Recent progress towards the Kobayashi and Green-Griﬃths-Lang conjectures

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Recent progress towards the Kobayashi and

Green-Griffiths-Lang conjectures

Jean-Pierre Demailly

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

November 28-29, 2015

16th Takagi Lectures, University of Tokyo

J.-P. Demailly (Grenoble), 16th Takagi Lectures, Tokyo On the Kobayashi and Green-Griffiths-Lang conjectures 1/34

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Kobayashi pseudodistance and infinitesimal metric

Let X be a complex space. Given two points p,q ∈X, consider a chain of analytic disks fromp to q, i.e. holomorphic maps

f_j : ∆ :=D(0,1)→X and points a_j,b_j ∈∆, 0≤j ≤k with p =f₀(a₀), q =f_k(b_k), f_j(b_j) = f_j₊₁(a_j+1), 0≤j ≤k−1.

One defines the Kobayashi pseudodistanced_Kob onX to be d_Kob(p,q) = inf

{f_j,a_j,b_j}dPoincar´e(a₁,b₁) +· · ·+dPoincar´e(a_k,b_k). The Kobayashi-Royden infinitesimal pseudometric onX is the Finsler pseudometric

k_x(ξ) = inf

λ >0 ; ∃f : ∆ →X,f(0) =x, λf⁰(0) =ξ , ξ ∈T_X_,x. The integrated pseudometric is precisely d_Kob.

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Kobayashi pseudodistance and infinitesimal metric

{f_j,a_j,b_j}dPoincar´e(a₁,b₁) +· · ·+dPoincar´e(a_k,b_k).

The Kobayashi-Royden infinitesimal pseudometric onX is the Finsler pseudometric

k_x(ξ) = inf

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Kobayashi pseudodistance and infinitesimal metric

{f_j,a_j,b_j}dPoincar´e(a₁,b₁) +· · ·+dPoincar´e(a_k,b_k).

The Kobayashi-Royden infinitesimal pseudometric onX is the Finsler pseudometric

k_x(ξ) = inf

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Kobayashi hyperbolicity and entire curves

Definition

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

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

Theorem (Brody, 1978)

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

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

(iii) The Kobayashi infinitesimal pseudometrick_x is everywhere non degenerate

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

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Kobayashi hyperbolicity and entire curves

Definition

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

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Kobayashi hyperbolicity and entire curves

Definition

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

(i) X is Kobayashi hyperbolic

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Kobayashi hyperbolicity and entire curves

Definition

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

(i) X is Kobayashi hyperbolic

Our interest is the study of hyperbolicity for projective varieties.

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

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Kobayashi-Eisenman measures

In a similar way, one can introduce the p-dimensional

Kobayashi-Eisenman infinitesimal metric on decomposable tensors ξ =ξ₁∧. . .∧ξ_p of Λ^pT_X_,x (i.e. on the tautological line bundle over the Grassmann bundle Gr(T_X,p)) by

e^p_x(ξ) = inf

λ >0 ; ∃f :B^p →X, f(0) =x, λf⁰(0)·τ =ξ , where B^p⊂C^p is the unit ball and τ = _∂t^∂

1 ∧. . .∧_∂t^∂

p.

Definition

A complex space X is said to be p-measure hyperbolic in the sense of Kobayashi-Eisenman if e^p is non degenerate on a dense Zariski open set.

Volume hyperbolicity refers to the case p=n= dimX.

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Kobayashi-Eisenman measures

In a similar way, one can introduce the p-dimensional

Kobayashi-Eisenman infinitesimal metric on decomposable tensors ξ =ξ₁∧. . .∧ξ_p of Λ^pT_X_,x (i.e. on the tautological line bundle over the Grassmann bundle Gr(T_X,p)) by

e^p_x(ξ) = inf

λ >0 ; ∃f :B^p →X, f(0) =x, λf⁰(0)·τ =ξ , where B^p⊂C^p is the unit ball and τ = _∂t^∂

1 ∧. . .∧_∂t^∂

p. Definition

A complex space X is said to be p-measure hyperbolic in the sense of Kobayashi-Eisenman if e^p is non degenerate on a dense Zariski open set.

Volume hyperbolicity refers to the case p =n= dimX.

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Main conjectures

Conjecture of General Type (CGT)

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

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

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

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

Consequence of CGT + GGL

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

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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 !

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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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Solution of the Bloch conjecture

The following has been proved by Ochiai 77, Noguchi 77, 81, 84, Kawamata 80 in the algebraic situation.

Theorem (Ochiai 77, Noguchi 77,81,84, Kawamata 80)

Let Z =Cⁿ/Λ be an abelian variety (resp. a complex torus). Then the (analytic) Zariski closure f(C)^Zar of the image of every entire curve f :C→Z is the translate of a subtorus.

