Algebraic properties of the algebra of differential operators

JEAN-PIERRE DEMAILLY

Abstract. In 2016, the conjecture has been settled in a different way by Damian Brotbek, making a more direct use of Wronskian differential operators and associated multiplier ideals; shortly after- wards, Ya Deng showed how the proof could be modified to yield an explicit value ofdn. We give here a short proof based on a drastic simplification of their ideas, along with a further improvement of Deng’s bound, namelydn=b¹₅(en)²ⁿ⁺²c.

Key words: Kobayashi hyperbolic variety, directed manifold, genus of a curve, jet bundle, jet differential, jet metric, Chern connection and curvature, negativity of jet curvature, variety of general type, Kobayashi conjecture, Green-Griffiths conjecture, Lang conjecture.

MSC Classification (2010): 32H20, 32L10, 53C55, 14J40

Contents

0. Introduction . . . 1

1. Semple tower associated to a directed manifold . . . 1

2. Algebraic properties of the algebra of differential operators . . . 7

3. Morse inequalities and the Green-Griffiths-Lang conjecture . . . 10

References . . . 28

0. Introduction

The goal of these lectures is to study the conjecture of Kobayashi [Kob70, Kob78] on the hyper- bolicity of generic hypersurfaces of high degree in projective space, and the related conjecture by Green-Griffiths [GG79] and Lang [Lan86] on the structure of entire curve loci.

1. Semple tower associated to a directed manifold 1.A. Category of directed manifolds

We start by recalling the main definitions concerning the category of directed varieties. For the sake of simplicity, we first assume that the objects under consideration are nonsingular.

1.1. Definition. A (complex) directed manifold is a pair (X, V) consisting of a n-dimensional complex manifold X equipped with a A morphism Φ : (X, V) →(Y, W) in the category of directed manifolds is a holomorphic map such thatΦ∗(V)⊂W.

It is eventually interesting to allow singularities forV. We then assume that there exists a dense Zariski open setX⁰ =XrY ⊂X such thatV_|X⁰ is a subbundle of (T_X))|X⁰ and the closureV_|X⁰ in the total space ofT_X is an anaytic subset. The rank r ∈ {0,1, . . . , n} of V is by definition the dimension ofVx at pointsx∈X⁰; the dimension may be larger at points x∈Y. This happens e.g.

onX=Cⁿ for the rank 1 linear spaceV generated by the Euler vector field: Vz =CP

16j6nzj ∂

∂zj

forz 6= 0, and V₀ = Cⁿ. The absolute situation is the case V = T_X and the relative situation is the case when V =T_X/S is the relative tangent space to a smooth holomorphic map X → S. In general, we can associate toV a sheafV =O(V)⊂ O(T_X) of holomorphic sections. No assumption

1

need be made on the Lie bracket tensor [•,•] :V × V → O(T_X)/V, i.e. we do not assume any kind of integrability forV. One of the most central conjectures in the theory is the

1.2. Generalized Green-Griffiths-Lang conjecture. Let (X, V) be a projective directed manifold whereV ⊂TX is nonsingular (i.e. a subbundle of TX). Assume that(X, V) is of “general type” in the sense thatK_V := detV^∗ is a big line bundle. Then there should exist a proper algebraic subvarietyY (X containing the imagesf(C) of all entire curves f :C→X tangent to V.

A similar statement can be made whenV is singular, but thenK_V has to be replaced by a certain (nonnecessarily invertible) rank 1 sheaf of “locally bounded” forms ofO(detV^∗), with respect to a smooth hermitian formω on T_X. The reader will find a more precise definition in [Dem18].

1.B. The 1-jet fonctor

The basic idea is to introduce a fonctorial process which produces a new complex directed manifold (X,e Ve) from a given one (X, V). The new structure (X,e Ve) plays the role of a space of 1-jets overX. We let

(1.3) Xe =P(V), Ve ⊂TXe

be the projectivized bundle of lines of V, together with a subbundle Ve of TXe defined as follows:

for every point (x,[v])∈Xe associated with a vector v∈Vxr{0}, (1.3⁰) Ve_(x,[v])=

η ∈TX,e (x,[v]);dπx(η)∈Cv , Cv⊂Vx ⊂T_X,x,

where π :Xe = P(V) → X is the natural projection and π∗ : TXe → π^∗TX is its differential. On Xe =P(V) we have a tautological line bundleOXe(−1)⊂π^∗V ⊂π^∗T_X such thatOXe(−1)_(x,[v]) =Cv.

The bundleVe is characterized by the exact sequences 0−→TX/Xe −→TXe dπ

−→π^∗T_X −→0,

|| ∪ ∪

0−→TX/Xe −→Ve −→ O^dπ Xe(−1)−→0, (1.4)

0−→ OXe −→π^∗V ⊗ OXe(1)−→TX/Xe −→0, (1.4⁰)

where TX/Xe denotes the relative tangent bundle of the fibration π : Xe → X. The first sequence is a direct consequence of the definition ofVe, whereas the second is a relative version of the Euler exact sequence describing the tangent bundle of the fibersP(Vx). From these exact sequences we infer

(1.5) dimXe =n+r−1, rankVe = rankV =r, and by taking determinants we find det(TX/Xe ) =π^∗detV ⊗ OXe(r), hence (1.6) detVe =π^∗detV ⊗ OXe(r−1).

Clearlyπ : (X,e Ve)→(X, V) is a morphism of complex directed manifolds and this construction is fonctorial with respect to morphisms Φ : (X, V)→(Y, W) for which Φ∗ is injective.

