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=b^{1}_{5}(en)^{2n+2}c.

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 setX^{0} =XrY ⊂X such thatV_{|X}^{0} is a subbundle of (T_{X}))|X^{0} and the closureV_{|X}^{0}
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∈X^{0}; the dimension may be larger at points x∈Y. This happens e.g.

onX=C^{n} for the rank 1 linear spaceV generated by the Euler vector field: Vz =CP

16j6nzj ∂

∂zj

forz 6= 0, and V_{0} = C^{n}. 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},
(1.3^{0}) 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,
(1.4^{0})

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., f^{0}(t) ∈ V_{f}_{(t)} for every t ∈ D(0, R). If f is nonconstant, there is a well
defined and unique tangent line [f^{0}(t)] for every t, even at stationary points, and the map

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

is holomorphic (at a stationary point t_{0}, we just write f^{0}(t) = (t−t_{0})^{s}u(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

t_{0}; even for t = t_{0}, we still denote [f^{0}(t_{0})] = [u(t_{0})] for simplicity of notation). By definition
f^{0}(t)∈ OXe(−1)_{f(t)}_{e} =Cu(t), hence the derivativef^{0} defines a section

(1.8) f^{0} :T_{D(0,R)}→fe^{∗}OXe(−1).

Moreover π◦fe=f, therefore

π∗fe^{0}(t) =f^{0}(t)∈Cu(t) =⇒fe^{0}(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 pointx_{0} ∈ X, there are local coordinates (z_{1}, . . . , z_{n}) on a neighborhood Ω of x_{0} 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 (v_{1}, . . . , v_{r}), and the affine chart v_{j} 6= 0 of P(V)_{Ω} can be
described by the coordinate system

(1.10)

z_{1}, . . . , z_{n};v_{1}

v_{j}, . . . ,vj−1

v_{j} ,v_{j+1}

v_{j} , . . . ,v_{r}
v_{j}

.

Let f ' (f_{1}, . . . , f_{n}) be the components of f in the coordinates (z_{1}, . . . , 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 (f_{1}, . . . , f_{r}). Indeed, as f^{0}(t) ∈ V_{f(t)}, we can recover the
other components by integrating the system of ordinary differential equations

(1.11) f_{j}^{0}(t) = X

16k6r

ajk(f(t))f_{k}^{0}(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 pointt_{0} ∈D(0, R), that is,m(f, t_{0}) is the smallest integerm∈N^{∗} such thatf_{j}^{(m)}(t_{0})6= 0
for some j. By (1.11), we can always suppose j ∈ {1, . . . , r}, for example f_{r}^{(m)}(t_{0}) 6= 0. Then
f^{0}(t) = (t−t0)^{m−1}u(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'

f_{1}, . . . , f_{n};f_{1}^{0}

f_{r}^{0}, . . . ,f_{r−1}^{0}
f_{r}^{0}

.

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_{X}_{k} 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_{X}_{k}_{/X}_{k−1} −→V_{k}−−−−→ O^{(π}^{k}^{)}^{∗} _{X}_{k}(−1)−→0,
(1.15)

0−→ O_{X}_{k} −→π_{k}^{∗}Vk−1⊗ O_{X}_{k}(1)−→T_{X}_{k}_{/X}_{k−1} −→0.

(1.15^{0})

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_{X}_{k}(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_{X}_{k}(−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_{s}^{0}_{1}

F_{s}^{0}_{r}, . . . ,F_{s}^{0}_{r−1}
F_{s}^{0}_{r}

whereN =n+k(r−1) and{s_{1}, . . . , s_{r}} ⊂ {1, . . . , N}. Ifk>1,{s_{1}, . . . , 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_{s}_{r},0) =m(f_{[k]},0), that is, F_{s}_{r} has the smallest vanishing
order among all componentsF_{s} (s_{r} may be vertical or not, and the choice of{s_{1}, . . . , s_{r}}need not
be unique).

