On the Diverio-Trapani Conjecture Ya Deng

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Ya Deng

To cite this version:

Ya Deng. On the Diverio-Trapani Conjecture. Annales Scientifiques de l’École Normale Supérieure, Société mathématique de France, In press. �hal-01338643v3�

by Ya Deng

Abstract. — In this paper we establish effective lower bounds on the degrees of the De- barre and Kobayashi conjectures. Then we study a more general conjecture proposed by Diverio-Trapani on the ampleness of jet bundles of general complete intersections in com- plex projective spaces.

R´esum´e (Autour de la conjecture de Diverio-Trapani). — Dans cet article, nous

´etablissons des bornes inf´erieures effectives sur les degr´es li´es aux conjectures de Debarre et Kobayashi. Ensuite, nous ´etudions une conjecture plus g´en´erale propos´ee par Diverio- Trapani sur l’amplitude des fibr´es de jets des intersections compl`etes g´en´erales dans les espaces projectifs complexes.

0. Introduction

A compact complex manifold X is said to be Kobayashi (Brody) hyperbolic if there exists no non-constant holomorphic mapf :C→X. As is well-known, a sufficient criteria for Kobayashi hyperbolicity is the ampleness of the cotangent bundle. Although the com- plex manifolds with ample cotangent bundles are expected to be reasonably abundant, there are few concrete constructions before the work of Debarre. In [Deb05], Debarre proved that the complete intersection of sufficiently ample general hypersurfaces in acom- plex abelian variety, whose codimension is at least as large as its dimension, has ample cotangent bundle. He further conjectured that this result should also hold for intersec- tion varieties of general hypersurfaces in complex projective spaces (the so-called Debarre conjecture). This conjecture was recently proved by Brotbek-Darondeau [BDa17] and in- dependently by Xie [Xie16, Xie18], based on the ideas and explicit methods in [Bro16].

Theorem 0.1 (Brotbek-Darondeau, Xie). — Let X be an n-dimensional projective manifold equipped with a very ample line bundleA. Then there existsd_Deb,n ∈Ndepending only on the dimensionn, such that for alld>d_Deb,n, the complete intersection ofc-general hypersurfacesH₁, . . . , H_c ∈ |A^d|has ample cotangent bundle, provided that ⁿ₂ 6c6n.

In [Xie18], Xie was able to obtain an effective lower bound d_Deb,n = nⁿ² by working with (much more elaborated) explicit expressions of some symmetric differential forms.

The result in [BDa17] is “almost” effective ond_Deb,n, because it depends on some constant

2000 Mathematics Subject Classification. — 32Q45, 14M10, 14M15, 14C20.

Key words and phrases. — Kobayashi hyperbolicity, ample cotangent bundle, Debarre conjecture, Kobayashi conjecture, Diverio-Trapani conjecture, Nakamaye’s theorem, Brotbek’s Wronskians.

Mots clefs. — hyperbolicit´e au sens de Kobayashi, fibr´e cotangent ample, conjecture de Debarre, conjecture de Kobayashi, conjecture de Diverio-Trapani, th´eor`eme de Nakamaye, Wronskians de Brotbek.

involved in some noetherianity argument, arising in their reduction to Nakamaye’s theorem [Nak00] for families of zero-dimensional subschemes.

One goal of the present paper is to provide an effective estimate for such a Nakamaye’s theorem (see Theorem 2.10). In particular, as a complement of [BDa17, Theorem 1.1], we can improve Xie’s effective lower boundd_Deb,n.

Theorem A. — In the same setting as Theorem 0.1, one can take d_Deb,n = (2n)ⁿ⁺³.

It is worth to mention that the techniques in [BDa17] are more intrinsic and the ideas of their proof brought new geometric insights in the understanding of the positivity of cotangent bundles. Later, Brotbek [Bro17] extended these techniques from the setting of symmetric differentials to that of higher order jet differentials, so that he was able to prove a long-standing conjecture of Kobayashi in [Kob70].

Theorem 0.2 (Brotbek). — Let X be a projective manifold of dimension n. For any very ample line bundleA on X, there exists d_Kob,n ∈Ndepending only on the dimension n such that for any d > d_Kob,n, a general smooth hypersurface H ∈ |A^d| is Kobayashi hyperbolic.

The proof of Theorem 0.2 in [Bro17] is also “almost” effective on d_Kob,n because of two noetherianity arguments: the first concerns the increasing sequences of Wronskians ideal sheaves; the second concerns a constant arising in Nakamaye’s theorem as that of [BDa17], which can be made effective by Theorem 2.2. Our second goal of the present paper is to give an intrinsic interpretation of Brotbek’s Wronskians (see§ 1.2), and as a byproduct, we can render the above-mentioned first noetherianity argument effective. This in turn provides effective lower bounds for the Kobayashi conjecture in combination with the explicit formula ofd_Kob,n in [Bro17].

Theorem B. — In the same setting as Theorem 0.2, one can take d_Kob,n =n²ⁿ⁺³(n+ 1).

Let us mention that in [Bro17] Brotbek obtained a much stronger result than Theo- rem 0.2. Indeed, he proved that for the hypersurfaceH in Theorem 0.2, the tautological line bundleO_H

k(a_k, . . . , a1) on theDemailly-Semplek-jet tower H_k of the direct manifold (H, T_H) is “almost ample” for some (a₁, . . . , a_k)∈N^kwhenk>n−1 = dimH. In view of the following vanishing theorem by Diverio in [Div08], the above-mentioned lower bound forkin [Bro17] is optimal.

Theorem 0.3 (Diverio). — Let Z ⊂Pⁿ be a smooth complete intersection of hypersur- faces of any degree inPⁿ. Then

H⁰(Z, E_k,m^GGT_Z^∗) = 0

for all m >1 and 1 6k < dim(Z)/codim(Z). Here E_k,m^GGT_Z^∗ denotes the Green-Griffiths jet bundleof order k and weighted degree m.

