Jet differentials on toroidal compactifications of ball quotients

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Jet differentials on toroidal compactifications of ball quotients

Benoit Cadorel

To cite this version:

Benoit Cadorel. Jet differentials on toroidal compactifications of ball quotients. Annales de l’Institut

Fourier, Association des Annales de l’Institut Fourier, In press. �hal-01567113v2�

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Jet differentials on toroidal compactifications of ball quotients

Benoˆıt Cadorel

Abstract

We give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.

1 Introduction

When dealing with complex hyperbolicity problems, finding global jet differentials on manifolds is an important question, since they permit to give restrictions on the geometry of the entire curves. Let us recall a few basic facts concerning Green-Griffiths jet differentials, which can be found in [Dem12]. Let X be a complex projective manifold. Then, for anyk, there exists a (singular) varietyX_k^GGÐ→^π^k X, and an orbifold line bundleOX_k^GG(1) on it, such that for anym, E_k,m^GGΩ_X = (π_k)∗O^k(m)is a vector bundle whose sections are holomorphic differential equations of orderk, and degreem, for some suitable notion of weighted degree.

In [Dem11], Demailly proves that if V is a complex projective manifold of general type, then for any k large enough, the Green-Griffiths jet differentials of order k will have maximal growth, or equivalently, OX_k^GG(1)is big. Finding an effectivekfor which this property holds is an interesting question, whose answer depends on the context: in [Dem11], Demailly uses his metric techniques to give an effective lower bound on k in the case of hypersurfaces ofPⁿ.

We propose here a method to obtain a similar effective result in the case oftoroidal compactifications of ball quotients (see [Mok12] for the main properties of these manifolds). Specifically, we will find a combinatorial lower bound on the volume ofE_k,^GG_●ΩX,valid for anyk:

Theorem 1. LetX be a toroidal compactification of a ball quotient by a lattice with only unipotent parabolic isometries. Then, for any k∈N, we have the following lower bound on the volume of thek-th order Green- Griffiths jet differentials:

vol(E_k,^GG_●Ω_X) ≥ 1 (k!)ⁿ

⎡⎢⎢⎢

⎢⎣

⎛

⎝

(K_X+D)ⁿ

(n+1)ⁿ ∑

{u₁≤...≤u_n}⊂S_k,n

1 u₁...u_n

⎞

⎠ +(−D)ⁿ ∑

1≤i1≤...≤in≤k

1 i1...in

⎤⎥⎥⎥

⎦, (1)

whereSk,nis the ordered set

Sk,n= {11<...<1n+1<21<...<2n+1<...<k1<...<kn+1}. Here, the fractions _u ¹

1...u_n are to be computed by forgetting the indexes on the integers in the setSk,n. In [Cad16], the particular case wherek=1 (i.e. the case ofsymmetric differentials) was already proved, under the additional assumption that Ω_X is nef. Our formula removes this hypothesis, and extends the result to any order k.

Using the results of [BT18], it is not hard to derive explicit orderskforE_k,^GG_●Ω_Xto have maximal growth:

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Corollary 1.1. Let X=Bⁿ/Γ be a toroidal compactification of a ball quotient. Letk∈N. Then, under any of the following hypotheses:

1. n∈ [∣4,5∣]et k>e⁻^γe⁻⁽⁻^D⁾ⁿ⁽⁽ⁿ⁻²⁾^n!⁺¹⁾ ; 2. n≥6 etk>e⁻^γe

π2 6(n−2)n!+1

n+1 2π−1 , the line bundle O_X^GG

k (1)is big. For the first values of n≥4, this yields the lower bounds for logkdisplayed in Table 1.

n 4 5 6 7 8

logk −γ+5(−D⁴) −γ+361(D⁵) 41 534 151 711 920 325 Table 1: Effective lower bounds on logkto haveO_X^GG

k (1)big

The starting point to prove Theorem1 consists in giving a more algebraic interpretation of the central metric construction of [Dem11]. Let us give the main ideas about this construction. For any complex manifold X, the Green-Griffiths jet differential spacesX_k^GGcan be deformed into a weighted projective bundle, using the standard construction of the Rees algebra. More specifically, there exists a family Xk^GGÐ→X×Csuch that for any λ∈C^∗, the specialization (Xk^GG)^λ Ð→X× {λ} is isomorphic to X_k^GG, and the specialization (Xk^GG)⁰Ð→X× {0}is isomorphic to theweighted projective bundle P(T_X⁽¹⁾⊕...⊕T_X⁽^k⁾). This last bundle is defined to be the quotient ofTX⊕...⊕TXÐ→X by the C^∗-actionλ⋅ (v1, ..., vk) = (λv1, ..., λ^kvk). Moreover, there is a natural orbifold line bundleO_X_k^GG(1)on the familyXk^GG whose restriction to the fibers(Xk^GG)_λ gives the tautological bundles ofX_k^GG and P(T_X⁽¹⁾⊕...⊕T_X⁽^k⁾). The metric used in [Dem11] can actually be seen as a singular metric onOX_k^GG(1); it is constructed in such a way that its specialization to the zero fiber P(T_X⁽¹⁾⊕...⊕T_X⁽^k⁾)is induced by some metric onTX.