By definition, there is a canonical injection O_{X}_{k}(−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_{X}_{k}(−1),−→π_{k}^{∗}Vk−1

(πk)^{∗}(dπk−1)

−−−−−−→ π^{∗}_{k}O_{X}_{k−1}(−1),

which admits precisely D_{k}=P(T_{X}_{k−1}_{/X}_{k−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_{X}_{k}(1) =π_{k}^{∗}O_{X}_{k−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 thatX^{0} ⊂Xis a Zariski open subset on whichV_{X}^{0} is a subbundle ofT_{X}^{0}. Then we consider the
injection of the nonsingular directed manifold (X^{0}, V^{0}) into the absolute structure (Z, W),W =TZ.
This yields an injection (X_{k}^{0}, V_{k}^{0}) ,→ (Z_{k}, W_{k}), and we simply define (X_{k}, V_{k}) to be the closure of
(X_{k}^{0}, V_{k}^{0}) 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= rankV^{0}.

1.E. Jet bundles and jet differentials

Following Green-Griffiths [GrGr79], we consider the bundleJ_{k}X →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_{k}X → X is simply f 7→ f(0). If (z_{1}, . . . , z_{n}) are local holomorphic coordinates on an open set
Ω⊂X, the elementsf of any fiber JkXx,x∈Ω, can be seen as C^{n}-valued maps

f = (f1, . . . , fn) : (C,0)→Ω⊂C^{n},

and they are completetely determined by their Taylor expansion of orderkatt= 0
f(t) =x+t f^{0}(0) + t^{2}

2!f^{00}(0) +· · ·+t^{k}

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

In these coordinates, the fiber J_{k}Xx can thus be identified with the set of k-tuples of vectors
(ξ_{1}, . . . , ξ_{k}) = (f^{0}(0), . . . , f^{(k)}(0)) ∈ (C^{n})^{k}. It follows that J_{k}X is a holomorphic fiber bundle
with typical fiber (C^{n})^{k} over X. However, J_{k}X 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_{k}X, given by a formula

(1.23) (Ψ◦f)^{(j)}= Ψ^{0}(f)·f^{(j)}+

s=j

X

s=2

X

j1+j2+···+j_{s}=j

c_{j}_{1}_{...j}_{s}Ψ^{(s)}(f)·(f^{(j}^{1}^{)}, . . . , f^{(j}^{s}^{)})

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_{k}V →X to be the bundle
of k-jets of curves f : (C,0)→X which are tangent toV, i.e., such that f^{0}(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_{k}V is actually a subbundle ofJ_{k}X. In fact, by using (1.11), we see that
the fibersJkVx are parametrized by

(f_{1}^{0}(0), . . . , f_{r}^{0}(0)); (f_{1}^{00}(0), . . . , f_{r}^{00}(0));. . .; (f_{1}^{(k)}(0), . . . , f_{r}^{(k)}(0))

∈(C^{r})^{k}

for allx∈Ω, hence J_{k}V is a locally trivial (C^{r})^{k}-subbundle of J_{k}X. 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_{λ}e_{λ} ∈ O(V)_{x} in a local trivializing frame (e_{1}, . . . , e_{r}) of V_{Ω} we have

(1.25) ∇_{w}v(x) = X

16j6n,16λ6r

w_{j}∂v_{λ}

∂zj

e_{λ}(x) + X

16j6n,16λ,µ6r

Γ^{µ}_{jλ}(x)w_{j}v_{λ}e_{µ}(x).

We can of course take the frame obtained from (1.9) by lifting the vector fields∂/∂z_{1}, . . . , ∂/∂z_{r},
and the “trivial connection” given by the zero Christoffel symbolds Γ = 0. One then obtains a
trivializationJ^{k}V_{Ω} 'V_{Ω}^{⊕k} by considering

(1.26) J_{k}V_{x}3f 7→(ξ_{1}, ξ_{2}, . . . , ξ_{k}) = (∇f(0),∇^{2}f(0), . . . ,∇^{k}f(0))∈V_{x}^{⊕k}
and computing inductively the successive derivatives∇f(t) =f^{0}(t) and ∇^{s}f(t) via

∇^{s}f = (f^{∗}∇)_{d/dt}(∇^{s−1}f) = X

16λ6r

d dt

∇^{s−1}f

λe_{λ}(f) + X

16j6n,16λ,µ6r

Γ^{µ}_{jλ}(f)f_{j}^{0}

∇^{s−1}f

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+a2t^{2}+· · ·+a_{k}t^{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_{k}V

(1.27) JkV ×Gk→JkV, (f, ϕ)7→f◦ϕ.