Motivated by the above vanishing theorem, in the same vein as the Debarre conjecture, Diverio-Trapani proposed the following generalized conjecture in [DT10].

Conjecture 0.4 (Diverio-Trapani). — Let Z ⊂Pⁿ be the complete intersection of c- general hypersurfaces of sufficiently high degree. Then the invariant jet bundle E_k,mT_Z^∗ is ample provided thatk> ⁿ_c −1 andm≫0.

The last aim of the present paper is to study Conjecture 0.4 using geometric methods in [BDa17, Bro17].

Theorem C. — Let X be an n-dimensional projective manifold equipped with a very ample line bundleA, and letZ⊂Xbe the complete intersection ofc-general hypersurfaces H₁. . . , H_c ∈ |A^d|. Then Z is almost k-jet ample (see Definition 1.2) if k> ⁿ_c −1, and d>2cn^c⌈nc⌉+1· ⌈ⁿ_c⌉^c⌈nc⌉+3. In particular, Z is Kobayashi hyperbolic.

Let us mention that we apply the results in the first part of the present paper to obtain the effective lower degree bounds in Theorem C.

In view of the correspondence between tautological line bundles on the Demailly-Semple jet towers and invariant jet bundles studied in [Dem97, Proposition 6.16], the following result on Conjecture 0.4 is a consequence of Theorem C.

Corollary D. — In the same setting as Theorem C, for any k> ⁿ_c −1, there exists a subbundle F ⊂E_k,mT_Z^∗ for some m≫0 such that

(i) F is ample.

(ii) For any regular germ of curve f : (C,0) → (Z, z), there is a global section P ∈ H⁰(Z,F ⊗A⁻¹) so that P([f]_k)(0)6= 0.

In other words, one can find a subbundleF of the invariant jet bundleE_k,mT_Z^∗, which is ample, and theDemailly-Semple locus (see [DR15, §2.1] for the definition) induced by F is empty.

Lastly, let us mention that the techniques in [BDa17, Bro17] were extended by Brotbek and the author to prove a logarithmic analogue of the Debarre conjecture in [BD17], and to prove the logarithmic (orbifold) Kobayashi conjecture in [BD18]. To achieve the effective lower degree bounds, both the articles [BD17, BD18] rely on the methods in the present paper.

This paper is organized as follows. In§1.1 we recall the fundamental tools of jet differ- entials by Demailly, Green-Griffiths and Siu, which can be seen as higher order analogues of symmetric differential forms and provide obstructions to the existence of entire curves.

§ 1.2 is devoted to the study of new techniques of Wronskians introduced by Brotbek in his proof of the Kobayashi conjecture [Bro17]. We bring a new perspective of Brotbek’s Wronskians, which we interpret as a certainmorphism of O-modules from the jet bundles of a line bundle to the invariant jet bundles. In view of this result one can immediately make the first noetherianity argument in [Bro17] effective. In§2, by means of an explicit construction of global sections with a “negative twist”, we obtain a slightly weaker but effective Nakamaye’s theorem for the universal families of zero-dimensional subschemes introduced in [BDa17, Bro17]. This in turn renders the second noetherianity argument in [Bro17] as well as that in [BDa17] effective, and in combination with the formulas for lower degree bounds in [BDa17, Bro17], we prove Theorems A and B. The aim of § 3 is to study Conjecture 0.4. In§ 3.1 we briefly recall the essential results in [Bro17], and we show in§ 3.2 and§ 3.3 how to deduce Theorem C from Brotbek’s techniques.

Acknowledgements. I would like to warmly thank my thesis supervisor Professor Jean- Pierre Demailly for his constant encouragements and supports, and Damian Brotbek for suggesting this problem and kindly sharing his ideas to me. I also thank Professors Steven Lu and Erwan Rousseau for their interests and suggestions on the work. I am indebted to Lionel Darondeau and Songyan Xie for their discussions. Lastly, I thank the anonymous referee for very helpful suggestions to improve the presentation in this paper. This work is supported by the ERC ALKAGE project.

1. Jet differentials and Brotbek’s Wronskians

By the work of Nadel [Nad89] and Demailly-El Goul [DEG97], theWronskians induced by meromorphic connections provide an abundant supply of invariant jet differentials.

In [Bro17] Brotbek introduced an alternative approach to construct Wronskian jet differ- entials associated to sections of a given line bundle. In§1.2 we give an intrinsic definition of Brotbek’s Wronskians via the jet bundles of line bundles.

1.1. Jet spaces and jet differentials. — In this subsection, we collect the main tech- niques of jet differentials in [Dem97]. A direct manifold is a pair (X, V) where X is a complex manifold andV ⊂T_X is a holomorphic subbundle of the tangent bundle. Denote byp_k:J_kV →X the bundle of k-jets of germs of parametrized curves in (X, V), that is, the set of equivalent classes of holomorphic maps f : (C,0) → (X, x) which are tangent to V, with the equivalence relation f ∼ g if and only if all derivatives f^(j)(0) = g^(j)(0) coincide for 0 6 j 6 k, when computed in some local coordinate system of X near x.

The class f in J_kV is denoted by [f]_k. The projection map p_k : J_kV → X is simply [f]_k7→f(0). WhenV =T_X, we simply writeJ_kXin place ofJ_kV. Note thatJ_kX→Xis a local trivial fibration with fibersC^nk. Indeed, local coordinates (z1, . . . , zn) for an open setU ⊂X induce coordinates

(z₁, . . . , z_n, z₁^′, . . . , z_n^′, . . . , z^(k)₁ , . . . , z_n^(k)) onp⁻¹_k (U), and anyk-jet [f]_k ∈p⁻¹_k (U) has coordinates

f1(0), . . . , fn(0), . . . , f₁^(k)(0), . . . , f_n^(k)(0) .