1.C. Lifting of curves to the 1-jet bundle

Suppose that we are given a holomorphic curve f : D(0, R) → X parametrized by the disk D(0, R) of centre 0 and radius R in the complex plane, and that f is a tangent curve of the directed manifold, i.e., f⁰(t) ∈ V_f(t) for every t ∈ D(0, R). If f is nonconstant, there is a well defined and unique tangent line [f⁰(t)] for every t, even at stationary points, and the map

(1.7) fe:D(0, R)→X,e t7→fe(t) := (f(t),[f⁰(t)])

is holomorphic (at a stationary point t₀, we just write f⁰(t) = (t−t₀)^su(t) with s ∈ N^∗ and u(t0) 6= 0, and we define the tangent line at t0 to be [u(t0)], hence fe(t) = (f(t),[u(t)]) near

t₀; even for t = t₀, we still denote [f⁰(t₀)] = [u(t₀)] for simplicity of notation). By definition f⁰(t)∈ OXe(−1)_f(t)e =Cu(t), hence the derivativef⁰ defines a section

(1.8) f⁰ :T_D(0,R)→fe^∗OXe(−1).

Moreover π◦fe=f, therefore

π∗fe⁰(t) =f⁰(t)∈Cu(t) =⇒fe⁰(t)∈Ve(f(t),u(t))=Vef(t)e

and we see that feis a tangent trajectory of (X,e Ve). We say that feis the canonical lifting of f to X. Conversely, ife g : D(0, R) → Xe is a tangent trajectory of (X,e Ve), then by definition of Ve we see that f =π◦g is a tangent trajectory of (X, V) and that g =fe(unless g is contained in a vertical fiberP(V_x), in which case f is constant).

For any pointx₀ ∈ X, there are local coordinates (z₁, . . . , z_n) on a neighborhood Ω of x₀ such that the fibers (Vz)z∈Ω can be defined by linear equations

(1.9) V_z =n

v= X

16j6n

v_j ∂

∂z_j ;v_j = X

16k6r

a_jk(z)v_k forj=r+ 1, . . . , no ,

where (ajk) is a holomorphic (n−r)×r matrix. It follows that a vector v ∈ Vz is completely determined by its first r components (v₁, . . . , v_r), and the affine chart v_j 6= 0 of P(V)_Ω can be described by the coordinate system

(1.10)

z₁, . . . , z_n;v₁

v_j, . . . ,vj−1

v_j ,v_j+1

v_j , . . . ,v_r v_j

.

Let f ' (f₁, . . . , f_n) be the components of f in the coordinates (z₁, . . . , z_n) (we suppose here R so small thatf(D(0, R))⊂Ω). It should be observed that f is uniquely determined by its initial value x and by the first r components (f₁, . . . , f_r). Indeed, as f⁰(t) ∈ V_f(t), we can recover the other components by integrating the system of ordinary differential equations

(1.11) f_j⁰(t) = X

16k6r

ajk(f(t))f_k⁰(t), j > r,

on a neighborhood of 0, with initial dataf(0) =x. We denote by m=m(f, t0) themultiplicity of f at any pointt₀ ∈D(0, R), that is,m(f, t₀) is the smallest integerm∈N^∗ such thatf_j^(m)(t₀)6= 0 for some j. By (1.11), we can always suppose j ∈ {1, . . . , r}, for example f_r^(m)(t₀) 6= 0. Then f⁰(t) = (t−t0)^m−1u(t) with ur(t0) 6= 0, and the lifting feis described in the coordinates of the affine chartv_r 6= 0 ofP(V)_Ω by

(1.12) fe'

f₁, . . . , f_n;f₁⁰

f_r⁰, . . . ,f_r−1⁰ f_r⁰

.

1.D. The Semple tower

Following [Dem95], we define inductively the projectivized k-jet bundle Xk (or Semple k-jet bundle) and the associated subbundleV_k⊂T_Xk by

(1.13) (X0, V0) = (X, V), (Xk, Vk) = (Xek−1,Vek−1).

In other words, (X_k, V_k) is obtained from (X, V) by iterating k-times the lifting construction (X, V)7→(X,e Ve) described in§1.B. By (1.3–1.5), we find

(1.14) dimX_k=n+k(r−1), rankV_k=r, together with exact sequences

0−→T_Xk/Xk−1 −→V_k−−−−→ O^(πk)∗ _Xk(−1)−→0, (1.15)

0−→ O_Xk −→π_k^∗Vk−1⊗ O_Xk(1)−→T_Xk/Xk−1 −→0.

(1.15⁰)

whereπ_k is the natural projection π_k :X_k →Xk−1 and (π_k)∗ its differential. Formula (1.6) yields (1.16) detVk=π_k^∗detVk−1⊗ O_Xk(r−1).

Every nonconstant tangent trajectoryf :D(0, R)→X of (X, V) lifts to a well defined and unique tangent trajectory f_[k] :D(0, R) → X_k of (X_k, V_k). Moreover, the derivative f_[k−1]⁰ gives rise to a section

(1.17) f_[k−1]⁰ :T_D(0,R)→f_[k]^∗ O_Xk(−1).

In coordinates, one can computef_[k] in terms of its components in the various affine charts (1.10) occurring at each step: we get inductively

(1.18) f_[k]= (F1, . . . , FN), f_[k+1] =

F1, . . . , FN,F_s⁰₁

F_s⁰_r, . . . ,F_s⁰_r−1 F_s⁰_r

whereN =n+k(r−1) and{s₁, . . . , s_r} ⊂ {1, . . . , N}. Ifk>1,{s₁, . . . , s_r}contains the lastr−1 indices of{1, . . . , N}corresponding to the “vertical” components of the projectionXk→Xk−1, and in general, s_r is an index such that m(F_sr,0) =m(f_[k],0), that is, F_sr has the smallest vanishing order among all componentsF_s (s_r may be vertical or not, and the choice of{s₁, . . . , s_r}need not be unique).