There is a semidirect decompositionGk=G^{0}_{k}n C^{∗} given by a split exact sequence
1→G^{0}k→Gk→C^{∗}→1

where Gk → C^{∗} is the obvious morphism ϕ 7→ ϕ^{0}(0), the commutator group G^{0}_{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, λ^{2}ξ2, . . . , λ^{k}ξk), ξs=∇^{s}f(0).

Following [GrGr79], we introduce the bundleE_{k,m}^{GG}V^{∗}→X of polynomialsP(x;ξ1, . . . , ξ_{k}) that are
homogeneous on the fibers ofJ_{k}V of weighted degree mwith respect to the C^{∗} action, i.e.

(1.28) P(x;λξ1, . . . , λ^{k}ξk) =λ^{m}P(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^{α}_{1}^{1}ξ_{2}^{α}^{2}· · ·ξ_{k}^{α}^{k}
where ξs = (ξs,1, . . . , ξs,r) ∈C^{r} 'Vx and ξ_{s}^{α}^{s} =ξ^{α}_{s,1}^{s,1}. . . ξs,r^{α}^{s,r}, |α_{s}|=P

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

(1.29^{0}) P(f)(t) = X

|α_{1}|+2|α_{2}|+···+k|α_{k}|=m

a_{α}_{1}_{...α}_{k}(f(t)) (∇f(t))^{α}^{1}(∇^{2}f(t))^{α}^{2}· · ·(∇^{k}f(t))^{α}^{k}
where thea_{α}_{1}_{...α}_{k}(x) are holomorphic inx. With the graded algebra bundleE_{k,•}^{GG}V^{∗}=L

mE_{k,m}^{GG}V^{∗}
we associate an analytic fiber bundle

(1.30) X_{k}^{GG}:= Proj(E_{k,•}^{GG}V^{∗}) = (J_{k}V 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_{X}GG

k (1) [the reader
should observe however thatO_{X}GG

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_{X}GG

k (m) =O(E_{k,m}^{GG}V^{∗})
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_{k}Vr{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,m}V^{∗} ⊂ E_{k,m}^{GG}V^{∗} 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) = (ϕ^{0})^{m}P(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_{k}V^{reg} be the bundle of regular
k-jets of maps f : (C,0)→(X, V), that is, jetsf such that f^{0}(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_{k}V^{reg}/Gk ,→X_{k} over X, which identifies J_{k}V^{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_{X}_{k}(m)' O(E_{k,m}V^{∗})
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_{X}_{k}(m)|is equal to the setX_{k}^{sing}
of singular k-jets [one has X_{k}^{sing} =∅ for k= 1]. Moreover,O_{X}_{k}(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, R^{0}) → D(0, R), they satisfy the
relations

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

(1.33) J_{k}V^{reg}/Gk→X_{k}^{reg}, f mod Gk7→f_{[k]}(0).

Given a holomorphic sectionσ∈H^{0}(π_{k,0}^{−1}(U),O_{X}_{k}(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}^{GG}V^{?}

of differential operators is fiberwise isomorphic to the polynomial ring
C[f_{1}^{0}, . . . , f_{r}^{0}, f_{1}^{00}, . . . , f_{r}^{00}, . . . , f_{1}^{(k)}, . . . , f_{r}^{(k)}]

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

F^{a}(E_{k,m}^{GG}V^{?}) =

P ∈E_{k,m}^{GG}V^{?}; deg_{f}(k)P(f)6a .

Then the graded pieces are polynomials of the formQ(f^{0}, . . . , f^{(k−1)})(f^{(k)})^{α}^{k},|α_{k}|=a, i.e.

G^{a}(E_{k,m}^{GG}V^{?})'E_{k−1,m−ka}^{GG} V^{?}⊗S^{a}V^{∗}.

We can then inductively combine the successive filtrations obtained via the partial degrees inf^{(k)},
f^{(k−1)}, . . . , f^{(1)} =f^{0} to get a full decomposition

(2.2) G^{•}(E_{k,m}^{GG}V^{?})' M

a=(a1,...,ak)∈Nk a1+2a2+···+kak=m

S^{a}^{1}V^{∗}⊗ · · · ⊗S^{a}^{k}V^{∗}.