One convenient feature about this family Xk^GG is the fact that it permits to interpret the intersection products on the jet spacesX_k^GGin terms of the intersection theory on P(T_X⁽¹⁾⊕...T_X⁽^k⁾). When dealing with Chow groups computations, these last spaces share many properties with the usual weightless projective spaces. In the first part of our work, we will recall some results about the intersection theory with rational coefficients for a weighted projective spaces P(E₁⁽^a¹⁾⊕...⊕Ep⁽^a^p⁾), which were proved first by Al-Amrani [AA97]. Since we work with rational coefficients instead of integer ones, the study is somewhat simplified; for the reader’s convenience, we will explain how we could prove these results by following [Ful98] in a standard way.

The other reason why studying the familyXk^GGis interesting is the fact that the main positivity properties (e.g. nefness, ampleness) of the tautological line bundleO(1) Ð→P(T_X⁽¹⁾⊕...⊕T_X⁽^k⁾)can be extended from the fiber over 0 to other fibers over λ ∈ C^∗, i.e. to the line bundle OX_k^GG(1). Moreover, the positivity properties of O(1) on P(T_X⁽¹⁾⊕...⊕T_X⁽^k⁾)are directly related to the ones of the vector bundle T_X. More generally, we will show in Section 3 that if E₁^∗, ..., E^∗_p are ample (resp. nef), then the orbifold line bundle O(1) Ð→P(E₁⁽^a¹⁾⊕...⊕E_p⁽^a^p⁾)is ample (resp. nef) in the orbifold sense for any choice of weightsa₁, ..., a_p. This will imply in particular the following result:

Proposition 1.2. Let X be a complex projective manifold. Assume thatΩ_X is ample (resp. nef ). Then for any k∈N^∗,OX^GG_k (1)is ample (resp. nef ) in the orbifold sense.

Getting back to the case of a ball quotient, we will use the logarithmic version of the previous discussion, and Riemann-Roch theorem in the orbifold case (see [To¨e99]) to obtain an estimate for the volume of the

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Green-Griffiths logarithmic jets differentialsE_k,^GG_●Ω_X(logD)in terms of the Segre class of the weighted direct sum T_X(−logD)⁽¹⁾⊕...⊕T_X(−logD)⁽^k⁾. This last Segre class can be in turn expressed in terms of the standard Segre class s_●(T_X(−logD)). An application of Hirzebruch proportionality principle in the non- compact case (see [Mum77]) will give our final estimate on vol(E_k,^GG_●Ω_X(logD)), which will be the first member of the estimate (1).

Finally, it will remain to relate the growth of the logarithmic jet differentials to the growth of the standard ones. To to this, we will simply bound from above the sections of the coherent sheavesQ^k,m, defined for any k andmby the exact sequence

0Ð→E^GG_k,mΩ_XÐ→E_k,m^GGΩ_X(logD) Ð→ Q^k,mÐ→0.

We will find a suitable filtration on the sheaves Q^k,m, in such a way that the graded terms are locally free above the boundary D, and can be expressed in terms of the vector bundles ΩD and N_D_/_X. Then, using Riemann-Roch computations and the fact that D is a disjoint union of abelian varieties, we will be able to boundh⁰(Q^k,m)from above, for a fixedk, andmgoing to+∞. This will give the second term in the estimate (1).

Acknowledgments

During the preparation of this work, the author was partially supported by the the French ANR project

”FOLIAGE”, ProjectID: ANR-16-CE40-0008.

I would like to thank my advisor Erwan Rousseau for his guidance and his support, and Julien Grivaux for many helpful and enlightening discussions. I thank also the anonymous referee for his suggestions which I hope have permitted to improve the quality and clarity of this article.

2 Segre classes of weighted projective bundles

We will now recall some results, first proved by Al-Amrani [AA97], permitting to construct Chern classes of weighted projective bundles. We will state the results in the simpler setting of Chow rings with rational coefficients.

Definition 2.1. LetX be a complex algebraic projective variety. Consider a family(Ei, ai)¹≤i≤p, where the E_i are vector bundles onX, and thea_i are positive integers. Theweighted projectivized bundle associated with the datum (Ei, ai) is the projectivized scheme of the graded O^X-algebra Sym(E₁⁽^a¹⁾⊕...⊕Ep⁽^a^p⁾)^∗, defined as

Sym(E₁⁽â¹⁾⊕...⊕E_p⁽â^p⁾)^∗=SymE₁^∗⁽â¹⁾⊗OX...⊗OX SymE_p^∗⁽â^p⁾,

where, for any i, SymE_i^∗⁽âⁱ⁾ is the graded O^X-algebra generated by sections of E_i^∗⁽âⁱ⁾ in degree a_i. We will denote this scheme by P(E₁⁽â¹⁾⊕...⊕Ep⁽â^p⁾); remark that we use here the geometric convention for projectivized bundles.

