Weak analytic hyperbolicity of generic hypersurfaces of high degree in <span class="mathjax-formula">$\mathbb{P}^{4}$</span>

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

ERWANROUSSEAU

Weak analytic hyperbolicity of generic hypersurfaces of high degree inP⁴

Tome XVI, n^o2 (2007), p. 369-383.

<http://afst.cedram.org/item?id=AFST_2007_6_16_2_369_0>

© Université Paul Sabatier, Toulouse, 2007, tous droits réservés.

L’accès aux articles de la revue « Annales de la faculté des sci- ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://afst.cedram.

org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

cedram

Article mis en ligne dans le cadre du

Centre de diffusion des revues académiques de mathématiques

pp. 369–383

Weak analytic hyperbolicity of generic hypersurfaces of high degree in

Erwan Rousseau⁽¹⁾

ABSTRACT.—In this article we prove that every entire curve in a generic hypersurface of degree d ^{593 in} P⁴_C is algebraically degenerated i.e there exists a proper subvariety which contains the entire curve.

R^´ESUM´E.—Dans cet article nous d´emontrons que toute courbe enti`ere dans une hypersurface g´en´erique de degr´e d ^{593 dans} P⁴_C ^{est alg´e-}

briquement d´eg´en´er´ee i.e il existe une sous-vari´et´e propre qui contient la courbe enti`ere.

1. Introduction

In 1970, S. Kobayashi conjectured in [8] that a generic hypersurfaceX in Pⁿ_C is hyperbolic provided that d = deg(X) 2n−1, for n 3. For n= 3, it was obtained by Demailly and El Goul in [4] thatd21 implies the hyperbolicity of very generic hypersurfacesX inP³_C.

In this paper, we would like to prove the following:

Theorem 1.1. — LetX ⊂P⁴_C be a generic hypersurface such that d= deg(X)593.Then every entire curvef :C→X is algebraically degener- ated, i.e there exists a proper subvarietyY ⊂X such thatf(C)⊂Y.

This result is a weaker version of the conjecture in dimension 3, because to obtain the full conjecture one needs to prove that the entire curve is constant.

The proof of the theorem is based on two techniques. Consider X ⊂ P⁴×P^Nd the universalhypersurface of degreedin P⁴.

(∗) Re¸cu le 21 juin 2005, accept´e le 3 novembre 2005

(1) D´epartement de Math´ematiques, IRMA, Universit´e Louis Pasteur, 7, rue Ren´e Descartes, 67084 Strasbourg, France.

rousseau@math.u-strasbg.fr

In the first section we construct meromorphic vector fields on the space J₃^v(X) of vertical3-jets ofX.This technique, initiated by Clemens [2], Ein [5], Voisin [14], was generalized by Y.-T. Siu [13] and detailed by M. Paun in dimension 2 [10]. Here we generalize M. Paun’s computations in dimension 3 and obtain that a pole order equal to 12 is enough to obtain a “large”

space of global sections of the twisted tangent bundle.

In the second section, we summarize the main facts about the bundle of jet differentials of orderkand degreem, Ek,mT_X^∗.The idea, in hyperbolicity questions, is that global sections of this bundle vanishing on ample divisors provide algebraic differential equations for any entire curve f : C → X.

Therefore, the main point is to produce enough algebraically independent global holomorphic jet differentials. In the case of surfaces in P³ of degree d15,one can produce global jet differentials of order 2 vanishing on ample divisors using a Riemann-Roch computation and a Bogomolov vanishing theorem (see [3]). Y.-T. Siu, in [13], described a way to produce global jet differentials vanishing on ample divisors for hypersurfaces of sufficiently large degreedinPⁿ,for anyn.One problem is that the bound obtained for dis quite high. If we are interested in the degreedfor smooth hypersurfaces of P⁴, an interesting result obtained in [12], is the existence of global jet differentials of order 3 vanishing on ample divisors ford97.

In the last section we complete the proof of the theorem using Siu’s approach [13], by taking the derivative of the jet differentialin the direction of the vector fields constructed in the first part, and an observation made by Mihai Paun in [10] to avoid the use of McQuillan results (cf. [9]).

Acknowledgements. We would like to thank Mihai Paun for his lec- tures about Siu’s ideas given at the summer schoolPragmatic in Catania, 2004.