By definition, there is a canonical injection O_Xk(−1) ,→ π_k^∗Vk−1, and a composition with the projection (πk−1)∗ (analogue for order k−1 of the arrow (πk)∗ in the sequence (1.15)) yields for allk>2 a canonical line bundle morphism

(1.19) O_Xk(−1),−→π_k^∗Vk−1

(πk)^∗(dπk−1)

−−−−−−→ π^∗_kO_Xk−1(−1),

which admits precisely D_k=P(T_{Xk−1/Xk−2})⊂P(Vk−1) =X_k as its zero divisor (clearly, D_k is a hyperplane subbundle ofX_k). Hence we find

(1.20) O_Xk(1) =π_k^∗O_Xk−1(1)⊗ O(D_k).

Now, we consider the composition of projections

(1.21) πk,j =πj+1◦ · · · ◦πk−1◦πk:Xk−→Xj.

Then π_k,0 : X_k → X0 = X is a locally trivial holomorphic fiber bundle over X, and the fibers X_k,x =π_k,0⁻¹(x) are k-stage towers ofP^r−1-bundles. Since we have (in both directions) morphisms (C^r, T_C^r) ↔ (X, V) of directed manifolds which are bijective on the level of bundle morphisms, the fibers are all isomorphic to a “universal” non singular projective algebraic variety of dimension k(r−1) which we will denote byRr,k; it is not hard to see thatRr,kis rational, since (1.18) provides affine charts ofRr,k that are isomorphic toC^k(r−1).

1.22. Remark. When (X, V) is singular, one can easily extend the construction of the Semple tower by fonctoriality. In fact, assume thatX is a closed analytic subset of some open setZ⊂C^N, and thatX⁰ ⊂Xis a Zariski open subset on whichV_X⁰ is a subbundle ofT_X⁰. Then we consider the injection of the nonsingular directed manifold (X⁰, V⁰) into the absolute structure (Z, W),W =TZ. This yields an injection (X_k⁰, V_k⁰) ,→ (Z_k, W_k), and we simply define (X_k, V_k) to be the closure of (X_k⁰, V_k⁰) into (Z_k, W_k). It is not hard to see that this is indeed a closed analytic subset of the same dimensionn+k(r−1), where r= rankV⁰.

1.E. Jet bundles and jet differentials

Following Green-Griffiths [GrGr79], we consider the bundleJ_kX →Xofk-jets of germs of para- metrized curves inX, i.e., the set of equivalence classes of holomorphic mapsf : (C,0)→(X, x), 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 projection map

J_kX → X is simply f 7→ f(0). If (z₁, . . . , z_n) are local holomorphic coordinates on an open set Ω⊂X, the elementsf of any fiber JkXx,x∈Ω, can be seen as Cⁿ-valued maps

f = (f1, . . . , fn) : (C,0)→Ω⊂Cⁿ,

and they are completetely determined by their Taylor expansion of orderkatt= 0 f(t) =x+t f⁰(0) + t²

2!f⁰⁰(0) +· · ·+t^k

k!f^(k)(0) +O(t^k+1).

In these coordinates, the fiber J_kXx can thus be identified with the set of k-tuples of vectors (ξ₁, . . . , ξ_k) = (f⁰(0), . . . , f^(k)(0)) ∈ (Cⁿ)^k. It follows that J_kX is a holomorphic fiber bundle with typical fiber (Cⁿ)^k over X. However, J_kX is not a vector bundle for k >2, because of the nonlinearity of coordinate changes: a coordinate changez7→w= Ψ(z) onX induces a polynomial transition automorphism on the fibers ofJ_kX, given by a formula

(1.23) (Ψ◦f)^(j)= Ψ⁰(f)·f^(j)+

s=j

X

s=2

X

j1+j2+···+j_s=j

c_j1...jsΨ^(s)(f)·(f^(j1), . . . , f^(js))

with suitable integer constants cj1...js (this is easily checked by induction on s). According to the above philosophy, we introduce the concept of jet bundle in the general situation of complex directed manifolds.

1.24. Definition.Let(X, V)be a complex directed manifold. We defineJ_kV →X to be the bundle of k-jets of curves f : (C,0)→X which are tangent toV, i.e., such that f⁰(t)∈V_f(t) for allt in a neighborhood of0, together with the projection map f 7→f(0) onto X.

It is easy to check thatJ_kV is actually a subbundle ofJ_kX. In fact, by using (1.11), we see that the fibersJkVx are parametrized by

(f₁⁰(0), . . . , f_r⁰(0)); (f₁⁰⁰(0), . . . , f_r⁰⁰(0));. . .; (f₁^(k)(0), . . . , f_r^(k)(0))

∈(C^r)^k

for allx∈Ω, hence J_kV is a locally trivial (C^r)^k-subbundle of J_kX. Alternatively, we can pick a local holomorphic connection ∇on V such that for any germsw =P

16j6nw_j∂z^∂

j ∈ O(T_X,x) and v=P

16λ6rv_λe_λ ∈ O(V)_x in a local trivializing frame (e₁, . . . , e_r) of V_Ω we have

(1.25) ∇_wv(x) = X

16j6n,16λ6r

w_j∂v_λ

∂zj

e_λ(x) + X

16j6n,16λ,µ6r

Γ^µ_jλ(x)w_jv_λe_µ(x).