We will say, by abuse of language, that E₁⁽^a¹⁾⊕...⊕Ep⁽^a^p⁾ is a weighted direct sum, or even a weighted vector bundle.

Proposition 2.2. The varietyP(E₁⁽^a¹⁾⊕...⊕Ep⁽^a^p⁾)has a natural orbifold structure (or a structure of Deligne- Mumford stack), for which the tautological line bundle O(1) is naturally defined as an orbifold line bundle.

Moreover, this orbifold line bundle is locally ample, in the sense that the local isotropy groups of the orbifold structure act transitively on the fibres of O(1) (see for example [RT11]). Besides, if lcm(a1, ..., ap)∣m, the bundle O(m)can be identified to a standard line bundle onP(E₁⁽^a¹⁾⊕...⊕E_r⁽^a^p⁾).

Proof. We can naturally endow P(E₁⁽^a¹⁾⊕...⊕E⁽r^a^p⁾) with a structure of Artin stack P, since it can be considered as a quotient stack

E1⊕...⊕Ep/C^∗,

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whereC^∗ acts byλ⋅ (v1, ..., vp) = (λ^a¹v1, ..., λ^a^pvp).

Locally onX, the weighted projectivized bundle can be trivialized as a product of the base with a weighted projectivized space P(a1, ..., ap), where each ai appears rkEi times. Consequently, the Artin stackP has locally an orbifold structure, which makes it an orbifold stack. The claims on O(1)are local, and they can be proved directly using [Dol82] and [RT11].

Let us start our review of the properties of the Chow groups with rational coefficients of the weighted projectivized bundles.

Proposition 2.3. Let E₁⁽â¹⁾⊕...⊕Ep⁽â^p⁾ be a weighted direct sum. Let us denote the natural projection by p∶PX(E⁽₁â¹⁾⊕...⊕Ep⁽â^p⁾) Ð→X. For anyk, there is an isomorphism

AkPX(E₁⁽^a¹⁾⊕...⊕E⁽_p^a^p⁾)

Q≅⊕^r

j=0(Ak−r+jX)_Q, (2) wherer= ∑^pj=1rkEj−1.

To prove this result, we can start by checking it in the case whereX is an affine scheme. In that case, the weighted projective bundle is a quotient of a standard (trivial) projective bundle by a finite group, and it suffices to use the fact that such a quotient induces an isomorphism on the Chow rings with rational coefficients. We can then use the localization exact sequence to prove the general result.

Using the isomorphism (2), we can now define the Segre classes associated with a weighted direct sum E₁⁽^a¹⁾⊕...⊕Ep⁽^a^p⁾.

Definition 2.4. Let X be a projective algebraic variety of dimension n, and let E⁽₁â¹⁾⊕...⊕E_p⁽â^p⁾ be a weighted direct sum onX. Letq∶P(E₁⁽â¹⁾⊕...⊕Ep⁽â^p⁾) Ð→X be the natural projection.

If k ∈ [∣0, n∣], the k-th Segre class of E₁⁽^a¹⁾⊕...⊕Ep⁽^a^p⁾ is defined as an endomorphism of (A_∗X)Q. If α∈ (AlX)Q, let

sk(E₁⁽^a¹⁾⊕...⊕E_p⁽^a^p⁾) ∩α= 1

m^k⁺^rq_∗(c1O(m)^r⁺^k∩q^∗α). wherer= ∑ⁱrkE_i−1, andm=lcm(a₁, ..., a_p).

Remark 2.5. In Definition 2.4, we could have replaced m by any integer divisible by lcm(a₁, ..., a_p). The important fact used here is that O(m)is a standard line bundle, which allows us to define its first Chern class in the usual way.

There is a Whitney formula for the weighted projective bundles, which permits to express the Segre classes sj(E₁⁽^a¹⁾⊕...⊕E⁽p^a^p⁾) in terms of thes_●(Ej)and of the weights(aj):

Proposition 2.6. Let E₁⁽â¹⁾⊕...⊕Ep⁽â^p⁾ be a weighted projective sum. We have s_●(E₁⁽â¹⁾⊕...⊕E_p⁽â^p⁾) = gcd(a1, ..., ap)

a1... ap ∏

1≤j≤p

s_●(E_j⁽^a^j⁾), (3)

where, for any vector bundleE and any weight a∈N, we haves_●(E⁽^a⁾) =_arkE¹−1∑^j≥0 s_j(E)

a^j .

