rΣ

(−1)^qθⁿ.

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Singular version of holomorphic Morse inequalities

j

f ∈ O_X_,x; ∃U 3x s.t.

Z

U

|f|²e^−mϕdV <+∞ .

X(θ,q)rΣ

(−1)^qθⁿ.

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Singular version of holomorphic Morse inequalities

j

f ∈ O_X_,x; ∃U 3x s.t.

Z

U

|f|²e^−mϕdV <+∞ .

X(θ,q)rΣ

(−1)^qθⁿ.

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Algebraic versions of Morse inequalities

Assume here that X is projective algebraic / C, and that L =O_X(A−B) whereA and B are ample (or nef)Q-divisors (such that A−B is integral).

Observation (D-, 1996)

In the above situation, the holomorphic Morse inequalities hold after replacing the q-index Morse integral by the intersection number ⁿ_q

A^n−q·B^q, and in particular (S. Trapani, 1995) h⁰(X,L^⊗m⊗ F)≥r mⁿ

n!(Aⁿ−nAⁿ⁻¹·B)−o(mⁿ). Proof. For (1,1)-forms α, β ≥0, elementary symmetric functions arguments yield

1l_X(α−β,≤q)(−1)^q(α−β)ⁿ≤

q

X

j=0

(−1)^q−j n

j

α^n−j ∧β^j.

Algebraic proof by F. Angelini (1996), via exact sequence arguments.

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Algebraic versions of Morse inequalities

n!(Aⁿ−nAⁿ⁻¹·B)−o(mⁿ).

Proof. For (1,1)-forms α, β ≥0, elementary symmetric functions arguments yield

1l_X(α−β,≤q)(−1)^q(α−β)ⁿ≤

q

X

j=0

(−1)^q−j n

j

α^n−j ∧β^j.

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Algebraic versions of Morse inequalities

Proof. For (1,1)-forms α, β ≥0, elementary symmetric functions

arguments yield q

X

n

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Algebraic versions of Morse inequalities

Proof. For (1,1)-forms α, β ≥0, elementary symmetric functions arguments yield

1l_X(α−β,≤q)(−1)^q(α−β)ⁿ≤

q

X

j=0

(−1)^q−j n

j

α^n−j ∧β^j.

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Algebraic Morse inequalities of Benoˆıt Cadorel

Definition of adapted stratifications (projective case)

An “adapted stratification” forL overX is a collection of non singular projective schemes S = (Sj), dimSj =j, Sn=X, together with proper birational morphisms ψ_j of S_j onto the support |D_j|=ψ_j(S_j) of a divisor D_j of S_j+1, such that, when putting Φj =ψn−1◦ · · · ◦ψj :Sj →X, the pull-back Φ^∗_jL satisfies Φ^∗_jL ' O_S_j(D_j−1) =O_S_j(D_j−1⁺ −D_j−1⁻ ).

The “truncated powers of the Chern class” c₁(L,S)^k_[q] are codim k cycles supported on Sn−k (= 0 ifq ∈/ [0,k]), defined inductively by c₁(L,S)⁰_[0]= [X], c₁(L,S)⁰_[q]= 0 for q 6= 0, and c₁(L,S)^k_[q]=ψ_n−k^∗ c₁(L,S)^k−1_[q] ·D_n−k⁺ −c₁(L,S)^k−1_[q−1]·D_n−k⁻

. Theorem (Cadorel, December 2019)

X

0≤j≤q

(−1)^q−jh^j(X,L^⊗m⊗ F)≤(−1)^qr mⁿ

n! degc₁(L,S)ⁿ_[≤q]+O(mⁿ⁻¹).

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Algebraic Morse inequalities of Benoˆıt Cadorel

The “truncated powers of the Chern class” c₁(L,S)^k_[q] are codim k cycles supported on Sn−k (= 0 ifq ∈/ [0,k]), defined inductively by c₁(L,S)⁰_[0] = [X],c₁(L,S)⁰_[q]= 0 for q 6= 0, and

c₁(L,S)^k_[q]=ψ_n−k^∗ c₁(L,S)^k−1_[q] ·D_n−k⁺ −c₁(L,S)^k−1_[q−1]·D_n−k⁻ . Theorem (Cadorel, December 2019)

X

0≤j≤q

(−1)^q−jh^j(X,L^⊗m⊗ F)≤(−1)^qr mⁿ

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Algebraic Morse inequalities of Benoˆıt Cadorel

The “truncated powers of the Chern class” c₁(L,S)^k_[q] are codim k cycles supported on Sn−k (= 0 ifq ∈/ [0,k]), defined inductively by c₁(L,S)⁰_[0] = [X],c₁(L,S)⁰_[q]= 0 for q 6= 0, and c₁(L,S)^k_[q]=ψ_n−k^∗ c₁(L,S)^k−1_[q] ·D_n−k⁺ −c₁(L,S)^k−1_[q−1]·D_n−k⁻

.

