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Algebraic properties of the algebra of differential operators


Academic year: 2022

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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


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 (TX))|X0 and the closureV|X0 in the total space ofTX 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



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



need be made on the Lie bracket tensor [,] :V × V → O(TX)/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 thatKV := 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 thenKV 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 TX. 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 ⊂TX,x,

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

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

−→πTX −→0,

|| ∪ ∪

0−→TX/Xe −→Ve −→ O Xe(−1)−→0, (1.4)

0−→ OXe −→πV ⊗ 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 ) =πdetV ⊗ OXe(r), hence (1.6) detVe =πdetV ⊗ 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) ∈ Vf(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 :TD(0,R)→feOXe(−1).

Moreover π◦fe=f, therefore

πfe0(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(Vx), in which case f is constant).

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

(1.9) Vz =n

v= X



∂zj ;vj = X


ajk(z)vk 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, . . . , vr), and the affine chart vj 6= 0 of P(V) can be described by the coordinate system


z1, . . . , zn;v1

vj, . . . ,vj−1

vj ,vj+1

vj , . . . ,vr vj


Let f ' (f1, . . . , fn) be the components of f in the coordinates (z1, . . . , zn) (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, . . . , fr). Indeed, as f0(t) ∈ Vf(t), we can recover the other components by integrating the system of ordinary differential equations

(1.11) fj0(t) = X


ajk(f(t))fk0(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 thatfj(m)(t0)6= 0 for some j. By (1.11), we can always suppose j ∈ {1, . . . , r}, for example fr(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 chartvr 6= 0 ofP(V) by

(1.12) fe'

f1, . . . , fn;f10

fr0, . . . ,fr−10 fr0


1.D. The Semple tower

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

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

In other words, (Xk, Vk) 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) dimXk=n+k(r−1), rankVk=r, together with exact sequences

0−→TXk/Xk−1 −→Vk−−−−→ Ok) Xk(−1)−→0, (1.15)

0−→ OXk −→πkVk−1⊗ OXk(1)−→TXk/Xk−1 −→0.



whereπk is the natural projection πk :Xk →Xk−1 and (πk) its differential. Formula (1.6) yields (1.16) detVkkdetVk−1⊗ OXk(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) → Xk of (Xk, Vk). Moreover, the derivative f[k−1]0 gives rise to a section

(1.17) f[k−1]0 :TD(0,R)→f[k] OXk(−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,Fs01

Fs0r, . . . ,Fs0r−1 Fs0r

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

By definition, there is a canonical injection OXk(−1) ,→ πkVk−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) OXk(−1),−→πkVk−1


−−−−−−→ πkOXk−1(−1),

which admits precisely Dk=P(TXk−1/Xk−2)⊂P(Vk−1) =Xk as its zero divisor (clearly, Dk is a hyperplane subbundle ofXk). Hence we find

(1.20) OXk(1) =πkOXk−1(1)⊗ O(Dk).

Now, we consider the composition of projections

(1.21) πk,jj+1◦ · · · ◦πk−1◦πk:Xk−→Xj.

Then πk,0 : Xk → X0 = X is a locally trivial holomorphic fiber bundle over X, and the fibers Xk,xk,0−1(x) are k-stage towers ofPr−1-bundles. Since we have (in both directions) morphisms (Cr, TCr) ↔ (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 toCk(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⊂CN, and thatX0 ⊂Xis a Zariski open subset on whichVX0 is a subbundle ofTX0. Then we consider the injection of the nonsingular directed manifold (X0, V0) into the absolute structure (Z, W),W =TZ. This yields an injection (Xk0, Vk0) ,→ (Zk, Wk), and we simply define (Xk, Vk) to be the closure of (Xk0, Vk0) into (Zk, Wk). 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 bundleJkX →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


JkX → X is simply f 7→ f(0). If (z1, . . . , zn) 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) +· · ·+tk

k!f(k)(0) +O(tk+1).

In these coordinates, the fiber JkXx can thus be identified with the set of k-tuples of vectors (ξ1, . . . , ξk) = (f0(0), . . . , f(k)(0)) ∈ (Cn)k. It follows that JkX is a holomorphic fiber bundle with typical fiber (Cn)k over X. However, JkX 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 ofJkX, given by a formula

(1.23) (Ψ◦f)(j)= Ψ0(f)·f(j)+






cj1...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 defineJkV →X to be the bundle of k-jets of curves f : (C,0)→X which are tangent toV, i.e., such that f0(t)∈Vf(t) for allt in a neighborhood of0, together with the projection map f 7→f(0) onto X.