To prove this result, we can use the ”splitting principle” to get back to the case where theEi are all line bundlesLi. Now, denoteP=P(L⁽₀â⁰⁾⊕...⊕L⁽râ^r⁾), andp∶P Ð→X the canonical projection. Then, for some m∈N, there exists a section of(p^∗L₀)^⊗^l⁰⊗O^P(m)cutting out the subvariety P(L⁽₁â¹⁾⊕...⊕L⁽râ^r⁾)with some computable multiplicity. As in [Ful98], we can use this fact to relate the Segre class s_●(L⁽₁â¹⁾⊕...⊕L⁽râ^r⁾) with the classes s_●(L⁽₀â⁰⁾⊕...⊕L⁽râ^r⁾)and c_●(L0). The formula then follows by induction.

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3 Positivity of weighted vector bundles

We now study the extension of the usual positivity properties of vector bundles to the case of weighted vector bundles.

Definition 3.1. LetE=E₁⁽â¹⁾⊕...⊕Ep⁽â^p⁾be a weighted direct sum. We say thatE^∗=E₁^∗⁽â¹⁾⊕...⊕E_p^∗⁽â^p⁾ isample (resp. nef) if for anym∈Ndivisible enough, the (standard) line bundle O(m)is ample (resp. nef) on P(E).

Remark 3.2. With the terminology of [RT11], saying that E^∗ is ample amounts to saying that O(1) is orbi-ample on P_X(E), the tautological orbifold line bundle being locally ample by [Dol82].

We will see that the positivity properties of weighted vector bundles are exactly similar to the ones of the usual vector bundles, and can be proved in the same manner, following [Laz04b].

Proposition 3.3. Assume thatE^∗₁, ...,E_p^∗ are ample onX. Then,

1. For any coherent sheafF on X, there existsm1∈Nsuch that, for any m≥m1, the sheaf F ⊗⎛

⎝ ⊕

a1l1+...+aplp=m

S^l¹E₁^∗⊗...S^l^pE_p^∗⎞

⎠ is globally generated.

2. For any ample divisor H onX, there existsm2∈Nsuch that for anym≥m2, the sheaf

a₁l₁+...⊕+a_pl_p=m

S^l¹E₁^∗⊗...⊗S^l^pE_p^∗ is a quotient of a direct sum of copies ofO^X(H).

3. Iflcm(a1, ..., ap)∣m, thenO(m)is ample onP(E₁⁽^a¹⁾⊕...⊕Ep⁽^a^p⁾). In particular, O(1)is orbi-ample.

Proof. 1. This is easy to prove by induction onpusing the similar characterization of ample vector bundles.

2. It suffices to apply the point 1. to the sheafF = O(−H). 3. Because of 2., there existm, N ∈Nand a surjective morphism

O^X(H)^⊕^N Ð→ ⊕

a₁l₁+...a_pl_p=m

S^l¹E^∗₁⊗...⊗S^l^pE_p^∗.

Besides, because of Lemma3.4, increasingmif necessary, we can suppose that for anyq∈N, the following natural morphism of vector bundles onX is surjective:

S^q⎡⎢

⎢⎢⎢⎣ ⊕

a1l1+...aplp=m

S^l¹E₁^∗⊗...⊗S^l^pE_r^∗⎤⎥

⎥⎥⎥⎦Ð→ ⊕

a1l1+...+aplp=mq

S^l¹E₁^∗⊕...⊕S^l^pE_p^∗.

We obtain a surjective morphism of gradedO^X-algebras

⊕q≥0

S^q(O^X(H)^⊕^N) Ð→ ⊕

q≥0

⎡⎢⎢⎢

⎢⎣ ⊕

a1l1+...+aplp=mq

S^l¹E^∗₁⊗...⊗S^l^pE^∗_p⎤⎥

⎥⎥⎥⎦. which determines an embedding P(E⁽₁^a¹⁾⊕...⊕Ep⁽^a^p⁾) ↪P(O^X(−H)^⊕^N), with

OP(OX(−H)^⊕N)(1)∣_P₍_E^(a1)

1 ⊕...⊕E⁽_p^ap⁾)≅ O(qm).

Since the tautological line bundle on P(O^X(−H)^⊕^N)is ample (cf. [Laz04b]), this ends the proof.

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Lemma 3.4. LetE1, ..., Ep beC-vector spaces, and leta1, ..., ap∈N^∗. Then, for any m∈Ndivisible enough, the natural linear maps

S^q⎡⎢⎢⎢

⎢⎣ ⊕

a1l1+...+aplp=m

S^a¹E1⊗...⊗S^a^pEp⎤⎥⎥⎥

⎥⎦Ð→ ⊕

a1l1+...+aplp=mq

S^a¹E1⊗...⊗S^a^pEp

are onto for all q≥1.