We can of course take the frame obtained from (1.9) by lifting the vector fields∂/∂z₁, . . . , ∂/∂z_r, and the “trivial connection” given by the zero Christoffel symbolds Γ = 0. One then obtains a trivializationJ^kV_Ω 'V_Ω^⊕k by considering

(1.26) J_kV_x3f 7→(ξ₁, ξ₂, . . . , ξ_k) = (∇f(0),∇²f(0), . . . ,∇^kf(0))∈V_x^⊕k and computing inductively the successive derivatives∇f(t) =f⁰(t) and ∇^sf(t) via

∇^sf = (f^∗∇)_d/dt(∇^s−1f) = X

16λ6r

d dt

∇^s−1f

λe_λ(f) + X

16j6n,16λ,µ6r

Γ^µ_jλ(f)f_j⁰

∇^s−1f

λe_µ(f).

This identification depends of course on the choice of∇and cannot be defined globally in general (unless we are in the rare situation whereV has a global holomorphic connection).

LetGk be the group of germs of k-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^∗, aj ∈C, j >2,

in which the composition law is taken modulo termst^j of degreej > k. ThenGk is ak-dimensional nilpotent complex Lie group, which admits a natural fiberwise right action onJ_kV

(1.27) JkV ×Gk→JkV, (f, ϕ)7→f◦ϕ.

There is a semidirect decompositionGk=G⁰_kn C^∗ given by a split exact sequence 1→G⁰k→Gk→C^∗→1

where Gk → C^∗ is the obvious morphism ϕ 7→ ϕ⁰(0), the commutator group G⁰_k = [Gk,Gk] is the group of k-jets of biholomorphisms tangent to the identity, and C^∗ ⊂ Gk is the (nonnormal) subgroup of homotheties ϕ(t) = λt. The corresponding action of C^∗ on k-jets is described in coordinates by

λ·(ξ1, ξ2, . . . , ξk) = (λξ1, λ²ξ2, . . . , λ^kξk), ξs=∇^sf(0).

Following [GrGr79], we introduce the bundleE_k,m^GGV^∗→X of polynomialsP(x;ξ1, . . . , ξ_k) that are homogeneous on the fibers ofJ_kV of weighted degree mwith respect to the C^∗ action, i.e.

(1.28) P(x;λξ1, . . . , λ^kξk) =λ^mP(x;ξ1, . . . ξk), in other words they are polynomials of the form

(1.29) P(x;ξ1, . . . ξ_k) = X

|α1|+2|α2|+···+k|α_k|=m

aα1...α_k(x)ξ^α₁¹ξ₂^α2· · ·ξ_k^αk where ξs = (ξs,1, . . . , ξs,r) ∈C^r 'Vx and ξ_s^αs =ξ^α_s,1^s,1. . . ξs,r^αs,r, |α_s|=P

16j6rαs,j. Sections of the sheaf O(E_k,m^GGV^∗) can also be viewed as algebraic differential operators acting on germs of curves f : (C,0)→X tangent toV, by putting

(1.29⁰) P(f)(t) = X

|α₁|+2|α₂|+···+k|α_k|=m

a_α1...αk(f(t)) (∇f(t))^α1(∇²f(t))^α2· · ·(∇^kf(t))^αk where thea_α1...αk(x) are holomorphic inx. With the graded algebra bundleE_k,•^GGV^∗=L

mE_k,m^GGV^∗ we associate an analytic fiber bundle

(1.30) X_k^GG:= Proj(E_k,•^GGV^∗) = (J_kV r{0})/C^∗

overX, which has weighted projective spaces P(1^[r],2^[r], . . . , k^[r]) as fibers; here JkV r{0} is the set of nonconstant jets of orderk. As such, it possesses a tautological sheaf O_XGG

k (1) [the reader should observe however thatO_XGG

k (m) is invertible only whenm is a multiple of lcm(1,2, . . . , k)].

1.31. Proposition. By construction, if π_k : X_k^GG → X is the natural projection, we have the direct image formula

(π_k)∗O_XGG

k (m) =O(E_k,m^GGV^∗) for allk and m.

In the geometric context, we are not really interested in the bundles (JkVr{0})/C^∗ themselves, but rather on their quotients (J_kVr{0})//Gk(would such nice complex space quotients exist!). In fact the following fundamental result from [Dem95] shows that the Semple bundleX_k constructed above plays the role of such a quotient.

1.32. Theorem and Definition. Let E_k,mV^∗ ⊂ E_k,m^GGV^∗ be the set of polynomial differential operators f 7→ P(f) that are invariant under arbitrary changes of parametrization, i.e., such that for everyϕ∈Gk

(∗) P(f ◦ϕ) = (ϕ⁰)^mP(f)◦ϕ

[the weighted degree condition (1.28) being the special case when ϕ(t) =λt, λ∈C^∗].

Let π_k,0 :X_k→ X be the Semple jet bundles defined above and let J_kV^reg be the bundle of regular k-jets of maps f : (C,0)→(X, V), that is, jetsf such that f⁰(0)6= 0. Then

(i) The quotient JkV^reg/Gk has the structure of a locally trivial bundle over X, and there is a holomorphic embedding J_kV^reg/Gk ,→X_k over X, which identifies J_kV^reg/Gk with X_k^reg (thus Xk is a relative compactification of JkV^reg/Gk over X).