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

(f10(0), . . . , fr0(0)); (f100(0), . . . , fr00(0));. . .; (f1(k)(0), . . . , fr(k)(0))


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


j ∈ O(TX,x) and v=P

16λ6rvλeλ ∈ O(V)x in a local trivializing frame (e1, . . . , er) of V we have

(1.25) ∇wv(x) = X




eλ(x) + X



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

(1.26) JkVx3f 7→(ξ1, ξ2, . . . , ξk) = (∇f(0),∇2f(0), . . . ,∇kf(0))∈Vx⊕k and computing inductively the successive derivatives∇f(t) =f0(t) and ∇sf(t) via

sf = (f∇)d/dt(∇s−1f) = X


d dt


λeλ(f) + X





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+· · ·+aktk, a1 ∈C, aj ∈C, j >2,

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

(1.27) JkV ×Gk→JkV, (f, ϕ)7→f◦ϕ.


There is a semidirect decompositionGk=G0kn 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 G0k = [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, . . . , λkξk), ξs=∇sf(0).

Following [GrGr79], we introduce the bundleEk,mGGV→X of polynomialsP(x;ξ1, . . . , ξk) that are homogeneous on the fibers ofJkV of weighted degree mwith respect to the C action, i.e.

(1.28) P(x;λξ1, . . . , λkξk) =λmP(x;ξ1, . . . ξk), in other words they are polynomials of the form

(1.29) P(x;ξ1, . . . ξk) = X


aα1...αk(x)ξα11ξ2α2· · ·ξkαk where ξs = (ξs,1, . . . , ξs,r) ∈Cr 'Vx and ξsαsαs,1s,1. . . ξs,rαs,r, |αs|=P

16j6rαs,j. Sections of the sheaf O(Ek,mGGV) 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


aα1...αk(f(t)) (∇f(t))α1(∇2f(t))α2· · ·(∇kf(t))αk where theaα1...αk(x) are holomorphic inx. With the graded algebra bundleEk,•GGV=L

mEk,mGGV we associate an analytic fiber bundle

(1.30) XkGG:= Proj(Ek,•GGV) = (JkV 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 OXGG

k (1) [the reader should observe however thatOXGG

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

1.31. Proposition. By construction, if πk : XkGG → X is the natural projection, we have the direct image formula


k (m) =O(Ek,mGGV) 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 (JkVr{0})//Gk(would such nice complex space quotients exist!). In fact the following fundamental result from [Dem95] shows that the Semple bundleXk constructed above plays the role of such a quotient.

1.32. Theorem and Definition. Let Ek,mV ⊂ Ek,mGGV 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 πk,0 :Xk→ X be the Semple jet bundles defined above and let JkVreg 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 JkVreg/Gk has the structure of a locally trivial bundle over X, and there is a holomorphic embedding JkVreg/Gk ,→Xk over X, which identifies JkVreg/Gk with Xkreg (thus Xk is a relative compactification of JkVreg/Gk over X).


(ii)The direct image sheaf

k,0)OXk(m)' O(Ek,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 |OXk(m)|is equal to the setXksing of singular k-jets [one has Xksing =∅ for k= 1]. Moreover,OXk(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]0f[k−1]0 ◦ϕ∈ OXk(−1)⊂πk,k−1Vk−1. We conclude that there is a well defined set-theoretic map

(1.33) JkVreg/Gk→Xkreg, f mod Gk7→f[k](0).

Given a holomorphic sectionσ∈H0k,0−1(U),OXk(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 JkVreg; 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) AGGk V?= M



of differential operators is fiberwise isomorphic to the polynomial ring C[f10, . . . , fr0, f100, . . . , fr00, . . . , f1(k), . . . , fr(k)]

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

Fa(Ek,mGGV?) =

P ∈Ek,mGGV?; degf(k)P(f)6a .

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

Ga(Ek,mGGV?)'Ek−1,m−kaGG V?⊗SaV.

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(Ek,mGGV?)' M

a=(a1,...,ak)∈Nk a1+2a2+···+kak=m

Sa1V⊗ · · · ⊗SakV.

Hence AGGk V? is just locally isomorphic to a k-fold tensor product of symmetric algebras SV. We define theSemple algebrato be the graded subalgebra of AGGk such that

(2.3) AkV? = (AGGk )G0k = M



in particular A1V? = AGG1 V? = SV. As G0k is a non reductive group, it is a priori unclear whetherAkV? is finitely generated fork>2.


the subalgebra ofG0k-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, . . . , zn) near a point x0 ∈X, such that Vx0 = Vect(∂/∂z1, . . . , ∂/∂zr). Let f = (f1, . . . , fn) be a regular k-jet tangent to V. Then there exists i ∈ {1,2, . . . , r} such that fi0(0)6= 0, and there is a unique reparametrizationt=ϕ(τ) such that f ◦ϕ=g = (g1, g2, . . . , gn) 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) = fi−1(τ)). Suppose i = r for the simplicity of notation. The space Xk is ak-stage tower of Pr−1-bundles. In the corresponding inhomogeneous coordinates on these Pr−1’s, the pointf[k](0) is given by the collection of derivatives

(g10(0), . . . , g0r−1(0)); (g100(0), . . . , g00r−1(0));. . .; (g1(k)(0), . . . , g(k)r−1(0)) .