Proof. Because of [Dol82], if m is sufficiently large and divisible by alla₁, ..., a_r, the (standard) line bundle O(m)on the weighted projective space Ppt(E₁^∗⁽^a¹⁾⊕...⊕E_p^∗⁽^a^p⁾)is very ample. Consequently, there exists an integerq∈Nsuch thatS^pH⁰(O(mq)) Ð→H⁰(O(mqp)) is onto for allp≥1, which gives the result.

We will now study the case of nef line bundles. We will prove the following result.

Proposition 3.5. Let E⁽₁â¹⁾⊕...⊕Ep⁽â^p⁾ be a weighted direct sum. Assume thatE₁^∗, ..., E_p^∗ are nef. Then, if m is sufficiently divisible, the line bundleO(m)is nef on P(E⁽₁â¹⁾⊕...⊕E_p⁽â^p⁾).

To this aim, we will use the formalism of vector bundles twisted by rational classes (see [Laz04b] for the definition and the positivity properties of these objects). As in the weightless case, we naturally define the notion of ampleness for a weighted sum of twisted vector bundles:

Definition 3.6. We say that a weighted direct sum of twisted vector bundles of the form E1<a1δ>⁽^a¹⁾⊕...⊕Ep<apδ>⁽^a^p⁾

is ample, if for any m, divisible by lcm(a1, ..., ap) the Q-line bundle O(m) ⊗π^∗O^X(mδ) is ample on P(E^∗₁⁽^a¹⁾⊕...E_p^∗⁽^a^p⁾).

Remark 3.7. In Definition3.6, we consider twists of the form a₁δ, ..., a_rδ with δ∈N¹(X)Q. This is related to the fact that if E₁, ..., E_rare vector bundles, and if Lis a line bundle, we have, for anym,

a₁l₁+...⊕+a_rl_r=m

S^l¹(E^∗₁⊗L^⊗^a¹) ⊗...⊗ (E_r^∗⊗L^⊗^a^r) = L^⊗^m⊗ ⊕

a₁l₁+...+a_rl_r=m

S^l¹E₁^∗⊗...⊗E_r^∗, which implies in particular that the weighted projective bundleP^′=

P((E₁⊗L^∗⊗â¹)⁽â¹⁾⊗...⊗ (E_p⊗L^∗⊗â^p)⁽â^p⁾) is identified to the variety P = P(E₁⁽â¹⁾ ⊕...⊕E_r⁽â^r⁾), with O^P^′(m) ≅ O^P(m) ⊗p^∗L^⊗^m.

Lemma 3.8. LetE₁<a₁δ>, ..., E_r<a_rδ>be twisted vector bundles onX. Assume that eachE_i^∗< −a_iδ>is ample. Then the weighted direct sum

E₁^∗< −a1δ>⁽^a¹⁾⊕...⊕E_r^∗< −arδ>⁽^a^r⁾ is ample.

Proof. We follow directly the proof presented in [Laz04b]. Because of Bloch-Gieseker theorem about ramified covers (see [Laz04b, Theorem 4.1.10]), there exist a finite, surjective, flat morphismf ∶Y Ð→X, where Y is a variety, and a divisorAsuch thatf^∗δ≡^linA. We have a fibered diagram

P^′=PY(f^∗E₁^∗⁽â¹⁾⊕...⊕f^∗E_r^∗⁽â^r⁾) ^g>PX(E₁^∗⁽â¹⁾⊕...⊕E^∗_r⁽â^r⁾) =P

Y∨ _f

> X∨ .

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Let Q=PY((f^∗E₁^∗⊗ O(a1A))⁽^a¹⁾⊕...⊕ (f^∗E_r^∗⊗ O(arA))⁽^a^r⁾). Then, we have a canonical identification Q ≅ P^′, which leads to identifying the line bundle O^Q(m) with O^P^′(m) ⊗π_Y^∗O^Y(mA), as mentioned in Remark 3.7.

Besides, theQ-line bundle

g^∗(O^P(m) ⊗π^∗O^X(mδ)) (4)

is canonically identified toO^P^′(m)⊗π_Y^∗O^Y(mA), thus toO^Q(m). However, since eachE_i^∗< −aiδ>is ample, and sincef is finite, each vector bundlef^∗E_i^∗⊗O(−aiA)is ample. Because of Proposition3.3, the line bundle O^Q(m)is ample, so the line bundle (4) is ample. Butg is finite and surjective, soO^P(m) ⊗π^∗O^X(mδ)is ample onP, which gives the result.