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_{k}V^{?} = (A^{GG}_{k} )^{G}^{0}^{k} = M

m∈Z

E_{k,m}V^{?},

in particular A_{1}V^{?} = A^{GG}_{1} V^{?} = S^{•}V^{∗}. As G^{0}_{k} is a non reductive group, it is a priori unclear
whetherA_{k}V^{?} is finitely generated fork>2.

the subalgebra ofG^{0}_{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 (z_{1}, . . . , z_{n}) near a point x_{0} ∈X, such that V_{x}_{0} = Vect(∂/∂z_{1}, . . . , ∂/∂z_{r}). Let
f = (f1, . . . , fn) be a regular k-jet tangent to V. Then there exists i ∈ {1,2, . . . , r} such that
f_{i}^{0}(0)6= 0, and there is a unique reparametrizationt=ϕ(τ) such that f ◦ϕ=g = (g_{1}, g_{2}, . . . , 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

(g_{1}^{0}(0), . . . , g^{0}_{r−1}(0)); (g_{1}^{00}(0), . . . , g^{00}_{r−1}(0));. . .; (g_{1}^{(k)}(0), . . . , g^{(k)}_{r−1}(0))
.

[Recall that the other components (g_{r+1}, . . . , g_{n}) can be recovered from (g_{1}, . . . , g_{r}) by integrating
the differential system (5.10)]. Thus the mapJ_{k}V^{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 = _{f}0^{1}

r(t) d

dt expresses all derivativesg_{i}^{(j)}(τ) =d^{j}gi/dτ^{j}in terms of the derivativesf_{i}^{(j)}(t) =d^{j}fi/dt^{j}
(g_{1}^{0}, . . . , g^{0}_{r−1}) =f_{1}^{0}

f_{r}^{0}, . . . ,f_{r−1}^{0}
f_{r}^{0}

;
(g_{1}^{00}, . . . , g^{00}_{r−1}) =f_{1}^{00}f_{r}^{0}−f_{r}^{00}f_{1}^{0}

f_{r}^{03} , . . . ,f_{r−1}^{00} f_{r}^{0} −f_{r}^{00}f_{r−1}^{0}
f_{r}^{03}

; . . . ; (3.12)

(g_{1}^{(k)}, . . . , g^{(k)}_{r−1}) =

f_{1}^{(k)}f_{r}^{0}−fr^{(k)}f_{1}^{0}
fr^{0k+1}

, . . . ,f_{r−1}^{(k)}f_{r}^{0} −f_{r}^{(k)}f_{r−1}^{0}
fr^{0k+1}

+ (order< k).

2.B. Wronskians

LetU be an open set of X, dimX =n, and s_{0}, . . . , 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) =

s_{0}(f) s_{1}(f) . . . s_{k}(f)
D(s0(f)) D(s1(f)) . . . D(sk(f))

... ...

D^{k}(s_{0}(f)) D^{k}(s_{1}(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_{`,s}of order6`acting on ϕsatisfying

D^{`}(s_{j}(f◦ϕ)) =ϕ^{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}(s_{0}, . . . , s_{k})(f◦ϕ) = (ϕ^{0})^{1+2+···+k}W_{k}(s_{0}, . . . , s_{k})(f)◦ϕ,

hence W_{k}(s_{0}, . . . , s_{k})(f) is an invariant differential operator of degree k^{0} = ^{1}_{2}k(k+ 1). Especially,
we get in this way a section that we denote

(4.2) W_{k}(s0, . . . , s_{k}) =

s_{0} s_{1} . . . s_{k}
D(s0) D(s1) . . . D(sk)

... ...

D^{k}(s_{0}) D^{k}(s_{1}) . . . D^{k}(s_{k})

∈H^{0}(U, E_{k,k}^{0}T_{X}^{∗}).

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

(a)W_{k}(s_{0}, . . . , s_{k}) isC-multilinear and alternate in (s_{0}, . . . , s_{k}).