[Recall that the other components (gr+1, . . . , gn) can be recovered from (g1, . . . , gr) by integrating the differential system (5.10)]. Thus the mapJkVreg/Gk → Xk is a bijection onto Xkreg, and the fibers of these isomorphic bundles can be seen as unions ofraffine charts '(Cr−1)k, associated with each choice of the axiszi used to describe the curve as a graph. The change of parameter formula

d = f01

r(t) d

dt expresses all derivativesgi(j)(τ) =djgi/dτjin terms of the derivativesfi(j)(t) =djfi/dtj (g10, . . . , g0r−1) =f10

fr0, . . . ,fr−10 fr0

; (g100, . . . , g00r−1) =f100fr0−fr00f10

fr03 , . . . ,fr−100 fr0 −fr00fr−10 fr03

; . . . ; (3.12)

(g1(k), . . . , g(k)r−1) =

f1(k)fr0−fr(k)f10 fr0k+1

, . . . ,fr−1(k)fr0 −fr(k)fr−10 fr0k+1

+ (order< k).

2.B. Wronskians

LetU be an open set of X, dimX =n, and s0, . . . , sk ∈ OX(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) . . . sk(f) D(s0(f)) D(s1(f)) . . . D(sk(f))

... ...

Dk(s0(f)) Dk(s1(f)) . . . Dk(sk(f))

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

D`(sj(f◦ϕ)) =ϕ0`D`(sj(f))◦ϕ+X



It follows easily from there that

Wk(s0, . . . , sk)(f◦ϕ) = (ϕ0)1+2+···+kWk(s0, . . . , sk)(f)◦ϕ,

hence Wk(s0, . . . , sk)(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) Wk(s0, . . . , sk) =

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

... ...

Dk(s0) Dk(s1) . . . Dk(sk)

∈H0(U, Ek,k0TX).


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

(a)Wk(s0, . . . , sk) isC-multilinear and alternate in (s0, . . . , sk).

(b)For any g∈ OX(U), we have

Wk(gs0, . . . , gsk) =gk+1Wk(s0, . . . , sk).

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




` k


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) Wk(s0, . . . , sk)∈H0(U, Ek,k0TX ⊗Lk+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 LU, 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) Wk(gs0, . . . , gsk) =gk+1Wk(s0, . . . , sk)∈H0(U, Ek,k0TX ⊗Lk+1⊗Gk+1).

2.C. Brackets

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


(2.5) DP(f) =P(f)0, i.e. DP(f)(t) = d


The operatorDP is then a section ofO(Ek,mGGV?). If P is a G0k-invariant operator in O(Ek,mV?), the relationP(f◦ϕ) =ϕ0mP(f)◦ϕimplies

DP(f ◦ϕ) = (ϕ0mP(f)◦ϕ)00m+1DP(f)◦ϕ+mϕ0m−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 G0k-invariant operators of respective degrees δP, δQ; it is easy to check that theirbracketdefined as

(2.6) [P, Q] =δPP(DQ)−δQQ(DP)

is againG0k-invariant, of degreeδPQ+ 1. This can be seen by observing that the termsϕ00(. . .) coming fromDP and DQcancel, or by noticing that

(2.60) [P, Q] =PδQ+1Q−δP+1D QδP


where QδP/PδQ 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, Ek,δPV⊗LδP), Q∈H0(X, Ek,δQV⊗LδQ)

we also get globally defined brackets

[P, Q]∈H0(X, Ek+1,δPQ+1V⊗LδPQ).


Another special case is the “degree 0” bracket associated with two sectionsσ1, σ2 ∈H0(X, L) (2.8) τ = [σ1, σ2] =σ12−σ2112D



∈H0(X, E1,1TX ⊗L2) =H0(X, TX ⊗L2).