Let G_k be the group of germs ofk-jets of biholomorphisms of (C,0), that is, the group of germs of biholomorphic maps

t7→ϕ(t) =a₁t+a₂t²+· · ·+a_kt^k, a₁ ∈C^∗, a_j ∈C,∀j>2,

in which the composition law is taken modulo termst^j of degree j > k. ThenG_k admits a natural fiberwise right action onJ_kX defined by ϕ·[f]_k := [f ◦ϕ]_k. Note that C^∗ can be seen as a subgroup ofG_k defined by (a₂=· · ·=a_k= 0).

In [GG80], Green-Griffiths introduced the vector bundle E_k,m^GGT_X^∗ → X whose fibres are complex valued polynomialsQ([f]_k) on the fibres of J_kX, of weighted degree mwith respect to theC^∗-action, that is, Q(λ·[f]_k) =λ^mQ([f]_k),for all λ∈C^∗ and [f]_k ∈J_kX.

Let U ⊂ X be an open set with local coordinates (z1, . . . , zn). Then any local section Q∈E_k,m^GGT_X^∗(U) can be written as

Q= X

|α1|+2|α2|+···+k|αk|=m

c_α(z)(d¹z)^α1(d²z)^α2· · ·(d^kz)^αk,

where c_α(z) ∈ O(U) for any α := (α₁, . . . , α_k) ∈ (Nⁿ)^k, such that for any holomorphic mapγ : Ω→U from an open set Ω⊂C, one has

Q [γ]_k

(t) = X

|α1|+2|α2|+···+k|αk|=m

c_α γ(t)

γ^′(t)α1

γ^′′(t)α2

· · · γ^(k)(t)αk

∈O(Ω), where [γ]_k(t) : Ω→J_kX_↾U is the lifted holomorphic curve on J_kX induced by γ.

The bundle E_k,•^GGT_X^∗ := L

m>0E_k,m^GGT_X^∗ is in a natural way a bundle of graded algebras (the product is obtained simply by taking the product of polynomials). There are natural inclusions E_k,•^GGT_X^∗ ⊂ E_k+1,•^GG T_X^∗ of algebras, hence E_∞,•^GGT_X^∗ := S

k>0E_k,•^GGT_X^∗ is also an algebra. It follows from [Dem97, §6] that the sheaf of holomorphic sections O(E_∞,•^GGT_X^∗) admits a canonical derivation Dgiven by a collection of C-linear maps

D:O(E_k,m^GGT_X^∗)→O(E^GG_k+1,m+1T_X^∗) (1.1.1)

constructed as follows. For any germ of curvef : (C,0) →X, and any Q∈O(E_k,m^GGT_X^∗), (DQ)([f]_k+1)(t) := d

dtQ([f]_k)(t).

We can also inductively defineD^k:=D◦D^k−1. In particular, for any holomorphic function s∈O(U),D^k(s)∈E_k,k^GGT_X^∗(U).

In this present paper, we are interested in the more geometric context introduced by Demailly in [Dem97]: the subbundle E_k,mT_X^∗ ⊂ E^GG_k,mT_X^∗ which consists of polynomial differential operators Q which are invariant under arbitrary changes of parametrization, that is, for anyϕ∈G_k and any [f]_k∈J_kX, one has

Q ϕ·[f]_k

=ϕ^′(0)^mQ([f]_k).

The bundleE_k,mT_X^∗ is called the invariant jet bundle of order k and weighted degree m.

It is noticeable that Wronskians provide a very natural construction for invariant jet differentials.

For any direct manifold (X, V) with rankV =r, Demailly [Dem97] introduced a fonc- torial construction of a sequence of direct manifolds

· · · →(P_kV, V_k)−→^πk (P_k−1, V_k−1)−−−→ · · ·^πk−1 −→^π2 (P₁V, V₁)−→^π1 (P₀V, V₀) = (X, V) (1.1.2)

so that P_kV := P(V_k−1) is a P^r−1-bundle over P_k−1V for each k > 1, and we say P_kV theDemailly-Semple k-jet tower of (X, V). In the absolute case (X, T_X), we simply write X_k := P_kV. In the case of smooth family of compact complex manifolds X → T, X^rel denotes to be the Demailly-Semple k-jet tower of the direct manifold (X, TX/T), wherek

TX/T denotes the relative tangent bundle. It follows from [Dem97,§6] that the Demailly- Semple jet tower has the following geometric properties.

1. Any germ of curve f : (C,0) →X tangent toV can be lifted tof_[k]: (C,0)→P_kV. 2. Denote by J_k^regV := {[f]_k | f^′(0) 6= 0} the set of regular k-jets tangent to V. Then

there exists a morphism J_k^regV → P_kV, which sends [f]_k to f_[k](0), whose image is a Zariski open subset P_kV^reg ⊂ P_kV which can be identified with the quotient J_k^regV /G_k. Moreover, the complement P_kV^sing:= P_kV \P_kV^reg is a divisor in P_kV. 3. For any k, m>0 one has

(π_0,k)_∗O_P

kV(m) =E_k,mV^∗, (1.1.3)

where we write π_j,k =π_j+1◦ · · · ◦π_k : P_kV → P_jV for any 06 j 6k, and O_P

kV(1) denotes the tautological line bundle over P_kV = P(V_k−1).

More generally, for ak-tuple (a₁, . . . , a_k)∈N^k, we write O_P

kV(a_k, . . . , a₁) :=O_P

kV(a_k)⊗π_k−1,k^∗ O_P

k−1V(a_k−1)⊗ · · · ⊗π_1,k^∗ O_P

1V(a₁).