Corollary 1

Let X be a complex analytic subvariety of a complex torus Z . Assume that X is of general type. Then every entire curve drawn in X is analytically degenerate.

Corollary 2

Let X be a complex analytic subvariety of a complex torus Z . Assume that X does not contain any translate of a positive dimensional subtorus. Then X isKobayashi hyperbolic.

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Solution of the Bloch conjecture

Corollary 1

Let X be a complex analytic subvariety of a complex torus Z . Assume that X is ofgeneral type. Then every entire curve drawn in X is analytically degenerate.

Corollary 2

Let X be a complex analytic subvariety of a complex torus Z . Assume that X does not contain any translate of a positive dimensional subtorus. Then X isKobayashi hyperbolic.

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Solution of the Bloch conjecture

Corollary 1

Let X be a complex analytic subvariety of a complex torus Z . Assume that X is ofgeneral type. Then every entire curve drawn in X is analytically degenerate.

Corollary 2

Let X be a complex analytic subvariety of a complex torus Z . Assume that X does not contain any translate of a positive dimensional subtorus. Then X is Kobayashi hyperbolic.

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Results on the Kobayashi conjecture

Kobayashi conjecture (1970)

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

• By a result of M. Zaidenberg (1987), the optimal bound must satisfy d_n≥2n+ 1, and one expects d_n = 2n+ 1.

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

Theorem (D., El Goul, 1998)

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

This was more recently improved to d ≥18(P˘aun, 2008).

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

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Results on the Kobayashi conjecture

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

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Results on the Kobayashi conjecture

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

Independently McQuillan got d ≥35.

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Results on the Kobayashi conjecture

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

Independently McQuillan got d ≥35.

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

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Results on the generic Green-Griffiths conjecture

By a combination of an algebraic existence theorem for jet differentials and of Siu’s technique of “slanted vector fields”

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

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

A generic hypersurface X ⊂Pⁿ⁺¹ of degree d ≥d_n := 2ⁿ⁵ satisfies the GGL conjecture.

The bound was improved by (D-, 2012) to d_n=j

n⁴

3 nlog(nlog(24n))nk

=O(exp(n^1+ε)), ∀ε >0.

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

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

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Results on the generic Green-Griffiths conjecture

n⁴

3 nlog(nlog(24n))nk

=O(exp(n^1+ε)), ∀ε >0.

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Results on the generic Green-Griffiths conjecture

n⁴

3 nlog(nlog(24n))nk

=O(exp(n^1+ε)), ∀ε >0.

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Category of directed varieties

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

Definition. Category of directed varieties :

– Objects : pairs (X,V),X variety/Cand V ⊂T_X

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

– “Relative case” (X,T_X_/S) where X →S – “Integrable case” when [V,V]⊂V (foliations) Fonctor “1-jet” : (X,V)7→( ˜X,V˜) where :

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

V˜_(x,[v]) =

ξ∈TX˜,(x,[v]); π∗ξ ∈Cv ⊂T_X_,x

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Category of directed varieties

– Arrows ψ : (X,V)→(Y,W) holomorphic s.t. ψ_∗V ⊂W

– “Absolute case”(X,T_X), i.e. V =T_X – “Relative case” (X,T_X_/S) where X →S – “Integrable case” when [V,V]⊂V (foliations) Fonctor “1-jet” : (X,V)7→( ˜X,V˜) where :

V˜_(x,[v]) =

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Category of directed varieties

– “Relative case”(X,T_X_/S) where X →S – “Integrable case” when [V,V]⊂V (foliations)

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

V˜_(x,[v]) =

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Category of directed varieties

– “Relative case”(X,T_X_/S) where X →S – “Integrable case” when [V,V]⊂V (foliations) Fonctor “1-jet” : (X,V)7→( ˜X,V˜) where :

V˜_(x,[v]) =

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Semple jet bundles (non singular case)

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

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

0→TX˜/X →V˜ ^π→ O^? X˜(−1)→0 ⇒rk ˜V =r = rkV 0→ OX˜ →π^?V ⊗ OX˜(1)→TX˜/X →0 (Euler)

0→T_X_k_/X_k−1 →Vk (πk)?

→ OX_k(−1)→0 ⇒rkVk =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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Semple jet bundles (non singular case)

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

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

Basic exact sequences

0→TX˜/X →V˜ ^π→ O^? X˜(−1)→0 ⇒rk ˜V =r = rkV 0→ OX˜ →π^?V ⊗ OX˜(1)→TX˜/X →0 (Euler)

0→T_X_k_/X_k−1 →Vk (πk)?