(b)For any g∈ O_{X}(U), we have

Wk(gs0, . . . , gsk) =g^{k+1}Wk(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}(s_{0}, . . . , s_{k})∈H^{0}(U, E_{k,k}^{0}T_{X}^{∗} ⊗L^{k+1})

for any (k+ 1)-tuple of sectionss0, . . . , sk∈H^{0}(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 ∈H^{0}(U, G) for some line bundle
G→X, we have

(2.4) W_{k}(gs_{0}, . . . , gs_{k}) =g^{k+1}W_{k}(s_{0}, . . . , s_{k})∈H^{0}(U, E_{k,k}^{0}T_{X}^{∗} ⊗L^{k+1}⊗G^{k+1}).

2.C. Brackets

IfP is a differential operator given by a section ofO(E^{GG}_{k,m}V^{?}), 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}^{GG}V^{?}). If P is a G^{0}_{k}-invariant operator in O(E_{k,m}V^{?}),
the relationP(f◦ϕ) =ϕ^{0}^{m}P(f)◦ϕimplies

DP(f ◦ϕ) = (ϕ^{0}^{m}P(f)◦ϕ)^{0}=ϕ^{0}^{m+1}DP(f)◦ϕ+mϕ^{0}^{m−1}ϕ^{00}P(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 G^{0}_{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 againG^{0}_{k}-invariant, of degreeδ_{P} +δ_{Q}+ 1. This can be seen by observing that the termsϕ^{00}(. . .)
coming fromDP and DQcancel, or by noticing that

(2.6^{0}) [P, Q] =P^{δ}^{Q}^{+1}Q^{−δ}^{P}^{+1}D
Q^{δ}^{P}

P^{δ}^{Q}

where Q^{δ}^{P}/P^{δ}^{Q} 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 (2.6^{0}) has the advantage that for any line bundle L(or even any Q-line bundleL) and
P ∈H^{0}(X, E_{k,δ}_{P}V^{∗}⊗L^{δ}^{P}), Q∈H^{0}(X, E_{k,δ}_{Q}V^{∗}⊗L^{δ}^{Q})

we also get globally defined brackets

[P, Q]∈H^{0}(X, E_{k+1,δ}_{P}_{+δ}_{Q}_{+1}V^{∗}⊗L^{δ}^{P}^{+δ}^{Q}).

Another special case is the “degree 0” bracket associated with two sectionsσ_{1}, σ_{2} ∈H^{0}(X, L)
(2.8) τ = [σ1, σ2] =σ1Dσ2−σ2Dσ1 =σ_{1}^{2}D

σ2

σ_{1}

∈H^{0}(X, E1,1T_{X}^{∗} ⊗L^{2}) =H^{0}(X, T_{X}^{∗} ⊗L^{2}).

In a similar way, given integers a, b > 1 and families of sections σj ∈ H^{0}(X, L^{a}), 16 j 6k−2,
τ_{j} ∈H^{0}(X, V^{∗}⊗L^{b}) =H^{0}(X, E_{1,1}V^{∗}⊗L^{b}), 16j 6k, we define inductively iterated brackets
(2.9) Bk(σ1, . . . , σk−2;τ1, . . . , τk)∈H^{0}(X, Ek,2k−1V^{∗}⊗L^{(k−2)a+kb}), k>2,

by putting

(2.9^{0}) B_{2}(τ_{1}, τ_{2}) = [τ_{1}, τ_{2}] =τ_{1}^{2}D
τ_{2}

τ_{1}

,

and, fork>3 andc_{k} = (k−3)−(k−2)b/a∈Q,
(2.9^{00}) Bk(σ1, . . . , σk−2;τ1, . . . , τk) =σ_{1}^{c}^{k}^{+1}τ_{1}^{2k−2}D

B_{k−1}(σ_{2}, . . . , σ_{k−2};τ_{2}, . . . , τ_{k})
σ^{c}_{1}^{k}τ_{1}^{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) ∈ H^{0}(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=−_{2π}^{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
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})6rm^{n}
n!

Z

X(L,h,q)

(−1)^{q}Θ^{n}_{L,h}+o(m^{n}) .
(3.1 SM) Strong Morse inequalities:

X

06j6q

(−1)^{q−j}h^{j}(X, E⊗L^{m})6rm^{n}
n!