The fundamental vanishing theorem shows that the jet differentials vanishing along any ample divisor gives rise to obstructions to the existence of entire curves.

Theorem 1.1 (Demailly, Green-Griffiths, Siu-Yeung). — Let (X, V) be any di- rect manifold equipped with an ample line bundle A. For any non-constant entire curve f : C → X tangent to V, and any ω ∈ H⁰ P_kV,O

PkV(a_k, . . . , a₁) ⊗π_0,k^∗ A⁻¹ with (a₁, . . . , a_k)∈N^k, one has f_[k](C)⊂(ω= 0).

Observe that for any non-constant entire curve f :C→ X tangent to V, the image of its lift f_[k]: C → P_kV is not entirely contained in P_kV^sing. In view of Theorem 1.1, we introduce the following definition.

Definition 1.2. — LetX be a projective manifold. We say that X is almostk-jet ample if there exists some(a₁, . . . , a_k)∈N^k so thatO_X

k(a_k, . . . , a₁)is big and its augmented base locusB+ O_X

k(a_k, . . . , a1)

⊂X_k^sing.In particular, X is Kobayashi hyperbolic.

Note that almost 1-jet ampleness is equivalent to the ampleness of cotangent bundle.

1.2. Brotbek’s Wronskians. — This subsection is devoted to the study of the Wron- skians defined by Brotbek in [Bro17, §2.2]. Let X be ann-dimensional compact complex manifold. Recall that for any holomorphic line bundleLon X, one can define the bundle J^kL of k-jet sections of L by J^kL_x =O(L)_x/ m^k+1_x ·O(L)_x

for every x ∈X, wherem_x is the maximal ideal ofO_x. Pick an open setU ⊂X with coordinates (z₁, . . . , z_n) so that L_↾U can be trivialized by a nowhere vanishing section e_U ∈L(U). The fiberJ^kL_x can be identified with the set of Taylor developments of orderk

X

|γ|6k

c_γ(z−x)^γ·e_U,

and the coefficients{c_γ}_γ∈Nⁿ_,|γ|6kdefine coordinates along the fibers ofJ^kL. This in turn gives rise to a natural local trivialization ofJ^kL defined by

ΨU :U ×C^In,k −^≃→ J^kL_↾U, (x, c_γ) 7→ X

γ∈In,k

c_γ(z−x)^γ·e_U,

where I_n,k := {γ = (γ₁, . . . , γ_n) ∈ Nⁿ | |γ| 6 k}. Observe that there exists a C-linear morphism

j_L^k :L→J^kL,

which is not a morphism ofO_X-modules, defined as follows. For any s∈L(U), define j_L^k(s)(x) := X

|γ|6k

1 γ!

∂^|γ|s_U

∂z^γ (x)(z−x)^γ·eU, (1.2.1)

where s_U ∈ O(U) so that s = s_U ·e_U. When L = O_X, we simply write j^k := j_O^k

X. The jet bundleJ^kL will be used to interpret the canonical derivativeD :O(E_k,m^GGT_X^∗) → O(E_k+1,m+1^GG T_X^∗) defined in (1.1.1) in an alternative way. Let us first give a more precise expression ofD.

Lemma 1.3. — Take any open setU ⊂X with coordinates (z₁, . . . , z_n). For any k>1, and any holomorphic functions∈O(U), one has

D^k(s)(z) = X

|α1|+2|α2|+···+k|αk|=k

c_k,α(z)(d¹z)^α1(d²z)^α2· · ·(d^kz)^αk ∈E_k,k^GGT_X^∗(U) (1.2.2)

such that for eachα:= (α₁, . . . , α_k)∈(Nⁿ)^k,c_k,α(z)∈O(U) is aZ-linear combination of

∂^|γ|s

∂z^γ (z) with |γ|=γ₁+· · ·+γ_n6k.

Proof. — We will prove the lemma by induction on k. For k= 1, we simply have D(s) =ds=

Xn i=1

∂s

∂z_i(z)dz_i∈T_X^∗(U), and thus (1.2.2) remains valid fork= 1.

Now we assume that D^k(s) has the form (1.2.2). By (1.1.1), one has D^k+1(s) =

X

|α1|+2|α2|+···+k|αk|=k

Xk−1 i=1

X

j=1,...n αi−e_j∈Nⁿ

c_k,α(z)(d¹z)^α1· · ·(dⁱz)^αi−ej(dⁱ⁺¹z)^αi+1+ej· · ·(d^kz)^αk

+ Xn j=1

∂c_k,α(z)

∂z_j (d¹z)^α1+ej· · ·(d^kz)^αk+ X

j=1,...n αk−e_j∈Nⁿ

c_k,α(z)(d¹z)^α1· · ·(d^kz)^αk−ej(d^k+1z)^ej

!

whereej denotes the vector inNⁿwith a 1 in thejth coordinate and 0’s elsewhere. By the assumption, for everyj= 1, . . . , n and every α, ^∂ck,α_∂z^(z)

j ∈O(U) is a Z-linear combination of ^∂_∂z^|γ|γ^s(z) with|γ|=γ₁+· · ·+γ_n6k+ 1. From the above expression we conclude that (1.2.2) also holds true forD^k+1(s). The lemma follows.

It follows from (1.2.1) and Lemma 1.3 that there exists a morphism of O_X-modules, denoted by j^kD : J^kO_X → E_k,k^GGT_X^∗, so that D^k : O_X → E_k,k^GGT_X^∗ factors through j^kD, that is,D^k=j^kD◦j^k.