2. Vector fields

In this first section, we generalize to dimension 3 the approach of Mihai Paun [10] which gives some precisions to Siu’s ideas [13] in dimension 2.

ConsiderX ⊂P⁴×P^Nd the universalhypersurface given by the equation

|α|=d

a_αZ^α= 0,where[a]∈P^Ndand[Z]∈P⁴. We use the notations: for α = (α₀, ..., α₄) ∈ N⁵, |α| =

iα_i and if Z = (Z₀, Z₁, ..., Z₄) are homogeneous coordinates onP⁴,thenZ^α=

Z_j^αj. X is a smooth hypersurface of degree (d,1) in P⁴ ×P^Nd. We denote by

J₃(X) the manifold of the 3-jets inX,andJ₃^v(X) the submanifold ofJ₃(X) consisting of 3-jets inX tangent to the fibers of the projectionπ:X →P^Nd. Let us consider the set Ω0 := (Z0 = 0)×(a0d000= 0)⊂P⁴×P^Nd.We assume that global coordinates are given onC⁴ and C^Nd. The equation of X becomes

X0:= (z₁^d+

|α|d,α1<d

aαz^α= 0)

Then the equations ofJ₃^v(X0) inC⁴×C^Nd×C⁴×C⁴×C⁴can be written:

|α|d,ad000=1

aαz^α= 0 (2.1)

4

j=1

|α|d,ad000=1

a_α∂z^α

∂z_jξ_j⁽¹⁾= 0 (2.2)

4

j=1

|α|d,ad000=1

a_α∂z^α

∂z_jξ⁽²⁾_j +

4

j,k=1

|α|d,ad000=1

a_α ∂²z^α

∂z_j∂z_kξ_j⁽¹⁾ξ_k⁽¹⁾= 0 (2.3)

4

j=1

|α|d,ad000=1

aα

∂z^α

∂zj

ξ_j⁽³⁾+

4

j,k=1

3

|α|d,ad000=1

aα

∂²z^α

∂zj∂zk

ξ_j⁽²⁾ξ_k⁽¹⁾

+

4

j,k,l=1

|α|d,ad000=1

a_α ∂³z^α

∂zj∂zk∂zl

ξ⁽¹⁾_j ξ⁽¹⁾_k ξ_l⁽¹⁾= 0 (2.4)

Consider now a vector field

V =

|α|d,α1<d

v_α ∂

∂aα

+

j

v_j ∂

∂zj

+

j,k

w_j,k ∂

∂ξ_j^(k)

on the vector spaceC⁴×C^Nd×C⁴×C⁴×C⁴.The conditions to be satisfied byV to be tangent toJ₃^v(X0) are