(ii)The direct image sheaf

(π_k,0)∗O_Xk(m)' O(E_k,mV^∗) can be identified with the sheaf of holomorphic sections of Ek,mV^∗.

(iii)For every m>1, the relative base locus of the linear system |O_Xk(m)|is equal to the setX_k^sing of singular k-jets [one has X_k^sing =∅ for k= 1]. Moreover,O_Xk(1) is relatively big over X.

Sketch of proof. We refer to [Dem95] for details. In order to prove (i) and (ii), the main point is that the lifts f_[k]:D(0, R) →(Xk, Vk) of a curve f :D(0, R) →(X, V) are defined inductively by f_[k] = (f_[k−1],[f_[k−1]⁰ ]), hence for any change of variable ϕ : D(0, R⁰) → D(0, R), they satisfy the relations

(f◦ϕ)_[k]=f_[k]◦ϕ, (f◦ϕ)⁰_[k−1] =ϕ⁰f_[k−1]⁰ ◦ϕ∈ O_Xk(−1)⊂π^∗_k,k−1Vk−1. We conclude that there is a well defined set-theoretic map

(1.33) J_kV^reg/Gk→X_k^reg, f mod Gk7→f_[k](0).

Given a holomorphic sectionσ∈H⁰(π_k,0⁻¹(U),O_Xk(m)), we can then associate a differential operator (1.34) P(f) =σ(f_[k])·(f_[k−1]⁰ )^m.

Clearly, condition (∗) is satisfied and in particularP is homogeneous of degreem on JkV^reg; such a holomorphic function must be a homogeneous polynomial on the fibers.

2. Algebraic properties of the algebra of differential operators 2.A. Green-Griffiths and Semple algebras

By construction, theGreen-Griffiths graded algebra

(2.1) A^GG_k V^?= M

m∈Z

E_k,m^GGV^?

of differential operators is fiberwise isomorphic to the polynomial ring C[f₁⁰, . . . , f_r⁰, f₁⁰⁰, . . . , f_r⁰⁰, . . . , f₁^(k), . . . , f_r^(k)]

and in particular it is finitely generated. More geometrically, we get a holomorphic filtration of E_k,m^GGV^? by considering the partial degree of P(f) in terms of the last derivativef^(k) and putting

F^a(E_k,m^GGV^?) =

P ∈E_k,m^GGV^?; deg_f(k)P(f)6a .

Then the graded pieces are polynomials of the formQ(f⁰, . . . , f^(k−1))(f^(k))^αk,|α_k|=a, i.e.

G^a(E_k,m^GGV^?)'E_k−1,m−ka^GG V^?⊗S^aV^∗.

We can then inductively combine the successive filtrations obtained via the partial degrees inf^(k), f^(k−1), . . . , f⁽¹⁾ =f⁰ to get a full decomposition

(2.2) G^•(E_k,m^GGV^?)' M

a=(a1,...,ak)∈Nk a1+2a2+···+kak=m

S^a1V^∗⊗ · · · ⊗S^akV^∗.

Hence A^GG_k V^? is just locally isomorphic to a k-fold tensor product of symmetric algebras S^•V^∗. We define theSemple algebrato be the graded subalgebra of A^GG_k such that

(2.3) A_kV^? = (A^GG_k )^G0k = M

m∈Z

E_k,mV^?,

in particular A₁V^? = A^GG₁ V^? = S^•V^∗. As G⁰_k is a non reductive group, it is a priori unclear whetherA_kV^? is finitely generated fork>2.

the subalgebra ofG⁰_k-invariant differential operators is finitely generated. This can be checked by hand ([Dem07a], [Dem07b]) forn= 2 andk64. Rousseau [Rou06] also checked the casen= 3, k= 3, and then Merker [Mer08, Mer10] proved the finiteness forn= 2,3,4,k64 andn= 2,k= 5.

Recently, B´erczi and Kirwan [BeKi12] made an attempt to prove the finiteness in full generality, but it appears that the general case is still unsettled.

Fix coordinates (z₁, . . . , z_n) near a point x₀ ∈X, such that V_x0 = Vect(∂/∂z₁, . . . , ∂/∂z_r). Let f = (f1, . . . , fn) be a regular k-jet tangent to V. Then there exists i ∈ {1,2, . . . , r} such that f_i⁰(0)6= 0, and there is a unique reparametrizationt=ϕ(τ) such that f ◦ϕ=g = (g₁, g₂, . . . , g_n) with gi(τ) = τ (we just express the curve as a graph over the zi-axis, by means of a change of parameter τ =fi(t), i.e. t = ϕ(τ) = f_i⁻¹(τ)). Suppose i = r for the simplicity of notation. The space X_k is ak-stage tower of P^r−1-bundles. In the corresponding inhomogeneous coordinates on these P^r−1’s, the pointf_[k](0) is given by the collection of derivatives

(g₁⁰(0), . . . , g⁰_r−1(0)); (g₁⁰⁰(0), . . . , g⁰⁰_r−1(0));. . .; (g₁^(k)(0), . . . , g^(k)_r−1(0)) .