Following [Bro17], given k+ 1 holomorphic functions g₀, . . . , g_k∈O(U), one can asso- ciate them to a jet differentials of order k and weighted degree k^′ := ^k(k+1)₂ , say Wron- skians, in the following way

W_U(g₀, . . . , g_k) :=

g0 g1 . . . g_k D(g₀) D(g₁) · · · D(g_k)

... ... . .. ... D^k(g0) D^k(g1) · · · D^k(g_k)

∈E_k,k^GG′T_X^∗(U).

(1.2.3)

It follows from [Bro17, Proposition 2.2] that Wronskians are indeedinvariant jet differen- tials. From its alternating property, WU induces a C-linear map, which we still denoted by W_U : Λ^k+1O(U) → E_k,k^′T_X^∗(U) abusively. By the factorization property of D^k, W_U gives rise to a morphism of O_U-module

W_J^kO_U : Λ^k+1J^kO_U →E_k,k^′T_U^∗ so that one has

W_U(g₀, . . . , g_k) =W_JkO_U j^k(g₀)∧ · · · ∧j^k(g_k) . In other words, Brotbek’s WronskiansW_U can be factorized as follows.

W_U : Λ^k+1O(U) ^Λ

k+1j^k

−−−−→Λ^k+1 J^kO_U(U)

→ Λ^k+1J^kO_U

(U)−−−−−→^{WJ kOU} E_k,k^′T_U^∗(U).

(1.2.4)

Now we consider the Demailly-Semple k-jet tower X_k of (X, T_X). For the open set U_k := π_0,k⁻¹(U) of X_k, the coordinate system (z₁, . . . , z_n) on U induces a trivialization U_k≃U×R_n,k,whereR_n,kis some smooth rational variety introduced in [Dem97, Theorem 9.1]. Hence

O_X

k(1)_↾Uk ≃pr^∗₂(O_R

n,k(1)), (1.2.5)

where pr₂ : U_k −^≃→ U ×R_n,k → R_n,k is the composition of the isomorphism with the projection map. By (1.1.3), we conclude that, under the above trivialization, the direct image (π_0,k)_∗ induces a local trivialization of the vector bundleE_k,k^′T_U^∗

ϕ_U :U×H⁰ R_n,k,O_R

n,k(k^′) ≃

−→E_k,k^′T_U^∗. (1.2.6)

Write F_n,k := H⁰ R_n,k,O_R

n,k(k^′)

. Therefore, under the trivializations ϕ_U and Ψ_U, the morphism of O_U-module W_J^kO_U is indeed constant, i.e. there is a C-linear map ν_n,k : Λ^k+1C^In,k →F_n,k such that one has the following diagram.

U×Λ^k+1C^In,k

ΨU

≃

1U×νn,k

//U×F_n,k

ϕU

≃

Λ^k+1J^kO_U

W_{J kO}

U //E_k,k^′T_U^∗

Denote by I_n,k ⊂ O_R

n,k the base ideal of the linear system |Im(ν_n,k)| ⊂ |O_R

n,k(k^′)|, and setw_k,U to be the ideal sheaf pr^∗₂(I_n,k) on U_k.

By [Bro17], Wronskians can also be associated to global sections of any line bundle L. Take an open set U ⊂X with coordinates (z₁, . . . , z_n) so that L_↾U can be trivialized by a nowhere vanishing section e_U ∈ L(U). Consider any s₀, . . . , s_k ∈ H⁰(X, L). There exists unique s_i,U ∈ O(U) so that s_i = s_i,U ·e_U for every i = 0, . . . , k. It was proved in [Bro17, Proposition 2.3] that the section

W_U(s_0,U, . . . , s_k,U)·e^k+1_U ∈(E_k,k^′T_X^∗ ⊗L^k+1)(U).

(1.2.7)

is intrinsically defined, i.e. it does not depend on the choice of e_U. Hence they can be glued together into a global section, denoted to beW(s₀, . . . , s_k)∈H⁰(X, E_k,k^′T_X^∗⊗L^k+1).

Set

ω(s₀, . . . , s_k) := (π_0,k)⁻¹_∗ W(s₀, . . . , s_k)∈H⁰ X_k,O_X

k(k^′)⊗π_0,k^∗ L^k+1 (1.2.8)

to be the inverse image of the WronskianW(s₀, . . . , s_k) under (1.1.3).

Following [Bro17, §2.3], define

W(X_k, L) : = Span{ω(s₀, . . . , s_n)|s₀, . . . , s_n∈H⁰(X, L)}

⊂H⁰ X_k,O_X

k(k^′)⊗π^∗_0,kL^k+1

and define thek-th Wronskian ideal sheaf ofL, denoted byw(X_k, L), to be the base ideal of W(X_k, L). It was also shown in [Bro17, §2.3] that if L is very ample, one has a chain of inclusions

w(X_k, L)⊂w(X_k, L²)⊂ · · · ⊂w(X_k, L^m)⊂ · · · .

By noetherianity, this increasing sequence stabilizes after somem_∞(X_k, L) ∈N, and the obtained asymptotic ideal sheaf is denoted byw_∞(X_k, L).Let us mention thatm_∞(X_k, L) concerns the first noetherianity argument in [Bro17], and in the rest of this subsec- tion we will apply our new interpretation of Brotbek’s Wronskians in (1.2.4) to render m∞(X_k, L) effective. The strategy is to compare the globally defined Wronskian ideal sheaves{w(X_k, L^m)}_m∈N to the intrinsic ideal sheafw_k,U.

One direction is easy to see from the very definition of w(X_k, L). By (1.2.7), for any s₀, . . . , s_k ∈H⁰(X, L), the Wronskian can be localized by

W(s₀, . . . , s_k)_↾U =W_U(s_0,U, . . . , s_k,U)·e^k+1_U ∈(E_k,k^′T_X^∗ ⊗L^k+1)(U).