Proof of Proposition3.5. It suffices to show that for any ample classh∈N₁(X)Q, the classO(m)⊗π^∗O(mh) is ample. Lethbe such a class. Then since eachE^∗_i is nef, the twisted vector bundlesE_i^∗<a_ih>are ample for anyi. Consequently, by Lemma3.8and Definition3.6, if lcm(a₁, ..., a_r)∣m, the line bundleO(m)⊗π^∗O^X(mh) is ample on P(E₁^∗⁽^a¹⁾⊕...⊕E_r^∗⁽^a^r⁾). This gives the result.

3.1 An example of combinatorial application

We present a simple example of application of the previous discussion, which will turn out to be useful in Section 5, where we deal with jet bundles on a toroidal compactification of a quotient of the ball.

Proposition 3.9. Let k, n∈N. We have the following asymptotic upper bound, asrÐ→ +∞ :

∑

j₁+2j₂+...+kj_k=r

(j₁+...+j_k)ⁿ

n! ≤ 1

k!

⎡⎢⎢⎢

⎣ ∑

1≤i₁≤...≤i_n≤k

1 i1... in

⎤⎥⎥⎥

⎦

rⁿ⁺^k⁻¹ (n+k−1)! +O(rⁿ⁺^k⁻²).

Proof. Let X be an abelian variety of dimension n, endowed with an ample line bundle L. Because of Proposition 3.3, the weighted direct sum L⁽¹⁾⊕...⊕L⁽^k⁾ is ample onX. This means that the orbifold line bundle O(1)is orbi-ample onP =PX(L^∗(¹⁾⊕...⊕L^∗(^k⁾). By orbifold asymptotic Riemann-Roch theorem ([To¨e99], see also [RT11]), we have then, for anym∈N,

h⁰_orb(P,O(m)) ≤ ∫_Pc1O(1)ⁿ⁺^k⁻¹ mⁿ⁺^k⁻¹

(n+k−1)!+O(mⁿ⁺^k⁻¹).

However, because of Definition2.4and Proposition 2.6,∫^Xc1O(1)ⁿ⁺^k⁻¹ can be computed as

∫_Pc1O(1)ⁿ⁺^k⁻¹= ∫_Xsn(L^∗(¹⁾⊕...⊕L^∗(^k⁾)

= 1

k!∫_X{(∑

i

Hⁱ)...(∑

i

Hⁱ

lⁱ )...(∑

i

Hⁱ kⁱ )}

n

,

whereH =c1(L). Expending the computation yields

∫_Pc1O(1)ⁿ⁺^k⁻¹= (Lⁿ) k!

⎡⎢⎢⎢

⎣ ∑

l₁+...+l_k=n

1 1^l¹... k^l^k

⎤⎥⎥⎥

⎦

= (Lⁿ) k!

⎡⎢⎢⎢

⎣ ∑

1≤i₁≤...≤i_n≤k

1 i1... ik

⎤⎥⎥⎥

⎦. To obtain the result, it suffices to remark that we can identify the vector space

H⁰(X, ⊕

j1+2j2+...+kj_k=m

L^⊗(^j¹⁺^...⁺^j^k⁾)

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to a subspace of the orbifold global sections ofO(m). Thus : h⁰(X, ⊕

j₁+2j₂+...+kj_k=m

L^⊗(^j¹⁺^...⁺^j^k⁾) ≤h⁰_orb(P,O(m)).

Besides, a direct application of Riemann-Roch-Hirzebruch theorem and Kodaira vanishing theorem on X gives

h⁰(X, L^⊗(^j¹⁺^...⁺^j^k⁾) = (j1+...+jk)ⁿ n! (Lⁿ) ifj1+...jk≠0. Combining all these equations, we get the inequality.

We can also get back the following classical result.

Proposition 3.10. Let a₀, ..., a_n ∈ N^∗. Let X = P(a₀, ..., a_n) be the associated weighted projective space, endowed with its tautological orbifold line bundleO^X(1). We then have the asymptotic estimate

h⁰_orb(X,O^X(m)) = gcd(a0, ..., an)

∏^ja_j

mⁿ

n! +O(mⁿ⁻¹).

Proof. It is clear that the weighted direct sum C⁽^a⁰⁾⊕...⊕C⁽^aⁿ⁾ Ð→Spec C is ample, which means that O^X(1)is orbi-ample. Then, using [RT11] and Definition2.4,

h⁰_orb(X,O^X(m)) = ∫_Xc1O(1)ⁿ⋅mⁿ

n! +O(mⁿ⁻¹)

=s0(C⁽^a⁰⁾⊕...⊕C⁽^aⁿ⁾)mⁿ

n! +O(mⁿ⁻¹). Besides, because of Proposition2.6, we have

s₀(C⁽^a⁰⁾⊕...⊕C⁽^aⁿ⁾) = gcd(a₀, ..., a_n)

∏^jaj

,

which gives the result.