Theorem (Cadorel, December 2019) X

0≤j≤q

(−1)^q−jh^j(X,L^⊗m⊗ F)≤(−1)^qr mⁿ

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Algebraic Morse inequalities of Benoˆıt Cadorel

The “truncated powers of the Chern class” c₁(L,S)^k_[q] are codim k cycles supported on Sn−k (= 0 ifq ∈/ [0,k]), defined inductively by c₁(L,S)⁰_[0] = [X],c₁(L,S)⁰_[q]= 0 for q 6= 0, and c₁(L,S)^k_[q]=ψ_n−k^∗ c₁(L,S)^k−1_[q] ·D_n−k⁺ −c₁(L,S)^k−1_[q−1]·D_n−k⁻

. Theorem (Cadorel, December 2019)

X

0≤j≤q

(−1)^q−jh^j(X,L^⊗m⊗ F)≤(−1)^qr mⁿ

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Considerations and questions about base loci

Let (L,h) be a hermitian line bundle over X. If we assume that θ = Θ_L,h satisfies R

X(θ,≤1)θⁿ >0, then we know thatL is big, i.e.

that h⁰(X,L^⊗m)≥c mⁿ, for m≥m₀ and c >0,

but this does not tell us anything about the base locus Bs(L) =T

σ∈H⁰(X,L^⊗m)σ⁻¹(0). Definition

The “iterated base locus” IBs(L) is obtained by picking inductively Z₀ =X and Z_k = zero divisor of a sectionσ_k of L^⊗m^k over the normalization of Zk−1, and taking T

k,m1,...,mk,σ1,...,σkZk. Unsolved problem

Find a condition, e.g. in the form of Morse integrals (or analogs) for θ = ΘL,h, ensuring for instance thatcodim IBs(L)>p. We would need for instance to be able to check the positivity of Morse integrals R

Z(θ|Z,≤1)θ^n−p for Z irreducible,codimZ =p.

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Considerations and questions about base loci

that h⁰(X,L^⊗m)≥c mⁿ, for m≥m₀ and c >0, but this does not tell us anything about the base locus Bs(L) =T

σ∈H⁰(X,L^⊗m)σ⁻¹(0).

Definition

The “iterated base locus” IBs(L) is obtained by picking inductively Z₀ =X and Z_k = zero divisor of a sectionσ_k of L^⊗m^k over the normalization of Zk−1, and taking T

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Considerations and questions about base loci

σ∈H⁰(X,L^⊗m)σ⁻¹(0).

Definition

The “iterated base locus” IBs(L)is obtained by picking inductively Z₀ =X and Z_k = zero divisor of a sectionσ_k of L^⊗m^k over the normalization of Zk−1, and taking T

k,m1,...,mk,σ1,...,σkZk.

Unsolved problem

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Considerations and questions about base loci

σ∈H⁰(X,L^⊗m)σ⁻¹(0).

Definition

Find a condition, e.g. in the form of Morse integrals (or analogs) for θ= ΘL,h, ensuring for instance thatcodim IBs(L)>p.

We would need for instance to be able to check the positivity of Morse integrals R

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Considerations and questions about base loci

σ∈H⁰(X,L^⊗m)σ⁻¹(0).

Definition

Find a condition, e.g. in the form of Morse integrals (or analogs) for θ= ΘL,h, ensuring for instance thatcodim IBs(L)>p.

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Transcendental holomorphic Morse inequalities

Morse inequalities were initially found as a strengthening of Siu’s solution of the Grauert-Riemenschneider conjecture characterizing Moishezon manifolds among compact complex manifolds.

In this general setting, we raised 25−30 years ago the following Conjecture

Let X be a compact complex manifold andα∈H_BC^1,1(X,R) a Bott-Chern class, represented by closed real (1,1)-forms modulo

∂∂ exact forms. Assumeα pseudoeffective, and set Vol(α) = sup

T=α+i∂∂ϕ≥0

Z

X

T_acⁿ , T ≥0 current, n= dimX. Then

Vol(α)≥ sup

θ∈{α}, θ∈C^∞

Z

X(θ,≤1)

θⁿ where

X(θ,q) =q-index set of θ=

x∈X;θ(x) has signature (n−q,q) .