|α|d,α1<d

vαz^α+

4

j=1

|α|d,ad000=1

aα

∂z^α

∂z_jvj= 0

4

j=1

|α|d,α1<d

vα

∂z^α

∂z_jξ_j⁽¹⁾+

4

j,k=1

|α|d,ad000=1

aα

∂²z^α

∂z_j∂z_kvjξ_k⁽¹⁾

+

4

j=1

|α|d,ad000=1

a_α∂z^α

∂zj

w⁽¹⁾_j = 0

|α|d,α1<d

(

4

j=1

∂z^α

∂z_jξ_j⁽²⁾+

4

j,k=1

∂²z^α

∂z_j∂z_kξ⁽¹⁾_j ξ_k⁽¹⁾)v_α +

4

j=1

|α|d,ad000=1

aα(

4

k=1

∂²z^α

∂z_j∂z_kξ_k⁽²⁾+

4

k,l=1

∂³z^α

∂z_j∂z_k∂z_lξ_k⁽¹⁾ξ⁽¹⁾_l )vj

+

|α|d,ad000=1

(

4

j,k=1

a_α ∂²z^α

∂zj∂zk

(w_j⁽¹⁾ξ_k⁽¹⁾+w⁽¹⁾_k ξ_j⁽¹⁾) +

4

j=1

a_α∂z^α

∂zj

w_j⁽²⁾) = 0

|α|d,α1<d

(

4

j=1

∂z^α

∂z_jξ⁽³⁾_j + 3

4

j,k=1

∂²z^α

∂z_j∂z_kξ⁽²⁾_j ξ_k⁽¹⁾

+

4

j,k,l=1

∂³z^α

∂zj∂zk∂zl

ξ⁽¹⁾_j ξ_k⁽¹⁾ξ_l⁽¹⁾)vα

+

4

j=1

|α|d,ad000=1

aα(

4

k=1

∂²z^α

∂z_j∂z_kξ_k⁽³⁾+ 3

4

k,l=1

∂³z^α

∂z_j∂z_k∂z_lξ_k⁽²⁾ξ⁽¹⁾_l

+

4

k,l,m=1

∂⁴z^α

∂zj∂zk∂zl∂zm

ξ⁽¹⁾_k ξ_l⁽¹⁾ξ_m⁽¹⁾)v_j

+

|α|d,ad000=1

(

4

j,k,l=1

aα

∂³z^α

∂z_j∂z_k∂z_l

(w⁽¹⁾_j ξ⁽¹⁾_k ξ_l⁽¹⁾+ξ_j⁽¹⁾w⁽¹⁾_k ξ_l⁽¹⁾+ξ_j⁽¹⁾ξ⁽¹⁾_k w⁽¹⁾_l ) +3

4

j,k=1

a_α ∂²z^α

∂z_j∂z_k(w_j⁽²⁾ξ_k⁽¹⁾+ξ_j⁽²⁾w_k⁽¹⁾) +

4

j=1

a_α∂z^α

∂z_jw_j⁽³⁾) = 0

Now we can introduce the first package of vector fields tangent toJ₃^v(X0).

We denote byδ_j ∈N⁴the multi-index whose j-component is equal to 1 and the other are zero.

Forα14 :

V_α⁴⁰⁰⁰:= ∂

∂aα −4z₁ ∂

∂aα−δ1

+ 6z₁² ∂

∂aα−2δ1

−4z₁³ ∂

∂aα−3δ1

+z₁⁴ ∂

∂aα−4δ1

.

Forα₁3, α₂1 : V_α³¹⁰⁰ : = ∂

∂aα −3z1

∂

∂aα−δ1

−z2

∂

∂aα−δ2

+ 3z1z2

∂

∂aα−δ1−δ2

+3z₁² ∂

∂a_α−2δ1 −3z₁²z₂ ∂

∂a_{α−2δ1−δ2} −z₁³ ∂

∂a_α−3δ1 +z₁³z₂ ∂

∂a_{α−3δ1−δ2}.

Forα₁2, α₂2 : V_α²²⁰⁰ : = ∂

∂aα−z2

∂

∂aα−δ2

−z1

∂

∂aα−δ1

+z1z₂² ∂

∂aα−δ1−2δ2

+z₁²z₂ ∂

∂a_{α−2δ1−δ2} −z₁²z₂² ∂

∂a_{α−2δ1−2δ2}.

Forα₁2, α₂1, α₃1 : V_α²¹¹⁰ : = ∂

∂aα−z3

∂

∂aα−δ3

−z2

∂

∂aα−δ2

−2z1

∂

∂aα−δ1

+z2z3

∂

∂aα−δ2−δ3

+2z₁z₃ ∂

∂a_{α−δ1−δ3} + 2z₁z₂ ∂

∂a_{α−δ1−δ2} +z₁² ∂

∂a_α−2δ1

−2z1z2z3

∂

∂aα−δ1−δ2−δ3

−z₁²z3

∂

∂aα−2δ1−δ3

−z₁²z2

∂

∂aα−2δ1−δ2

+z₁²z₂z₃ ∂

∂a_{α−2δ1−δ2−δ3}.

Forα₁1, α₂1, α₃1, α₄1 : V_α¹¹¹¹ : = ∂

∂aα −z1

∂

∂aα−δ1

−z2

∂

∂aα−δ2

−z3

∂

∂aα−δ3

−z₄ ∂

∂a_α−δ4 +z₂z₃z₄ ∂

∂a_{α−δ2−δ3−δ4} +z₁z₃z₄ ∂

∂a_{α−δ1−δ3−δ4}

+z1z2z3

∂

∂aα−δ1−δ2−δ3

+z1z2z4

∂

∂aα−δ1−δ2−δ4

−z1z2z3z4

∂

∂a_{α−δ1−δ2−δ3−δ4}.

Similar vector fields are constructed by permuting the z-variables, and changing the index α as indicated by the permutation. The pole order of the previous vector fields is equal to 4.