4 Green-Griffiths jet bundles

4.1 Deformation of the jet spaces

We first remark that for any projective complex manifold, there is a natural deformation of its Green-Griffiths jets spaces to a weighted projectivized bundles, which will permit us to apply the previous discussion to the study of jet differentials.

LetXbe a projective complex manifold. Fork∈N, we consider the Green-Griffiths jet differentials algebra E_k,^GG_●Ω_X. Recall (cf. [Dem12]) thatE_k,^GG_●Ω_X is endowed with a natural filtration, which can be described as follows.

For each(n1, ..., nk) ∈N^k, and any coordinate chartU⊂X, we define the(n1, ..., nk)-graded term as the following space of local jet differentials:

F⁽^l¹^,...,l^k⁾E_k,m^GGΩX(U)

=⎧⎪⎪

⎨⎪⎪⎩ ∑

I=(I₁,...,I_k)

aI(f^′)^I¹...(f⁽^k⁾)^I^kRRRRR RRRRR

RR (∣I1∣, ...,∣Ik∣) ≤ (l1, ..., lk)⎫⎪⎪

⎬⎪⎪⎭ .

where for each I_l= (p₁, ..., p_n), we write(f⁽^l⁾)^I^l= (f₁⁽^l⁾)^r¹...(fn⁽^l⁾)^rⁿ. In the above formula, the lexicographic order onN^kis defined so that(p1, ..., pk) < (n1, ..., nk)means that eitherpk<nk, orpk=nkand(p1, ..., pk−1) <

(n1, ..., nk−1)in the lexigraphic order forN^k⁻¹.

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The formula of derivatives of composed maps implies that these local definitions glue together to give a well defined N^k-filtration F^●E_k,m^GGΩX, compatible with the O^X-algebra structure on E_k,^GG_●ΩX, and which is increasing with respect to the lexicographic order. Moreover, the graded terms occur only for(l1, ..., lk)such that l1+2l2+...+klk=m, and we have, in this case:

Gr⁽_F^l¹^,...,l^k⁾(E_k,m^GG) =Sym^l¹ΩX⊗...⊗Sym^l^kΩX. By Definition2.1, this means exactly that, as anO^X-algebra,

Gr_F(E_k,^GG_●) ≅Sym Ω⁽_X¹⁾⊗...⊗Sym Ω⁽_X^k⁾. (5) We can use the Rees deformation construction (see for example [BG96]), to construct a O^X×C-algebra Ek,^GG● onX×C, such that for anyλ∈C^∗, Ek,^GG●∣_X_×{_λ_} is identified toE^GG_k,_●ΩX, and Ek,^GG●∣_X_×{₀_} is identified to Gr_F(E_k,^GG_●).

Let us give a few details about the construction. We first define a gradedO^X×C^k-algebraẼk,^GG●, as follows:

Ẽk,m^GG(U×C^k) =⎧⎪⎪

⎨⎪⎪⎩ ∑

l₁,...,l_k

u_l₁_,...,l_kt^l₁¹...t^l_k^kRRRRR RRRRR R

u_l₁_,...,l_k∈F⁽^l¹^,...,l^k⁾E_k,m^GGΩ_X(U)⎫⎪⎪

⎬⎪⎪⎭

where(t1, ..., tk)are coordinates onC^k. It is easy to check that for any(λ1, ..., λk) ∈C^ksuch that eachλj≠0, we have Ẽk,m^GG∣^X×{(λ₁,...,λ_k)} ≅E_k,m^GGΩ_X, and that Ẽk,m^GG∣^X×{(0,...,0)}≅ Gr_F(E_k,m^GGΩ_X). Now, define Ek,^GG● to be the pullback of Ẽk,^GG● by the embedding(x, t) ∈X×C↦ (x, t, ..., t) ∈X×C^k.

Remark 4.1. WhileẼk,^GG● seems to be the natural object arising in the construction above, it is more tractable to work over X ×C with the sheaf Ek,^GG●. To define the latter, we could have used any embedding t ∈ C z→ (tα1, ..., tαk) ∈C^k, with αi ≠ 0, so our choice (α1, ..., αk) = (1, ...,1) is rather arbitrary. The same phenomenon occurs in [Dem11], where the metric on OX_k^GG(m)constructed by Demailly depends on some auxiliary parameters1, ..., k∈R^∗₊.

Applying theProjfunctor, we obtain the following result.

Proposition 4.2. For any complex projective manifold X, and for anyk∈N^∗, there exists a morphism of varietiesXk^GGÐ→X×C, and an orbifold line bundleO_X_k^GG(1)onXk^GG, such that :

1. for anyλ∈C^∗, Xk^GG∣_λ is identified to X_k^GG, and OX_k^GG(1)∣_λ identified to OX^GG_k (1);

2. the fibreXk^GG∣₀is identified to the varietyProj_X(Gr_F(E^GG_k,_●)) ≅P_X(T_X⁽¹⁾⊕...⊕T_X⁽^k⁾), and OX_k^GG(1)∣₀ is identified to the tautological line bundle of this weighted projective bundle.