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Semple jet bundles (non singular case)

0→TX˜/X →V˜ ^π→ O^? X˜(−1)→0 ⇒rk ˜V =r = rkV 0→ O_X_˜ →π^?V ⊗ O_X_˜(1)→T_X_˜_/X →0 (Euler)

0→T_X_k_/X_k−1 →Vk (πk)?

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Semple jet bundles (non singular case)

0→TX˜/X →V˜ ^π→ O^? X˜(−1)→0 ⇒rk ˜V =r = rkV 0→ O_X_˜ →π^?V ⊗ O_X_˜(1)→T_X_˜_/X →0 (Euler)

0→T_X_k_/X_k−1 →Vk (πk)?

→ OX_k(−1)→0 ⇒rkVk =r 0→ O_X_k →π_k^?V_k−1⊗ O_X_k(1) →T_X_k_/X_k−1 →0 (Euler)

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k -jets of curves

For n= dimX and r = rkV, one gets a tower ofP^r⁻¹-bundles π_k_,0 :X_k ^π→^k Xk−1 → · · · →X₁ ^π→¹ X₀ =X

with dimX_k =n+k(r −1), rkV_k =r,

and tautological line bundlesO_X_k(1) on X_k =P(Vk−1).

We define the bundle J^kV of k-jets of curves tangent to V by taking J^kV_x to be the set of equivalence classes of germs f : (C,0)→(X,V) such that in some coordinates f(t) = (f₁(t), . . . ,f_n(t)) has a Taylor expansion

f(t) = x+tξ₁+. . .+t^kξ_k+O(t^k+1). Here we take ξ_s = _s!¹∇^sf(0) with respect to some local

holomorphic connection on V (obtained e.g. from a trivialization). Thus ξ_s ∈V_x and

J^kV_x 'V_x^⊕k 'C^kr (non intrinsically).

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k -jets of curves

f(t) = x+tξ₁+. . .+t^kξ_k+O(t^k+1).

Here we take ξ_s = _s!¹∇^sf(0) with respect to some local

holomorphic connection on V (obtained e.g. from a trivialization). Thus ξ_s ∈V_x and

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k -jets of curves

f(t) = x+tξ₁+. . .+t^kξ_k+O(t^k+1).

Here we take ξ_s = _s!¹∇^sf(0) with respect to some local

holomorphic connection on V (obtained e.g. from a trivialization).

Thus ξ_s ∈V_x and

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Semple bundles and reparametrization of curves

Consider the group G^k of k-jets of germs of biholomorphisms ϕ: (C,0)→(C,0), i.e.

ϕ(t) =α₁t +α₂t²+. . .+α_kt^k +O(t^k+1) and the natural G^k action on the right:

J^kV ×Gk →J^kV, (f, ϕ)7→f ◦ϕ.

The action is free on germs J^kV^reg of regular curveswith ξ₁ =f⁰(0) 6= 0.

Theorem

X_k is a smooth compactification ofJ_kV^reg/Gk.

Now we want to deal with possibly singular directed varieties (X,V), i.e. X and V both possibly singular.

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Semple bundles and reparametrization of curves

Theorem

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Semple bundles and reparametrization of curves

Theorem

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Singular directed varieties

Definition

A singular directed variety is a pair (X,V) where X is a reduced complex space, and V ⊂T_X is a closed linear subspace of T_X.

This means that we have an irreducible component Vj lying over each irreducible component X_j of X.

Assume X to be irreducible, dimX =n. Every point x ∈X has a neighborhood U with an embedding U ,→Ω as a closed analytic subset in a smooth open set Ω⊂C^N. Then T_X|U is taken to be the closure of T_U_reg in T_Ω, and V is always assumed to be the closure of Vreg|U (part of V that is a subbundle of T_X_reg|U). If X is non singular and V ⊂T_X is singular, V is a subbundle of T_X over a Zariski open set X⁰ =X rY, and we have at least an absolute Semple tower (X_k^a,V_k^a) associated with (X,TX).

We then define (X_k,V_k) to be theclosure of (X_k⁰,V_k⁰) [associated with (X⁰,V⁰), V⁰ =V_|X⁰] in (X_k^a,V_k^a).

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Singular directed varieties

Definition

A singular directed variety is a pair (X,V) where X is a reduced complex space, and V ⊂T_X is a closed linear subspace of T_X. This means that we have an irreducible component V_j lying over each irreducible component X_j of X.

Assume X to be irreducible, dimX =n. Every point x ∈X has a neighborhood U with an embedding U ,→Ω as a closed analytic subset in a smooth open set Ω⊂C^N. Then T_X|U is taken to be the closure of T_U_reg in T_Ω, and V is always assumed to be the closure of Vreg|U (part of V that is a subbundle of T_X_reg|U). If X is non singular and V ⊂T_X is singular, V is a subbundle of T_X over a Zariski open set X⁰ =X rY, and we have at least an absolute Semple tower (X_k^a,V_k^a) associated with (X,TX).