Lemma 2.1. — For any (v_i)_1i4∈C⁴, there existv_α(a),with degree at most 1 in the variables (a_γ), such that V :=

αv_α(a)_∂a^∂

α +

jv_j∂z^∂

j is tangent to J₃^v(X0)at each point.

Proof. — We impose the additionalconditions of vanishing for the coeffi- cients of ξ⁽¹⁾_j in the second equation (respectively of ξ_j⁽¹⁾ξ_k⁽¹⁾ in the third equation andξ_j⁽¹⁾ξ⁽¹⁾_k ξ_l⁽¹⁾ in the fourth equation) for any 1jkl4.

Then the coefficients ofξ_j⁽²⁾(respectivelyξ_j⁽²⁾ξ_k⁽¹⁾andξ_j⁽³⁾) are automatically zero in the third (respectively fourth) equation. The resulting 35 equations

are

|α|d,α1<d

vαz^α+

4

j=1

|α|d,ad000=1

aα

∂z^α

∂z_jvj= 0

|α|d,α1<d

vα

∂z^α

∂zj

+

4

k=1

|α|d,ad000=1

aα

∂²z^α

∂zj∂zk

vk= 0

α

∂²z^α

∂z_j∂z_kv_α+

4

l=1

|α|d,ad000=1

a_α ∂³z^α

∂z_j∂z_k∂z_lv_l= 0

α

∂³z^α

∂z_j∂z_k∂z_lv_α+

4

m=1

|α|d,ad000=1

a_α ∂⁴z^α

∂z_j∂z_k∂z_l∂z_mv_m= 0

Now we can observe that if thevα(a) satisfy the first equation, they au- tomatically satisfy the other ones because thevαare constants with respect to z. Therefore it is sufficient to find (v_α) satisfying the first equation. We identify the coefficients ofz^ρ =z₁^ρ1 z^ρ₂² z₃^ρ3 z₄^ρ4 :

v_ρ+

4

j=1

a_ρ+δjv_j(ρ_j+ 1) = 0

Another family of vector fields can be obtained thanks to the generaliza- tion to dimension 3 of a lemma (cf. [10]) given by Mihai Paun in dimension 2. Consider a 4×4-matrixA= (A^k_j)∈ M4(C) and let V :=

j,k

w^(k)_j ∂

∂ξ^(k)_j , wherew^(k):=Aξ^(k),fork= 1,2,3.

Lemma 2.2. — There exist polynomialsv_α(z, a) :=

|β|3v^α_β(a)z^βwhere each coefficient v_β^α has degree at most 1 in the variables(a_γ)such that

V :=

α

vα(z, a) ∂

∂a_α +V is tangent to J₃^v(X0)at each point.

Proof. — We impose the additionalconditions of vanishing for the coeffi- cients of ξ⁽¹⁾_j in the second equation (respectively of ξ_j⁽¹⁾ξ_k⁽¹⁾ in the third equation andξ⁽¹⁾_j ξ_k⁽¹⁾ξ_l⁽¹⁾ in the fourth equation) for any 1jkl4.

Then the coefficients ofξ_j⁽²⁾(respectivelyξ_j⁽²⁾ξ_k⁽¹⁾andξ_j⁽³⁾) are automatically zero in the third (respectively fourth) equation. The resulting 35 equations

are

|α|d,α1<d

v_αz^α= 0 (5)

|α|d,α1<d

vα

∂z^α

∂z_j +

4

k=1

|α|d,ad000=1

aα

∂z^α

∂z_kA^j_k= 0 (6j)

α

∂²z^α

∂z_j∂z_kv_α+

α,p

a_α ∂²z^α

∂z_j∂z_pA^k_p+

α,p

a_α ∂²z^α

∂z_k∂z_pA^j_p= 0 (7_jk)

α

∂³z^α

∂z_j∂z_k∂z_lvα+

α,p

aα

∂³z^α

∂z_p∂z_k∂z_lA^j_p+

α,p

aα

∂³z^α

∂z_j∂z_p∂z_lA^k_p

+

α,p

aα

∂³z^α

∂zj∂zk∂zp

A^l_p= 0 (8jkl)

The equations for the unknownsv^α_β are obtained by identifying the coef- ficients of the monomialsz^ρin the above equations. We can do the following reductions: using equations 6j v_β^α = 0 if |α|+|β| d+ 1, and using the equation of X0, we can assume that degree in the z₁ variable is at most d−1.