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Transcendental holomorphic Morse inequalities

T=α+i∂∂ϕ≥0

Z

X

T_acⁿ , T ≥0 current, n= dimX.

Then

Vol(α)≥ sup

θ∈{α}, θ∈C^∞

Z

X(θ,≤1)

θⁿ where

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Transcendental holomorphic Morse inequalities

T=α+i∂∂ϕ≥0

Z

X

T_acⁿ , T ≥0 current, n= dimX. Then

Vol(α)≥ sup

θ∈{α}, θ∈C^∞

Z

X(θ,≤1)

θⁿ where

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Conjecture on volumes of (1,1)-classes

Conjectural corollary (transcendental volume estimate)

Let X be compact K¨ahler, dimX =n, and α, β ∈H^1,1(X,R) be nef classes. Then Vol(α−β)≥αⁿ−nαⁿ⁻¹·β.

By BDPP 2004, this conjecture yields a characterization of thedual of the pseudoeffective cone on arbitrary compact K¨ahler manifolds. Observation (BDPP, 2004)

The volume estimate holds if X hasdeformation approximations by projective manifolds X_ν of maximal Picard numberρ(X_ν) = h^1,1. Theorem 1 (Xiao 2015, Popovici 2016)

If αⁿ−nαⁿ⁻¹·β >0, then α−β is a big class, i.e.Vol(α−β)>0. Theorem 2 (Witt-Nystr¨om & Boucksom 2019)

The transcendental volume estimate holds if X is projective.

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Conjecture on volumes of (1,1)-classes

By BDPP 2004, this conjecture yields a characterization of thedual of the pseudoeffective cone on arbitrary compact K¨ahler manifolds.

Observation (BDPP, 2004)

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Conjecture on volumes of (1,1)-classes

The volume estimate holds if X hasdeformation approximations by projective manifolds X_ν of maximal Picard numberρ(X_ν) = h^1,1.

Theorem 1 (Xiao 2015, Popovici 2016)

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Conjecture on volumes of (1,1)-classes

If αⁿ−nαⁿ⁻¹·β >0, then α−β is a big class, i.e.Vol(α−β)>0.

Theorem 2 (Witt-Nystr¨om & Boucksom 2019)

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Conjecture on volumes of (1,1)-classes

If αⁿ−nαⁿ⁻¹·β >0, then α−β is a big class, i.e.Vol(α−β)>0.

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Entire curves in projective varieties and hyperbolicity

The goal is to study (nonconstant) entire curves f :C→X drawn in a projective variety/C. The varietyX is said to be Brody (⇔ Kobayashi) hyperbolic if there are no such curves.

More generally, if ∆ = P

∆_j is a reduced normal crossing divisor in X, we want to study entire curves f :C→Xr∆ drawn in the complement of ∆.

If there are none, the log pair (X,∆) is said Brody hyperbolic. The strategy is to show that under suitable conditions, such entire curves must satisfy algebraic differential equations.

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Entire curves in projective varieties and hyperbolicity

If there are none, the log pair (X,∆) is said Brody hyperbolic. The strategy is to show that under suitable conditions, such entire curves must satisfy algebraic differential equations.

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Entire curves in projective varieties and hyperbolicity

If there are none, the log pair (X,∆) is said Brody hyperbolic.

The strategy is to show that under suitable conditions, such entire curves must satisfy algebraic differential equations.

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Entire curves in projective varieties and hyperbolicity

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k -jets of curves and k-jet bundles

Let X be a nonsingularn-dimensional projective variety overC.

Definition of k-jets

For k ∈N^∗, a k-jet of curve f_[k] : (C,0)_k →X is an equivalence class of germs of holomorphic curves f : (C,0)→X, written f = (f₁, . . . ,f_n) in local coordinates (z₁, . . . ,z_n) on an open subset U ⊂X, where two germs are declared to be equivalent if they have the same Taylor expansion of order k at 0 :

f(t) = x+tξ₁ +t²ξ₂+· · ·+t^kξ_k +O(t^k⁺¹), t ∈D(0, ε)⊂C, and x =f(0) ∈U, ξ_s ∈Cⁿ, 1≤s ≤k.