We denote byωU(s0,U, . . . , s_k,U)∈O_X

k(k^′)(U_k) the corresponding element ofWU(s0,U, . . . , s_k,U) under the isomorphism (1.1.3), where U_k := π⁻¹_0,k(U). In view of (1.2.5), one has O_X

k(k^′)(U_k)≃H⁰(U, U×F_n,k), or more precisely, H⁰(U,Λ^k+1J^kO_U)

Ψ⁻_U¹

≃

W_{J kO}

U //H⁰(U, E_k,k^′T_U^∗)

ϕ⁻¹_U

≃

H⁰(U, U×Λ^k+1C^In,k)^1U×νn,k//H⁰(U, U×F_n,k).

By (1.2.4),W_U(s_0,U, . . . , s_k,U) =W_JkO_U(j^ks_0,U∧ · · · ∧j^ks_k,U). Hence ω_U(s_0,U, . . . , s_k,U)≃(1U×ν_n,k)◦Ψ⁻¹_U (j^ks_0,U ∧ · · · ∧j^ks_k,U).

(1.2.9)

Recall that I_n,k ⊂ O_R

n,k is the base ideal of the linear system |Im(ν_n,k)|, and w_k,U is defined to be the ideal sheaf pr^∗₂(I_n,k) on U_k ≃ U ×R_n,k. By (1.2.9), the base ideal of ωU(s0,U, . . . , s_k,U) is contained inw_k,U. As s0, . . . , s_k∈H⁰(X, L) are arbitrary, this leads to

w(X_k, L)_↾Uk ⊂w_k,U. (1.2.10)

Now we further assume that the line bundle L separates k-jets everywhere, i.e. the C-linear map

H⁰(X, L) ^j

k

−→L H⁰(X, J^kL)→J^kLx

is surjective for anyx∈X. Then

Λ^k+1H⁰(X, L)→Λ^k+1O(U)−→^jk Λ^k+1J^kO_U(U)→Λ^k+1J^kO_x≃Λ^k+1C^In,k is also surjective for anyx∈U. By (1.2.9) again,

Im(ν_n,k) = Span{ω_U(s_0,U, . . . , s_k,U)_↾{x}×Rn,k |s₀, . . . , s_k ∈H⁰(X, L)},

where we identify {x} × R_n,k with the fiber π_0,k⁻¹(x). Write ι_x to be the composition R_n,k → {x} ×R_n,k ֒→U ×R_n,k −→^≃ U_k֒→X_k. This in turn implies that

ι^∗_xw(X_k, L) :=ι⁻¹_x w(X_k, L)⊗_ι⁻¹

x O

Xk

O_R

n,k =I_n,k

It follows from w_k,U := pr^∗₂I_n,k that w(X_k, L)_↾Uk = w_k,U. By the inclusive relation (1.2.10), one has

w_k,U =w(X_k, L)_↾Uk =w(X_k, L²)_↾Uk =· · ·=w(X_k, L^k)_↾Uk =· · ·. (1.2.11)

As is well-known, A^k separates k-jets everywhere once A is very ample. By (1.2.11), we conclude the following result.

Theorem 1.4. — Let X be a projective manifold, and letA be a very ample line bundle on X. Then w(X_k, A^k) =w_∞(X_k, A) and m_∞(X_k, A) =k.

Moreover, it follows from the relation (1.2.11) that the asymptotic Wronskian ideal sheaf is intrinsically defined,i.e. w_∞(X_k, L) does not depend on the very ample line bundleL.

This reproves [Bro17, Lemma 2.8]. It also allows us to denote byw_∞(X_k) the asymptotic Wronskian ideal sheaf.

Remark 1.5. — In a joint work with Brotbek [BD18], we generalize the alternative in- terpretation of Wronskians by jets of sections of line bundles in this subsection to the logarithmic settings.

1.3. Blow-up of the Wronskian ideal sheaf. — This subsection is mainly borrowed from [Bro17]. We will state some important results without proof, and the readers who are interested in further details are encouraged to refer to [Bro17,§2.4]. Let us first recall the following crucial property of the Wronskian ideal sheaf in [Bro17].

Lemma 1.6 ([Bro17, Lemma 2.4]). — Let X be a projective manifold equipped with a very ample line bundleL. Then

Supp O_X

k/w(X_k, L^k)

⊂X_k^sing, where X_k^sing is the set of singular k-jets of X_k.

Based on the above lemma, as was shown in [Bro17], Brotbek introduced a fonctorial birational morphism of the Demailly-Semple k-jet tower ν_k : ˆX_k → X_k by blowing-up the asymptotic Wronskian ideal sheaf w_∞(X_k), so that he was able to establish astrong Zariski open property for hyperbolicity. Indeed, Brotbek even built the strong Zariski open property for almost k-jet ampleness. We require the following results in [Bro17] to proceed further.

Theorem 1.7 ([Bro17, Propositions 2.10, 2.11 and 2.13]) Let X be a projective manifold.

(i) For any smooth closed submanifold Y ⊂X, the inclusion Y_k⊂X_k induces an inclu- sion Yˆ_k⊂Xˆ_k. Moreover, Yˆ_k is the strict transform of Y_k in Xˆ_k.

(ii) If

(∗) ∃a₀, . . . , a_k ∈N s.t. ν_k^∗O_X

k(a_k, . . . , a₁)⊗O_ˆ

Xk(−a₀F) is ample,

then X is almost k-jet ample. Here F is an effective divisor on Xˆ_k defined by O_ˆ

Xk(−F) =ν_k^∗w_∞(X_k).

(iii) Let X −→^ρ T be a smooth and projective morphism between non-singular varieties.