[Recall that the other components (g_r+1, . . . , g_n) can be recovered from (g₁, . . . , g_r) by integrating the differential system (5.10)]. Thus the mapJ_kV^reg/Gk → X_k is a bijection onto X_k^reg, and the fibers of these isomorphic bundles can be seen as unions ofraffine charts '(C^r−1)^k, associated with each choice of the axisz_i used to describe the curve as a graph. The change of parameter formula

d dτ = _f0¹

r(t) d

dt expresses all derivativesg_i^(j)(τ) =d^jgi/dτ^jin terms of the derivativesf_i^(j)(t) =d^jfi/dt^j (g₁⁰, . . . , g⁰_r−1) =f₁⁰

f_r⁰, . . . ,f_r−1⁰ f_r⁰

; (g₁⁰⁰, . . . , g⁰⁰_r−1) =f₁⁰⁰f_r⁰−f_r⁰⁰f₁⁰

f_r⁰³ , . . . ,f_r−1⁰⁰ f_r⁰ −f_r⁰⁰f_r−1⁰ f_r⁰³

; . . . ; (3.12)

(g₁^(k), . . . , g^(k)_r−1) =

f₁^(k)f_r⁰−fr^(k)f₁⁰ fr^0k+1

, . . . ,f_r−1^(k)f_r⁰ −f_r^(k)f_r−1⁰ fr^0k+1

+ (order< k).

2.B. Wronskians

LetU be an open set of X, dimX =n, and s₀, . . . , s_k ∈ O_X(U) be holomorphic functions. To these functions, we can associate a Wronskian operator of orderk defined by

(4.1) Wk(s0, . . . , sk)(f) =

s₀(f) s₁(f) . . . s_k(f) D(s0(f)) D(s1(f)) . . . D(sk(f))

... ...

D^k(s₀(f)) D^k(s₁(f)) . . . D^k(s_k(f))

wheref :t7→f(t)∈U ⊂X is a germ of holomorphic curve (or a k-jet of curve), andD= _dt^d. For a biholomorphic change of variableϕof (C,0), we find by induction on` a polynomial differential operatorQ<sub>`,s</sub>of order6`acting on ϕsatisfying

D^{`</sup>(s<sub>j</sub>(f◦ϕ)) =ϕ<sup>0`}D^`(s_j(f))◦ϕ+X

s<`

p_`,s(ϕ)D^s(s_j(f))◦ϕ.

It follows easily from there that

W_k(s₀, . . . , s_k)(f◦ϕ) = (ϕ⁰)^1+2+···+kW_k(s₀, . . . , s_k)(f)◦ϕ,

hence W_k(s₀, . . . , s_k)(f) is an invariant differential operator of degree k⁰ = ¹₂k(k+ 1). Especially, we get in this way a section that we denote

(4.2) W_k(s0, . . . , s_k) =

s₀ s₁ . . . s_k D(s0) D(s1) . . . D(sk)

... ...

D^k(s₀) D^k(s₁) . . . D^k(s_k)

∈H⁰(U, E_k,k⁰T_X^∗).

2.2. Proposition.These Wronskian operators satisfy the following properties.

(a)W_k(s₀, . . . , s_k) isC-multilinear and alternate in (s₀, . . . , s_k).

(b)For any g∈ O_X(U), we have

Wk(gs0, . . . , gsk) =g^k+1Wk(s0, . . . , sk).

Property 2.2 (b) is an easy consequence of the Leibniz formula D^`(g(f)sj(f)) =

`

X

k=0

` k

D^k(g(f))D^`−k(sj(f)),

by performing linear combinations of rows in the determinants. This property implies in its turn that one can define more generally an operator

(2.3) W_k(s₀, . . . , s_k)∈H⁰(U, E_k,k⁰T_X^∗ ⊗L^k+1)

for any (k+ 1)-tuple of sectionss0, . . . , sk∈H⁰(U, L) of a holomorphic line bundleL→X. In fact, when we compute the Wronskian in a local trivialization of L_U, Property 4.3 (b) shows that the determinant is independent of the trivialization. Moreover, if g ∈H⁰(U, G) for some line bundle G→X, we have

(2.4) W_k(gs₀, . . . , gs_k) =g^k+1W_k(s₀, . . . , s_k)∈H⁰(U, E_k,k⁰T_X^∗ ⊗L^k+1⊗G^k+1).

2.C. Brackets

IfP is a differential operator given by a section ofO(E^GG_k,mV^?), we defineDP to be its “obvious”

derivative

(2.5) DP(f) =P(f)⁰, i.e. DP(f)(t) = d

dtP(f)(t).

The operatorDP is then a section ofO(E_k,m^GGV^?). If P is a G⁰_k-invariant operator in O(E_k,mV^?), the relationP(f◦ϕ) =ϕ^0mP(f)◦ϕimplies

DP(f ◦ϕ) = (ϕ^0mP(f)◦ϕ)⁰=ϕ^0m+1DP(f)◦ϕ+mϕ^0m−1ϕ⁰⁰P(f)◦ϕ,

therefore DP is no longer an invariant operator (unless m = 0, in which case DP is of degree 1).

However, if P, Q are G⁰_k-invariant operators of respective degrees δP, δQ; it is easy to check that theirbracketdefined as

(2.6) [P, Q] =δPP(DQ)−δQQ(DP)

is againG⁰_k-invariant, of degreeδ_P +δ_Q+ 1. This can be seen by observing that the termsϕ⁰⁰(. . .) coming fromDP and DQcancel, or by noticing that

(2.6⁰) [P, Q] =P^δQ+1Q^−δP+1D Q^δP

P^δQ

where Q^δP/P^δQ is homogeneous of degree 0 (i.e. Gk-invariant). A straightforward albeit tedious calculation shows that this bracket satisfies the usual Jacobi identity

(2.7) [P,[Q, R]] + [Q,[R, P]] + [R,[P, Q]] = 0.

Formula (2.6⁰) has the advantage that for any line bundle L(or even any Q-line bundleL) and P ∈H⁰(X, E_k,δPV^∗⊗L^δP), Q∈H⁰(X, E_k,δQV^∗⊗L^δQ)

we also get globally defined brackets

[P, Q]∈H⁰(X, E_{k+1,δP+δQ+1}V^∗⊗L^δP+δQ).