The monomialsz^ρin (5) arez₁^ρ1z₂^ρ2z₃^ρ3z^ρ₄⁴with

ρ_idandρ₁d−1.

If all the components ofρare greater than 3, then we obtain the following system

9. The coefficient ofz^ρ in (5) impose the condition

α+β=ρ

v_β^α= 0

10_j.The coefficient of the monomialz^ρ−δj in (6_j) impose the condition

α+β=ρ

αjv^α_β =lj(a)

wherelj is a linear expression in thea-variables.

11jj. For j = 1, ...,4 the coefficient of the monomial z^ρ−2δj in (7jj) impose the condition

α+β=ρ

αj(αj−1)v_β^α=ljj(a)

11jk. For 1 j < k 4 the coefficient of the monomial z^{ρ−δj−δk} in (7jk) impose the condition

α+β=ρ

α_j(α_j−1)v^α_β =l_jk(a)

12jjj. For j = 1, ...,4 the coefficient of the monomial z^ρ−3δj in (8jjj) impose the condition

α+β=ρ

α_j(α_j−1)(α_j−2)v_β^α=l_jjj(a)

12jjk. For 1 j < k 4 the coefficient of the monomial z^{ρ−2δj−δk} in (8jjk) impose the condition

α+β=ρ

α_j(α_j−1)α_kv^α_β =l_jjk(a)

12_jkl.For 1j < k < l4 the coefficient of the monomialz^{ρ−δj−δk−δl} in (8_jjk) impose the condition

α+β=ρ

αjαkαlv^α_β =ljkl(a)

As in the case of dimension 2 (cf. [10]) we obtain that the determinant of the matrix associated to the system is not zero. Indeed, for eachρthe matrix whose column C_β consists of the partialderivatives of order at most 3 of the monomialz^ρ−βhas the same determinant, at the pointz₀= (1,1,1,1), as our system. Therefore if the determinant is zero, we would have a non- identically zero polynomial

Q(z) =

β

aβz^ρ−β

such that allits partialderivatives of order less or equalto 3 vanish atz₀. Thus the same is true for

P(z) =z^ρQ(1 z1

, ..., 1 z4

) =

β

aβz^β.

But this impliesP ≡0.

Finally, we conclude by Cramer’s rule. The systems we have to solve are never over determined as well. The lemma is proved.

Proposition 2.3. — Let Σ₀ :={(z, a, ξ⁽¹⁾, ξ⁽²⁾, ξ⁽³⁾)∈ J₃^v(X) / ξ⁽¹⁾∧ ξ⁽²⁾∧ξ⁽³⁾= 0}. Then the vector spaceT_J^v

3(X)⊗ O_P⁴(12)⊗ O_PNd(∗) is gen- erated by its global sections on J₃^v(X)\Σ, whereΣis the closure of Σ₀. Proof. — From the preceding lemmas, we are reduced to consider V =

|α|3

v_α ∂

∂aα

.The conditions forV to be tangent toJ₃^v(X0) are

|α|3

v_αz^α= 0

4

j=1

|α|d,α1<d

vα

∂z^α

∂zj

ξ_j⁽¹⁾= 0

|α|3

(

4

j=1

∂z^α

∂zj

ξ⁽²⁾_j +

zj∂zk

ξ_j⁽¹⁾ξ_k⁽¹⁾)vα= 0

|α|3

(

4

j=1

∂z^α

∂zj

ξ_j⁽³⁾+ 3

zj∂zk

ξ⁽²⁾_j ξ_k⁽¹⁾+

4

j,k,l=1

∂³z^α

∂zj∂zk∂zl

ξ_j⁽¹⁾ξ⁽¹⁾_k ξ_l⁽¹⁾)v_α= 0

We denote byW_jkl the wronskian operator corresponding to the vari- ables zj, zk, zl.We can supposeW123:= det(ξ_j⁽ⁱ⁾)1i,j3= 0.Then we can solve the previous system withv0000, v1000, v0100, v0010as unknowns. By the Cramer rule, each of the previous quantity is a linear combination of the v_α, |α| 3, α = (0000), (1000), (0100), (0010) with coefficients rational functions in z, ξ⁽¹⁾, ξ⁽²⁾, ξ⁽³⁾. The denominator isW₁₂₃ and the numerator is a polynomial whose monomials verify either:

i) degree inz at most 3 and degree in eachξ⁽ⁱ⁾ at most 1.