We denote by X^rel

k the Demailly-Semple k-jet tower of the relative directed variety (X, TX/T). Take ν_k : Xˆ^rel

k → X^rel

k to be the blow-up of the asymptotic Wron- skian ideal sheaf w_∞(X^rel

k ). Then for any t0 ∈ T writing Xt0 := ρ⁻¹(t0), we have ν_k⁻¹(X_t0,k) = ˆX_t0,k.

(iv) Property (∗) is a Zariski open property. Precisely speaking, in the same setting as above, if there exists t∈T such that X_t satisfies (∗), then there exists a non-empty Zariski open subset T₀ ⊂ T such that for any s ∈ T₀, X_s satisfies (∗) as well. In particular, X_s is almost k-jet ample for all s∈T₀.

2. An effective Nakamaye’s theorem

As mentioned in § 0, both [BDa17] and [Bro17] applied Nakamaye’s Theorem on the augmented base locus [Nak00] for families of zero-dimensional subschemes to provide a geometric control on base locus. In this section we render their noetherianity arguments effective.

We start by setting notations as in [Bro17, §3]. Consider V :=H⁰ P^N,O

P^N(δ)

, which can be identified with the space of homogeneous polynomials of degreeδ inC[z₀, . . . , z_N].

For anyJ ⊂ {0, . . . , N} we set

P_J :={[z₀, . . . , z_N]∈P^N |z_j = 0 if j ∈J}.

Given any ∆∈Gr_k+1(V) and [z]∈P^N, we denote by ∆([z]) = 0 onceP([z]) = 0 for any P ∈∆⊂V. Define the universal family of complete intersections to be

Y :={(∆,[z])∈Gr_k+1(V)×P^N |∆([z]) = 0}.

(2.1)

For anyJ ⊂ {0, . . . , N}, set

Y_J :=Y ∩(Gr_k+1(V)×P_J).

(2.2)

Let us denote by p : Y → Gr_k+1(V) and q : Y → P^N the projection maps. The next lemma is our starting point.

Lemma 2.1. — For any J ⊂ {0, . . . , N}, Y_J → P_J is a locally trivial holomorphic fibration with fibers isomorphic to the GrassmannianGr k+ 1,dim(V)−1

. In particular, Y_J is a smooth projective manifold.

Proof. — Any linear transformation g ∈GL_N+1(C) induces a natural action ˜g∈GL(V), hence also induces a biholomorphism ˆg of Gr_k+1(V). Observe that for any [z] ∈ P^N, ˆg maps the fiberq⁻¹([z]) toq⁻¹([g·z]) bijectively. Since GL_N+1(C) acts transitively onP^N, the fibrationq :Y →P^N can thus be trivialized locally.

Take a special point [e₀] := [1,0, . . . ,0]∈P^N. For anyP =P

|I|=δa_Iz^I ∈V,P([e₀]) = 0 if and only if the coefficient ofz₀^δinP is zero. If we denote byV₀the subspace ofV spanned by{z^I | |I|=δ, z^I 6=z₀^δ}, thenq⁻¹([e₀]) = Gr_k+1(V₀)≃Gr k+1,dim(V)−1

. The lemma is thus proved.

Observe that whenk+1>N,p:Y →Gr_k+1(V) is agenerically finite to onemorphism.

Let us denote byL be the very ample line bundle on Gr_k+1(V) which is the pull back of O(1) on P(Λ^k+1V) under the Pl¨ucker embedding Gr_k+1(V) ֒→ P(Λ^k+1V). Then p^∗L_↾Y

J

is a big and nef line bundle on Y_J for any J ⊂ {0, . . . , N}. Write pJ : Y_J → Gr_k+1(V) andq_J :Y_J →P_J for the natural projections, and define

EJ :={y ∈Y |dimy p⁻¹_J (pJ(y))

>0}

G^∞_J :=p_J(E_J)⊂Gr_k+1(V).

When J = ∅ we simply write E := E∅ and G^∞ :=G^∞_∅. By the definition of null locus [Laz04, Definition 10.3.4], E_J = Null(p^∗_JL). It then follows from Nakamaye’s theorem [Nak00] that

B₊(p^∗_JL) = Null(p^∗_JL) =E_J.

Observe that the line bundle L ⊠O_PN(1) on Gr_k+1(V) ×P^N is ample, and so is its restriction to Y_J. Hence by the definition of augmented base locus and noetherianity, there exists m_J ∈Nsuch that

Bs L^m⊠O_P

J(−1)_↾Y_J

=B₊(p^∗_JL) =E_J ⊂p⁻¹_J (G^∞), ∀m>m_J. (2.3)

We emphasize that the value M := max{mJ | J ⊂ {0, . . . , N}} concerns the second noetherianity argument in [Bro17] resulting in the loss of effective lower degree bounds d_Kob,n in Theorem 0.2.

Instead of requiring (2.3), we will provide a slightly weaker base control but with an effective estimate onM, which still remains valid in Brotbek’s proof (see [Bro17, Remark 3.13]).

Theorem 2.2. — When m>δ^k, for any J ⊂ {0, . . . , N}, one has Bs L^m⊠O_P

J(−1)_↾Y_J

⊂p⁻¹_J (G^∞_J ).

(2.4)

To prove Theorem 2.2, we construct sufficiently many global sections of L^m ⊠ O_P

J(−1)↾Y_J in an explicit manner to control their base locus. Precisely speaking, for any

∆∈/ G^∞_J , by definition p⁻¹_J (∆) is a finite set. We will show that for each m > δ^k there exists an effective divisorD_∆∈ |L^m⊠O_P

J(−1)_↾Y_J|so that D_∆∩p⁻¹_J (∆) =∅.

Let us first recall a version of projection formula inintersection theory, which is indeed a direct consequence of [Ful98, Example 8.1.7].