Another special case is the “degree 0” bracket associated with two sectionsσ₁, σ₂ ∈H⁰(X, L) (2.8) τ = [σ1, σ2] =σ1Dσ2−σ2Dσ1 =σ₁²D

σ2

σ₁

∈H⁰(X, E1,1T_X^∗ ⊗L²) =H⁰(X, T_X^∗ ⊗L²).

In a similar way, given integers a, b > 1 and families of sections σj ∈ H⁰(X, L^a), 16 j 6k−2, τ_j ∈H⁰(X, V^∗⊗L^b) =H⁰(X, E_1,1V^∗⊗L^b), 16j 6k, we define inductively iterated brackets (2.9) Bk(σ1, . . . , σk−2;τ1, . . . , τk)∈H⁰(X, Ek,2k−1V^∗⊗L^(k−2)a+kb), k>2,

by putting

(2.9⁰) B₂(τ₁, τ₂) = [τ₁, τ₂] =τ₁²D τ₂

τ₁

,

and, fork>3 andc_k = (k−3)−(k−2)b/a∈Q, (2.9⁰⁰) Bk(σ1, . . . , σk−2;τ1, . . . , τk) =σ₁^ck+1τ₁^2k−2D

B_k−1(σ₂, . . . , σ_k−2;τ₂, . . . , τ_k) σ^c₁^kτ₁^2k−3

.

IfL is very ample, by (2.8), there are many such sections when we take a= 1,b= 2,ck = 1−k, and we then get sections Bk(σ1, . . . , σk−2;τ1, . . . , τk) ∈ H⁰(X, Ek,2k−1V^∗⊗L^3k−2) whose degrees m= 2k−1 grow linearly withk, as well as the correcting twist L^3k−2.

3. Morse inequalities and the Green-Griffiths-Lang conjecture 3.A. Statement of Morse inequalities

One of the main purpose of holomorphic Morse inequalities is to provide estimates of cohomology groups with values in high tensor powers of a given line bundleL, once a smooth hermitian metric hon L is given. We denote by ΘL,h=−_2πⁱ ∂∂logh the (1,1)-curvature form ofh.

3.1. Holomorphic Morse inequalities ([Dem85]). Let X be a compact complex manifolds, E→X a holomorphic vector bundle of rankr, and(L, h) a hermitian line bundle. The dimensions h^q(X, E⊗L^m) of cohomology groups of the tensor powers E⊗L^m satisfy the following asymptotic estimates asm→+∞ :

(3.1 WM)Weak Morse inequalities:

h^q(X, E⊗L^m)6rmⁿ n!

Z

X(L,h,q)

(−1)^qΘⁿ_L,h+o(mⁿ) . (3.1 SM) Strong Morse inequalities:

X

06j6q

(−1)^q−jh^j(X, E⊗L^m)6rmⁿ n!

Z

X(L,h,6q)

(−1)^qΘⁿ_L,h+o(mⁿ) . (3.1 RR) Asymptotic Riemann-Roch formula:

χ(X, E⊗L^m) := X

06j6n

(−1)^jh^j(X, E⊗L^m) =rmⁿ n!

Z

X

Θⁿ_L,h+o(mⁿ) .

Moreover (cf. Bonavero’s PhD thesis [Bon93]), if h = e^−ϕ is a singular hermitian metric with analytic singularities of pole set Σ = ϕ⁻¹(−∞), the estimates still hold provided all cohomology groups are replaced by cohomology groupsH^q(X, E⊗L^m⊗ I(h^m)) twisted with the corresponding L² multiplier ideal sheaves

I(h^m) =I(kϕ) =

f ∈ O_X,x, ∃V 3x, Z

V

|f(z)|²e^−mϕ(z)dλ(z)<+∞ ,

and provided the Morse integrals are computed on the regular locus of h, namely restricted to X(L, h, q)rΣ :

Z

X(L,h,q)rΣ

(−1)^qΘⁿ_L,h.

The special case of 3.1 (SM) whenq= 1 yields a very useful criterion for the existence of sections of large multiples ofL.

3.2. Corollary. Let L → X be a holomorphic line bundle equipped with a singular hermitian metric h=e^−ϕ with analytic singularities of pole set Σ =ϕ⁻¹(−∞). Then we have the following lower bounds

(a)at the h⁰ level :

h⁰(X, E⊗L^m)>h⁰(X, E⊗L^m⊗ I(h^m))

>h⁰(X, E⊗L^m⊗ I(h^m))−h¹(X, E⊗L^m⊗ I(h^m))

>rkⁿ n!

Z

X(L,h,61)rΣ

Θⁿ_L,h−o(kⁿ) . Especially Lis big as soon asR

X(L,h,61)rΣΘⁿ_L,h >0 for some singular hermitian metrichonL.

(b)at the h^q level :

h^q(X, E⊗L^m⊗ I(h^m))>rkⁿ n!

X

j=q−1,q,q+1

(−1)^q Z

X(L,h,j)rΣ

Θⁿ_L,h−o(kⁿ) .

The goal of this section is to study the existence and properties of entire curvesf :C→Xdrawn in a complex irreducible n-dimensional variety X, and more specifically to show that they must satisfy certain global algebraic or differential equations as soon asX is projective of general type.

By means of holomorphic Morse inequalities and a probabilistic analysis of the cohomology of jet spaces, it is possible to prove a significant step of the generalized Green-Griffiths-Lang conjecture.