Notation

Let J^kX be the bundle of k-jets of curves, and π_k :J^kX →X the natural projection, where the fiber (J^kX)_x =π_k⁻¹(x) consists of k-jets of curves f_[k] such that f(0) =x.

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k -jets of curves and k-jet bundles

Let X be a nonsingularn-dimensional projective variety overC. Definition of k-jets

f(t) = x+tξ₁+t²ξ₂+· · ·+t^kξ_k +O(t^k⁺¹), t ∈D(0, ε)⊂C, and x =f(0) ∈U, ξ_s ∈Cⁿ, 1≤s ≤k.

Notation

Let J^kX be the bundle of k-jets of curves, and π_k :J^kX →X the natural projection, where the fiber (J^kX)_x =π_k⁻¹(x) consists of k-jets of curves f_[k] such that f(0) =x.

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k -jets of curves and k-jet bundles

Let X be a nonsingularn-dimensional projective variety overC. Definition of k-jets

f(t) = x+tξ₁+t²ξ₂+· · ·+t^kξ_k +O(t^k⁺¹), t ∈D(0, ε)⊂C, and x =f(0) ∈U, ξ_s ∈Cⁿ, 1≤s ≤k.

Notation

Let J^kX be the bundle of k-jets of curves, and πk :J^kX →X the natural projection, where the fiber (J^kX)_x =π_k⁻¹(x) consists of k-jets of curves f_[k] such that f(0) =x.

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Algebraic differential operators

Let t 7→z =f(t) be a germ of curve, f_[k]= (f⁰,f⁰⁰, . . . ,f^(k⁾) its k-jet at any pointt = 0. Look at the C^∗-action induced by dilations λ·f(t) :=f(λt), λ∈C^∗, for f_[k] ∈J^kX.

Taking a (local) connection∇onT_Xand putting ξ_s=f^(s)(0) =∇^sf(0), we get a trivialization J^kX '(T_X)^⊕k and the C^∗ action is given by (∗) λ·(ξ₁, ξ₂, . . . , ξ_k) = (λξ₁, λ²ξ₂, . . . , λ^kξ_k).

We consider the Green-Griffiths sheaf Ek,m(X) of homogeneous polynomials of weighted degree m on J^kX defined by

P(x; ξ₁, . . . , ξ_k) =P

a_α₁_α₂_...α_k(x)ξ₁^α¹. . . ξ_k^α^k, Pk

s=1s|α_s|=m. Here, we assume the coefficients aα1α2...α_k(x) to be holomorphic in x, and view P as a differential operator P(f) =P(f ; f⁰,f⁰⁰, . . . ,f^(k)),

P(f)(t) =P

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

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Algebraic differential operators

We consider the Green-Griffiths sheaf Ek,m(X) of homogeneous polynomials of weighted degree m on J^kX defined by

P(x; ξ₁, . . . , ξ_k) =P

a_α₁_α₂_...α_k(x)ξ₁^α¹. . . ξ_k^α^k, Pk

s=1s|α_s|=m. Here, we assume the coefficients aα1α2...α_k(x) to be holomorphic in x, and view P as a differential operator P(f) =P(f ; f⁰,f⁰⁰, . . . ,f^(k)),

P(f)(t) =P

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Algebraic differential operators

We consider the Green-Griffiths sheaf E_k,m(X) of homogeneous polynomials of weighted degree m on J^kX defined by

P(x; ξ₁, . . . , ξ_k) =P

a_α₁_α₂_...α_k(x)ξ₁^α¹. . . ξ_k^α^k, Pk

s=1s|α_s|=m.

Here, we assume the coefficients aα1α2...α_k(x) to be holomorphic in x, and view P as a differential operator P(f) =P(f ; f⁰,f⁰⁰, . . . ,f^(k)),

P(f)(t) =P

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Algebraic differential operators

We consider the Green-Griffiths sheaf E_k,m(X) of homogeneous polynomials of weighted degree m on J^kX defined by

P(x; ξ₁, . . . , ξ_k) =P

a_α₁_α₂_...α_k(x)ξ₁^α¹. . . ξ_k^α^k, Pk

s=1s|α_s|=m.

Here, we assume the coefficients aα1α2...α_k(x) to be holomorphic in x, and view P as a differential operator P(f) =P(f ; f⁰,f⁰⁰, . . . ,f^(k)),

P(f)(t) =P