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Singular directed varieties

Definition

Assume X to be irreducible, dimX =n. Every point x ∈X has a neighborhood U with an embedding U ,→Ω as a closed analytic subset in a smooth open set Ω⊂C^N. Then T_X|U is taken to be the closure of TUreg in TΩ, and V is always assumed to be the closure of Vreg|U (part of V that is a subbundle of T_X_reg|U).

If X is non singular and V ⊂T_X is singular, V is a subbundle of T_X over a Zariski open set X⁰ =X rY, and we have at least an absolute Semple tower (X_k^a,V_k^a) associated with (X,TX).

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Singular directed varieties

Definition

If X is non singular and V ⊂T_X is singular, V is a subbundle of T_X over a Zariski open set X⁰ =X rY, and we have at least an absolute Semple tower (X_k^a,V_k^a) associated with (X,T_X).

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Singular directed varieties

Definition

If X is non singular and V ⊂T_X is singular, V is a subbundle of T_X over a Zariski open set X⁰ =X rY, and we have at least an absolute Semple tower (X_k^a,V_k^a) associated with (X,T_X).

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Base resolution of singularities

Let (X,V) be singular pair. By Hironaka, there exists a

modification µ:Xe →X (in the form of a composition of blow-ups with smooth centers), such that Xe is non singular.

Let µ∗ :T_X_e →µ^∗T_X be the differential dµ. We define Ve =µ⁻¹V ⊂T_X_e to be the closure of (µ∗)⁻¹(V|X⁰), where X⁰ ⊂X_reg is a Zariski open set over which V_|X⁰ is a subbundle of T_X_reg and µ:µ⁻¹(X⁰)→X⁰ is a biregular.

We can then construct a Semple tower (Xek,Vek) by taking the closure over regular points of (X_k⁰,V_k⁰) in the (regular) absolute tower (Xe_kâ,Ve_kâ), where Ve₀â =T_X_e.

Big caution !

In general, for dimX ≥2, one can never make Ve non singular, even by blowing up further !

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Base resolution of singularities

We can then construct a Semple tower (Xek,Vek) by taking the closure over regular points of (X_k⁰,V_k⁰) in the (regular) absolute tower (Xe_kâ,Ve_kâ), where Ve₀â =T_X_e.

Big caution !

In general, for dimX ≥2, one can never make Ve non singular, even by blowing up further !

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Base resolution of singularities

We can then construct a Semple tower (Xek,Vek) by taking the closure over regular points of (X_k⁰,V_k⁰) in the (regular) absolute tower (Xe_kâ,Ve_kâ), where Ve₀â=T_X_e.

Big caution !

In general, for dimX ≥2, one can never makeVe non singular, even by blowing up further !

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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-jetat any point t = 0. We first look at the C^∗-action induced by dilationsϕ(t) = η_λ(t) =λt.

Putting ξs = ∆^sf(0), the C^∗ action is obtained by computing the derivatives of f(λt), hence it is given on J^kV_x 'V_x^⊕k by

(∗) λ·(ξ₁, ξ₂, . . . , ξ_k) = (λξ₁, λ²ξ₂, . . . , λ^kξ_k).

We consider the Green-Griffiths bundle E_k,m^GGV^∗ of polynomials of weighted degree m onJ^kV_x written locally in coordinate charts as

P(x; ξ₁, . . . , ξ_k) = P

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

Take P to be a holomorphic section inx. It can then be viewed as an algebraic differential operatorP(f_[k]) = P(f ;f⁰,f⁰⁰, . . . ,f^(k)),

P(f_[k])(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

Let t 7→z =f(t) be a germ of curve, f_[k]= (f⁰,f⁰⁰, . . . ,f^(k⁾) its k-jetat any point t = 0. We first look at the C^∗-action induced by dilationsϕ(t) = η_λ(t) =λt.

Putting ξs = ∆^sf(0), the C^∗ action is obtained by computing the derivatives of f(λt), hence it is given on J^kV_x 'V_x^⊕k by

(∗) λ·(ξ₁, ξ₂, . . . , ξ_k) = (λξ₁, λ²ξ₂, . . . , λ^kξ_k).

We consider the Green-Griffiths bundle E_k,m^GGV^∗ of polynomials of weighted degree m onJ^kV_x written locally in coordinate charts as

P(x; ξ₁, . . . , ξ_k) = P

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

Take P to be a holomorphic section inx. It can then be viewed as an algebraic differential operatorP(f_[k]) = P(f ;f⁰,f⁰⁰, . . . ,f^(k)),

P(f_[k])(t) = P

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