Remark 4.3. By construction, the identifications mentioned above already occur at the level of sheaves of algebras. To obtain the identifications when takingProjfunctors, we just need to check that the gradings on these sheaves of algebras are compatible under these identifications.

Proof. Let us prove the second point, the first one being similar. By construction, we have a natural identification between

Ek,^GG●∣^X×{0}= ⊕

m≥0Ek,m^GG∣^X×{0}

and

GrF(E_k,^GG_●ΩX) = ⊕

m≥0

GrF(E_k,m^GGΩX)

Moreover, this identification is compatible with the grading inm. Besides, by (5), the latter sheaf is identified, as a sheaf ofgraded algebras, with

Sym Ω⁽_X¹⁾⊗...⊗Sym Ω⁽_X^k⁾= ⊕

m≥0( ⊕

l1+2l2+...+klk=m

Sym^l¹Ω_X⊗...⊗Sym^l^kΩ_X).

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Now, by Definition2.1, the projectivized bundle associated to the latter sheaf of graded algebras, with respect to the grading inm, is P(T_X⁽¹⁾⊕...⊕T_X⁽^k⁾). This implies immediately the identifications of varieties and orbifold line bundles mentioned in the second point.

We can now show that some usual positivity properties of the cotangent bundle can be transmitted to the higher order jet differentials.

Proposition 4.4. If Ω_X is ample (resp. nef ), then for any k∈N^∗, E^GG_k,_●Ω_X is ample (resp. nef ), meaning that OX^GG_k (1)is ample (resp. nef ) as an orbifold line bundle.

Proof. Let Xk^GGÐ→X×C be the variety given by Proposition4.2, endowed with its orbifold line bundle O_X_k^GG(1).

Assume first that ΩX is ample. Then, because of Proposition 3.3, a suitable power of the tautological line bundleO(m)is ample on P(T_X⁽¹⁾⊕...⊕T_X⁽^k⁾)ifmis sufficiently divisible. Because of Proposition4.2, it means thatO_X_k^GG(1)∣⁰ is ample. By semi-continuity of the ampleness property, for any λ∈C^∗ in a Zariski neighborhood of 0, the orbifold line bundleO_X_k^GG(1)∣^λis ample. Again because of Proposition4.2, this means exactly thatE_k,^GG_●ΩX is ample.

The case where ΩX is nef is dealt with in the same manner, using Proposition 3.5, and the fact that if OX_k^GG(1)∣⁰ is nef, thenOX_k^GG(1)∣^λis nef for any very generalλ∈C(see [Laz04a]).

The previous discussion extends naturally to the case of logarithmic jet differentials. We then have the following proposition.

Proposition 4.5. Let(X, D)be a smooth log-pair. For anyk∈N^∗, there exists a morphismXk^GG,logÐ→X×C and an orbifold line bundle O_X^GG,log

k (1)onXk^GG,log such that

1. for anyλ∈C^∗, Xk^GG,log∣_λ is identified toX_k^GG,log, and O_X_k^GG,log(1)∣_λ identified to OX^GG_k (1); 2. the fibre Xk^GG,log∣₀ is identified to

Proj_XGrF(E_k,^GG_●) ≅PX(TX(−logD)⁽¹⁾⊕...⊕TX(−logD)⁽^k⁾),

and O_X_k^GG,log(1)∣₀ is identified to the tautological orbifold line bundle of this weighted projectivized bundle.

Proof. As before, it suffices to use the fact that E_k,m^GGΩ_X(logD)admits a filtration whose graded algebra is Sym ΩX(logD)⁽¹⁾⊗...⊗Sym ΩX(logD)⁽^k⁾.

In this setting, Proposition4.4extends naturally:

Proposition 4.6. Let (X, D)be a smooth log-pair. IfΩX(logD)is nef (resp. ample), then for any k∈N^∗, E_k,^GG_●ΩX(logD)is nef (resp. ample), meaning thatOX_k^GG,log(1)is nef (resp. ample) as orbifold line bundle.

Remark 4.7. Proposition4.6is actually only relevant for the nef property. Indeed, except whenXis a curve, the bundle Ω_X(logD)cannot be ample in general: ifD is smooth and dimX≥2, we have the residue map

ΩX(logD) Ð→ O^DÐ→0,

and the restriction of this map to D shows that ΩX(logD)∣^D has a trivial quotient. This implies that ΩX(logD)is not ample.

The following result, combining two theorems of Campana and P˘aun [CP15], and Demailly [Dem11], shows that the bigness of the canonical orbifold line bundle O(1)on P(T_X⁽¹⁾⊕...⊕T_X⁽^k⁾)forklarge enough suffices to characterize the manifolds of general type.