The use of holomorphic Morse inequalities was first suggested in [Dem07a], and then carried out in an algebraic context by S. Diverio in his PhD work ([Div08, Div09]). The general more analytic and more powerful results presented here first appeared in [Dem11, Dem12].

3.B. Positively curved hermitian metric on the Semple tautological bundle To start with, we consider the directed variety (C^r, T_C^r) when C^r is equipped with its standard hermitian metric. Pick a randomk-jet

(3.3) f(t) =x+tξ₁+t²ξ₂+· · ·+t^kξ_k+O(t^k+1), ξ_s∈C^r, 16s6k.

Given ε₁, . . . , ε_k > 0, we get a natural Finsler metric on the tautological Green-Griffiths bundle O_GG,(Cr)k(−1) of (C^r, T_C^r) by putting

(3.4) kfk^GG_ε,p =

k

X

s=1

(ε_s|ξ_s|)^2p/s 1/2p

.

In fact ifλ·f denotes thet7→f(λt), we do have the required homogeneity property under theC^∗ action, namelykλ·fk^GG_ε,p =|λ| kfk^GG_ε,p. The choice of a suitable integerp (e.g.p= lcm(1,2, . . . , k) or a multiple) yields a smooth metric onO_GG,(Cr)k(−1). However, formula (3.4) is notG⁰_kinvariant and therefore cannot be used to construct a metric on the Semple tautological bundleO_(Cr)k(−1) of (C^r, T_C^r). Let us assume that we have a regular k-jet, i.e. that ξ1 6= 0. By composing f with a suitable element ϕ(t) = t+a2t² +. . . +akt^k+O(t^k+1) and applying a Gram-Schmidt orthogonalization argument, we can always obtain ξs ∈ (ξ1)^⊥ for s > 2 (proceeding inductively

with changes of variablest7→t+a_st^s). In fact, there is a unique ϕ∈G⁰_k achieving this condition.

In view of (??), it is natural to define a Finsler metric onf (taken modG⁰_k) by

(3.5) kfk_ε,β,p =

k

X

s=1

ε_s|ξ₁|^1−sβs|ξ_s|^βs2p

!1/2p

,

under the assumption that ξs ∈ (ξ1)^⊥ for s > 2 ; here β = (β1, . . . , βs) is a positive weight with β₁ = 1 and 0 < β_s 61/s fors>2 (the first term in the sum is equal to (ε₁|ξ₁|)^2p and we will in fact take ε1 = 1). The curvature of O_(Cr)k(1) fiberwise (i.e. on the rational variety Rr,k) is given by _2πⁱ ∂∂logkfk²_ε,p, provided holomorphic coordinates are used. We compute the curvature form at a point ξ⁰= (ξ_s⁰)_16s6k such that ξ₁⁰ 6= 0. By applying a dilation t 7→ f(λt), λ ∈ C^∗, we can assume that |ξ₁⁰|= 1. A nearby point ξ = (ξ_s)_16s6k can be written as ξ₁ =ξ₁⁰+ζ and ξ_s, s>2, with ζ, ξ_s ∈ (ξ₁⁰)^⊥. Notice that ((ξ⁰₁)^⊥)^k is a k(r−1)-dimensional complex subspace that defines an affine chart ofRr,k containing ξ⁰. The difficulty is that we do not necessarily have hξ_s, ξ1i= 0 any more, howeverhξ_s, ξ₁i=hξ_s, ζi=O(|ζ|) for s>2, and we have to correct this by applying an element ϕ∈G⁰_k close to identity. When computing i∂∂(...) at ζ = 0, all terms O(ζ², ζ²,|ζ|³) can be neglected, and the calculations performed below will be made modulo such terms. In particular, higher powers of hξ_s, ζi can be be neglected as they are of the formO(ζ²). A suitable choice is

ϕ(t) =t−

k

X

s=2

hξ_s, ζit^s.

Then

f ◦ϕ(t) =x+

k

X

s=1

t^sξ˜s+O(t^k+1) where ˜ξ1 =ξ1 and

ξ˜_s =ξ_s− hξ_s, ζiξ₁−

s−1

X

j=2

jhξs−j+1, ζiξ_j modO(ζ², ζ²,|ζ|³), s>2 (the final summation is obtained by expanding the terms (t−P

`>2hξ<sub>`</sub>, ζit<sup>`</sup>)<sup>j</sup>ξj for 26j 6s−1 and`=s−j+ 1). Observe that |ξ₁|² = 1 +|ζ|² and hξ_j, ξ₁i=hξ_j, ξ⁰₁i+hξ_j, ζi=hξ_j, ζi forj>2, hence

hξ˜_s, ξ₁i=hξ_s, ζi − hξ_s, ζi(1 +|ζ|²)−

s−1

X

j=2

jhξs−j+1, ζihξ_j, ζi=O(ζ², ζ²,|ζ|³), so we do not need any more accurate correction. Moreover, by a straightforward calculation

|ξ˜_s|² =

ξ_s−

s−1

X

j=2

jhξ_s−j+1, ζiξ_j

2

− |hξ_s, ζi|² modO(ζ², ζ²,|ζ|³).

Therefore, forε1 = 1, we find

kfk_ε,β,p = |ξ₁|^2p+

k

X

s=2

ε_s|ξ₁|^1−sβs|ξ˜_s|^βs2p

!1/2p

= (1 +|ζ|²)^p+

k

X

s=2

ε²_s(1 +|ζ|²)^1−sβs|ξ˜_s|^2βsp

!1/2p

(3.6)