Theorem 2.3 (Projection formula). — Let f :X→ Y be a generically finite to one and surjective morphism between non-singular irreducible varieties, and x (resp. y) be cycle on X (resp. Y) of dimension k (resp. dim(X)−k). Then

deg

f_∗ f^∗(y)·x

= deg y·f_∗(x) ,

wheref^∗ and f_∗ are defined in the Chow group. When the scheme-theoretic inverse image f⁻¹(y) is of pure dimension dim(X)−k, one has f^∗(y) = [f⁻¹(y)].

Proof of Theorem 2.2. — We first deal with the case k+ 1 = N, and then reduce the general setting k+ 1>N to this case.

Claim 2.4. — When k+ 1 =N, H⁰ Y,L^m⊠O_PN(−1)↾Y

6=∅for all m>δ^N−1. Proof. — Let us pick a smooth curve C in GrN(V) of degree 1 with respect toL, given by

∆([t₀, t₁]) := Span(z^δ₁, z₂^δ, . . . z_N^δ₋₁, t₀z_N^δ +t₁z₀^δ),

where [t₀, t₁] ∈ P¹. Indeed, the curve C is the line in the Pl¨ucker embedding P(Λ^NV) defined by two vectorsz^δ₁∧ · · · ∧z_N^δ₋₁∧z^δ₀ and z₁^δ∧ · · · ∧z_N^δ in Λ^NV. HenceL ·C= 1.

Consider a hyperplane D in P^N given by {[z₀, . . . , z_N]|z₀+z_N = 0}. Since p :Y → Gr_N(V) is a generically finite to one and surjective morphism,p_∗q^∗Dis an effective divisor in GrN(V).

Since p⁻¹(C) has pure dimension 1, thenp^∗C is a 1-cycle in Y. An easy computation shows thatp^∗C andq^∗Dintersect only at the point

Span(z₁^δ, z₂^δ, . . . z^δ_N−1, z_N^δ + (−1)^δ+1z₀^δ)×[1,0, . . . ,0,−1]∈Y with multiplicity δ^N−1. Hencep^∗C·q^∗D=δ^N−1. By Theorem 2.3, one has

C·p∗(q^∗D) =p∗(p^∗C·q^∗D) =δ^N−1. Note that the Picard group Pic Gr_N(V)

≃Z is generated byL, which in turn implies p∗q^∗D∈ |L^δN−1|

(2.5)

by the fact thatL ·C= 1. It follows from Lemma 2.1 that q^∗Dis a smooth hypersurface inY. Since Supp(q^∗D) ⊂ Supp(p^∗p_∗q^∗D), p^∗p_∗q^∗D−q^∗D is thus an effective divisor of Y, and by (2.5)

p^∗p_∗q^∗D−q^∗D∈ |L^δN−1 ⊠O_PN(−1)_↾Y|.

(2.6)

The claim follows from the fact thatL is very ample.

The base locus of |L^δN−1 ⊠O

P^N(−1)_↾Y|can be well understood.

Claim 2.5. — For any m>δ^N−1, the base locus Bs L^m⊠O

P^N(−1)_↾Y

⊂p⁻¹(G^∞).

(2.7)

Proof. — For given any ∆₀ ∈/ G^∞,p⁻¹(∆₀) is a finite set by the definition of G^∞. Then one can take a general hyperplaneD∈ |O_PN(1)|such thatD∩q p⁻¹(∆₀)

=∅. By (2.6), Dgives rise to an effective divisor

p^∗p∗q^∗D−q^∗D∈ |L^δN−1 ⊠O_PN(−1)↾Y|.

For any ∆∈Gr_N(V), if we denote by

Int(∆) :={[z]∈P^N |∆([z]) = 0}, thenq p⁻¹(∆)

= Int(∆). Hence the conditionD∩q p⁻¹(∆₀)

=∅is equivalent to that Int(∆₀)∩D=∅. On the other hand, for any ∆∈Supp(p_∗q^∗D), one has Int(∆)∩D6=∅, and thus we conclude that ∆₀∈/ Supp(p_∗q^∗D). In particular,

p⁻¹(∆₀)∩Supp(p^∗p_∗q^∗D−q^∗D) =∅.

As ∆₀ is an arbitrary point outsideG^∞, we conclude that Bs L^δN−1 ⊠O

P^N(−1)_↾Y

⊂p⁻¹(G^∞).

SinceL is very ample, we have Bs L^m⊠O

P^N(−1)_↾Y

⊂Bs L^δN−1 ⊠O

P^N(−1)_↾Y

⊂p⁻¹(G^∞) for any m>δ^N−1. The claim is thus proved.

Let us deal with the general caseJ ) ∅. Without loss of generality we can assume that J ={n+ 1, . . . , N}. For any ∆₀ ∈p_J(Y_J)\G^∞_J , the setp⁻¹_J (∆₀) = Int(∆₀)∩P_J is finite.

We can also take a general hyperplane D ∈ |O_PN(1)| such that Int(∆₀)∩D∩P_J = ∅.

One can further choose a proper coordinate forP^N such that D= (z_n= 0).

By Lemma 2.1,q^∗_J(D∩P_J) is a smooth hypersurface inY_J. SetF :=p_J q⁻¹_J (D∩P_J) set- theoretically. Then for any effective divisor ˜H∈ |L^m|on Gr_N(V) such thatF ⊂Supp( ˜H) andpJ(Y_J)6⊂Supp( ˜H),

p^∗_J( ˜H)−q_J^∗(D∩P_J)∈ |L^m⊠O

P^N(−1)_↾Y_J| (2.8)

is an effective divisor ofY_J. However, it may happen that for any hyperplane ˜D∈ |O

P^N(1)|, all constructed divisors of the formp∗q^∗( ˜D) will always contain ∆0.