ii) degree inzat most 2 and degree in ξ⁽¹⁾ at most 3, degree inξ⁽²⁾ at most 0, degree inξ⁽³⁾ at most 1.

iii) degree inz at most 2 and degree inξ⁽¹⁾ at most 2, degree inξ⁽²⁾ at most 2, degree inξ⁽³⁾ at most 0.

iv) degree inzat most 1 and degree inξ⁽¹⁾ at most 4, degree inξ⁽²⁾ at most 1, degree inξ⁽³⁾ at most 0.

ξ⁽¹⁾ has a pole of order 2,ξ⁽²⁾ has a pole of order 3 andξ⁽³⁾ has a pole of order 4, therefore the previous vector field has order at most 12.

Remark 2.4. — If the third derivative of f : (C,0) → X lies inside Σ0

then the image off is contained in a hyperplane section ofX.

3. Jet differentials

In this section we recall the basic facts about jet differentials following J.-P. Demailly [3].

Let X be a complex manifold. We start with the directed manifold (X, TX). We defineX1:=P(T_X), andV1⊂TX1:

V_1,(x,[v]):={ξ∈T_X1,(x,[v]);π_∗ξ∈Cv}

whereπ:X1→X is the naturalprojection. Iff : (C,0)→(X, x) is a germ of holomorphic curve then it can be lifted toX1as f[1].

By induction, we obtain a tower of varieties (X_k, V_k). π_k :X_k → X is the natural projection. We have a tautological line bundle OXk(1) and we denoteu_k:=c₁(OXk(1)).

Let’s consider the direct imageπk∗(OXk(m)). It’s a vector bundle over X which can be described with local coordinates. Let z = (z1, ..., zn) be

local coordinates centered in x ∈X. A local section of π_k∗(OXk(m)) is a polynomial

P =

|α1|+2|α2|+...+k|αk|=m

R_α(z)dz^α1...d^kz^αk

which acts naturally on the fibers of the bundle J_kX → X of k-jets of germs of curves inX, i.e the set of equivalence classes of holomorphic maps f : (C,0)→(X, x) with the equivalence relation which identifies two such maps if their derivatives agree up to orderk, and which is invariant under reparametrization i.e

P((f ◦φ), ...,(f◦φ)^(k))t=φ(t)^mP(f, ..., f^(k))_φ(t)

for every φ ∈ G, the group of k-jets of biholomorphisms of (C,0). The vector bundle π_k∗(OXk(m)) is denoted E_k,mT_X^∗. This bundle of invariant jet differentials is a subbundle of the bundle of jet differentials, of order k and degree m,E_k,m^GGT_X^∗ →X whose fibres are complex-valued polynomials Q(f, f, ..., f^(k)) on the fibers ofJkX,of weightmunder the action ofC^∗:

Q(λf, λ²f, ..., λ^kf^(k)) =λ^mQ(f, f, ..., f^(k)) for anyλ∈C^∗ and (f, f, ..., f^(k))∈J_kX.

It turns out that we have an embeddingJ_k^regX/Gk ,→Xk,whereJ_k^regX denotes the space of non-constants jets.

Fork= 1, E_1,mT_X^∗ =S^mT_X^∗.

IfX is a surface we have the following description of E_2,mT_X^∗. Let W be the wronskian,W =dz₁d²z₂−dz₂d²z₁,then every invariant differential operator of order 2 and degree mcan be written

P =

|α|+3k=m

R_α,k(z)dz^αW^k.

The following theorem makes clear the use of jet differentials in the study of hyperbolicity:

Theorem 3.1 ([7], [3]). — Assume that there exist integers k, m > 0 and an ample line bundle Lon X such that

H⁰(Xk,OXk(m)⊗π_k^∗L⁻¹)H⁰(X, Ek,mT_X^∗ ⊗L⁻¹)

has non zero sectionsσ1, ..., σN. Let Z⊂Xk be the base locus of these sec- tions. Then every entire curve f : C → X is such that f_[k](C) ⊂ Z. In

other words, for every global Gk−invariant polynomial differential opera- tor Pwith values in L⁻¹, every entire curve f : C → X must satisfy the algebraic differential equationP(f) = 0.