In a similar way, given integers a, b > 1 and families of sections σj ∈ H0(X, La), 16 j 6k−2, τj ∈H0(X, V⊗Lb) =H0(X, E1,1V⊗Lb), 16j 6k, we define inductively iterated brackets (2.9) Bk1, . . . , σk−21, . . . , τk)∈H0(X, Ek,2k−1V⊗L(k−2)a+kb), k>2,

by putting

(2.90) B21, τ2) = [τ1, τ2] =τ12D τ2



and, fork>3 andck = (k−3)−(k−2)b/a∈Q, (2.900) Bk1, . . . , σk−21, . . . , τk) =σ1ck+1τ12k−2D

Bk−12, . . . , σk−22, . . . , τk) σc1kτ12k−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 Bk1, . . . , σk−21, . . . , τk) ∈ H0(X, Ek,2k−1V⊗L3k−2) whose degrees m= 2k−1 grow linearly withk, as well as the correcting twist L3k−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=−i ∂∂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 hq(X, E⊗Lm) of cohomology groups of the tensor powers E⊗Lm satisfy the following asymptotic estimates asm→+∞ :

(3.1 WM)Weak Morse inequalities:

hq(X, E⊗Lm)6rmn n!



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



(−1)q−jhj(X, E⊗Lm)6rmn n!



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

χ(X, E⊗Lm) := X


(−1)jhj(X, E⊗Lm) =rmn n!



ΘnL,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 groupsHq(X, E⊗Lm⊗ I(hm)) twisted with the corresponding L2 multiplier ideal sheaves

I(hm) =I(kϕ) =

f ∈ OX,x, ∃V 3x, Z


|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 :




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⊗Lm)>h0(X, E⊗Lm⊗ I(hm))

>h0(X, E⊗Lm⊗ I(hm))−h1(X, E⊗Lm⊗ I(hm))

>rkn n!



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

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

(b)at the hq level :

hq(X, E⊗Lm⊗ I(hm))>rkn n!



(−1)q Z


ΘnL,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 (Cr, TCr) when Cr is equipped with its standard hermitian metric. Pick a randomk-jet

(3.3) f(t) =x+tξ1+t2ξ2+· · ·+tkξk+O(tk+1), ξs∈Cr, 16s6k.

Given ε1, . . . , εk > 0, we get a natural Finsler metric on the tautological Green-Griffiths bundle OGG,(Cr)k(−1) of (Cr, TCr) by putting

(3.4) kfkGGε,p =




ss|)2p/s 1/2p


In fact ifλ·f denotes thet7→f(λt), we do have the required homogeneity property under theC action, namelykλ·fkGGε,p =|λ| kfkGGε,p. The choice of a suitable integerp (e.g.p= lcm(1,2, . . . , k) or a multiple) yields a smooth metric onOGG,(Cr)k(−1). However, formula (3.4) is notG0kinvariant and therefore cannot be used to construct a metric on the Semple tautological bundleO(Cr)k(−1) of (Cr, TCr). 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 +. . . +aktk+O(tk+1) and applying a Gram-Schmidt orthogonalization argument, we can always obtain ξs ∈ (ξ1) for s > 2 (proceeding inductively


with changes of variablest7→t+asts). In fact, there is a unique ϕ∈G0k achieving this condition.

In view of (??), it is natural to define a Finsler metric onf (taken modG0k) by

(3.5) kfkε,β,p =







under the assumption that ξs ∈ (ξ1) for s > 2 ; here β = (β1, . . . , βs) is a positive weight with β1 = 1 and 0 < βs 61/s fors>2 (the first term in the sum is equal to (ε11|)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 i ∂∂logkfk2ε,p, provided holomorphic coordinates are used. We compute the curvature form at a point ξ0= (ξs0)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 = (ξs)16s6k can be written as ξ110+ζ and ξs, s>2, with ζ, ξs ∈ (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 hξs, ξ1i= 0 any more, howeverhξs, ξ1i=hξs, ζi=O(|ζ|) for s>2, and we have to correct this by applying an element ϕ∈G0k 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 hξs, ζi can be be neglected as they are of the formO(ζ2). A suitable choice is

ϕ(t) =t−




s, ζits.


f ◦ϕ(t) =x+




tsξ˜s+O(tk+1) where ˜ξ11 and

ξ˜ss− hξs, ζiξ1




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

`>2`, ζit`)jξj for 26j 6s−1 and`=s−j+ 1). Observe that |ξ1|2 = 1 +|ζ|2 and hξj, ξ1i=hξj, ξ01i+hξj, ζi=hξj, ζi forj>2, hence

hξ˜s, ξ1i=hξs, ζi − hξs, ζi(1 +|ζ|2)−




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

|ξ˜s|2 =





jhξs−j+1, ζiξj


− |hξs, ζi|2 modO(ζ2, ζ2,|ζ|3).

Therefore, forε1 = 1, we find

kfkε,β,p = |ξ1|2p+






= (1 +|ζ|2)p+




ε2s(1 +|ζ|2)1−sβs|ξ˜s|sp




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