Remark 3.2. — In fact, this theorem is true for global sections ofE_k,m^GGT_X^∗ vanishing on an ample divisor.

A complex compact manifold is hyperbolic if there is no non constant entire curve f : C → X. Thus, the problem reduces to produce enough independant algebraic differential equations.

IfX⊂P⁴is a smooth hypersurface,we have established the next result:

Theorem 3.3 [12]. — Let X be a smooth hypersurface of P⁴ such that d = deg(X) 97, and A an ample line bundle, then E_3,mT_X^∗ ⊗A⁻¹ has global sections for m large enough and every entire curve f :C→X must satisfy the corresponding algebraic differential equation.

The proof relies on the filtration ofE3,mT_X^∗ [11]:

Gr^•E_3,mT_X^∗ = ⊕

0γ^m5

( ⊕

{λ1+2λ2+3λ3=m−γ;λi−λjγ,i<j}Γ^{(λ1,λ2,λ3)}T_X^∗) where Γ is the Schur functor. This filtration provides a Riemann-Roch com- putation of the Euler characteristic [11]:

χ(X, E_3,mT_X^∗) = m⁹

81648×10⁶d(389d³−20739d²+185559d−358873)+O(m⁸).

In dimension 3 there is no Bogomolov vanishing theorem (cf. [1]) as it is used in dimension 2 to controlthe cohomology groupH², therefore we need the following proposition:

Proposition 3.4 [12] . — Letλ= (λ1, λ2, λ3)be a partition such that λ1> λ2> λ3 and|λ|=

λi>4(d−5) + 18. Then : h²(X,Γ^λT_X^∗)g(λ)d(d+ 13) +q(λ) where g(λ) = ^3|λ₂^|3

λi>λj

(λ_i−λ_j)and q is a polynomial inλ with homoge- neous components of degrees at most 5.

This proposition provides the estimate

h²(X, Gr^•E3,mT_X^∗)Cd(d+ 13)m⁹+O(m⁸) where C is a constant.

4. Proof of the theorem

Let us consider an entire curvef :C→X in a generic hypersurface of P⁴.By Riemann-Roch and the proposition of the previous section we obtain the following lemma:

Lemma 4.1. — Let X be a smooth hypersurface of P⁴ of degree d,0 <

δ < ₁₈¹ thenh⁰(X, E3,mT_X^∗ ⊗K_X^−δm)α(d, δ)m⁹+O(m⁸), with α(d, δ) = − 1

408240000000d(−1945d³−784080δ²d³+ 105030d³δ+ 1058400δ³d³+ 103695d²+ 7075491d+ 105837083 +322256880δ²d−1260083250δ

−435002400δ³d−6819271200δ³+ 5051827440δ²

−2255850d²δ−15876000δ³d²−81814050δd+ 11761200δ²d²).

Proof. —E_3,mT_X^∗ ⊗K_X^−δm admits a filtration with graded pieces Γ^{(λ1,λ2,λ3)}T_X^∗ ⊗K_X^−δm= Γ^{(λ1−δm,λ2−δm,λ3−δm)}T_X^∗

forλ₁+ 2λ₂+ 3λ₃=m−γ;λ_i−λ_jγ, i < j,0γ^m₅.We compute by Riemann-Roch

χ(X, E_3,mT_X^∗ ⊗K_X^−δm) =χ(X, Gr^•E_3,mT_X^∗ ⊗K_X^−δm).

We use the proposition of the previous section to control h²(X, E_3,mT_X^∗ ⊗K_X^−δm) :

h²(X,Γ^{(λ1−δm,λ2−δm,λ3−δm)}T_X^∗) g(λ₁−δm, λ₂−δm, λ₃−δm)d(d+ 13) +q(λ1−δm, λ2−δm, λ3−δm) under the hypothesis

λ_i−3δm >4(d−5) + 18. The conditions verified byλimply

λ_i^m₆ therefore the hypothesis will be verified if m(1

6 −3δ)>4(d−5) + 18.

We conclude with the computation

χ(X, E_3,mT_X^∗⊗K_X^−δm)−h²(X, Gr^•E_3,mT_X^∗⊗K_X^−δm)h⁰(X, E_3,mT_X^∗⊗K_X^−δm).