Z

X(L,h,6q)

(−1)^{q}Θ^{n}_{L,h}+o(m^{n}) .
(3.1 RR) Asymptotic Riemann-Roch formula:

χ(X, E⊗L^{m}) := X

06j6n

(−1)^{j}h^{j}(X, E⊗L^{m}) =rm^{n}
n!

Z

X

Θ^{n}_{L,h}+o(m^{n}) .

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
L^{2} multiplier ideal sheaves

I(h^{m}) =I(kϕ) =

f ∈ O_{X,x}, ∃V 3x,
Z

V

|f(z)|^{2}e^{−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)^{q}Θ^{n}_{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 h^{0} level :

h^{0}(X, E⊗L^{m})>h^{0}(X, E⊗L^{m}⊗ I(h^{m}))

>h^{0}(X, E⊗L^{m}⊗ I(h^{m}))−h^{1}(X, E⊗L^{m}⊗ I(h^{m}))

>rk^{n}
n!

Z

X(L,h,61)rΣ

Θ^{n}_{L,h}−o(k^{n}) .
Especially Lis big as soon asR

X(L,h,61)rΣΘ^{n}_{L,h} >0 for some singular hermitian metrichonL.

(b)at the h^{q} level :

h^{q}(X, E⊗L^{m}⊗ I(h^{m}))>rk^{n}
n!

X

j=q−1,q,q+1

(−1)^{q}
Z

X(L,h,j)rΣ

Θ^{n}_{L,h}−o(k^{n}) .

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+tξ_{1}+t^{2}ξ_{2}+· · ·+t^{k}ξ_{k}+O(t^{k+1}), ξ_{s}∈C^{r}, 16s6k.

Given ε_{1}, . . . , ε_{k} > 0, we get a natural Finsler metric on the tautological Green-Griffiths bundle
O_{GG,(}_{C}r)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,(}_{C}r)k(−1). However, formula (3.4) is notG^{0}_{k}invariant
and therefore cannot be used to construct a metric on the Semple tautological bundleO_{(}_{C}r)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+a2t^{2} +. . . +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_{s}t^{s}). In fact, there is a unique ϕ∈G^{0}_{k} achieving this condition.

In view of (??), it is natural to define a Finsler metric onf (taken modG^{0}_{k}) by

(3.5) kfk_{ε,β,p} =

k

X

s=1

ε_{s}|ξ_{1}|^{1−sβ}^{s}|ξ_{s}|^{β}^{s}2p

!1/2p

,

under the assumption that ξs ∈ (ξ1)^{⊥} for s > 2 ; here β = (β1, . . . , βs) is a positive weight with
β_{1} = 1 and 0 < β_{s} 61/s fors>2 (the first term in the sum is equal to (ε_{1}|ξ_{1}|)^{2p} and we will in
fact take ε1 = 1). The curvature of O_{(}_{C}r)k(1) fiberwise (i.e. on the rational variety Rr,k) is given
by _{2π}^{i} ∂∂logkfk^{2}_{ε,p}, provided holomorphic coordinates are used. We compute the curvature form
at a point ξ^{0}= (ξ_{s}^{0})_{16s6k} such that ξ_{1}^{0} 6= 0. By applying a dilation t 7→ f(λt), λ ∈ C^{∗}, we can
assume that |ξ_{1}^{0}|= 1. A nearby point ξ = (ξ_{s})_{16s6k} can be written as ξ_{1} =ξ_{1}^{0}+ζ and ξ_{s}, s>2,
with ζ, ξ_{s} ∈ (ξ_{1}^{0})^{⊥}. Notice that ((ξ^{0}_{1})^{⊥})^{k} is a k(r−1)-dimensional complex subspace that defines
an affine chart ofRr,k containing ξ^{0}. The difficulty is that we do not necessarily have hξ_{s}, ξ1i= 0
any more, howeverhξ_{s}, ξ_{1}i=hξ_{s}, ζi=O(|ζ|) for s>2, and we have to correct this by applying an
element ϕ∈G^{0}_{k} close to identity. When computing i∂∂(...) at ζ = 0, all terms O(ζ^{2}, ζ^{2},|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(ζ^{2}). 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^{s}ξ˜s+O(t^{k+1})
where ˜ξ1 =ξ1 and

ξ˜_{s} =ξ_{s}− hξ_{s}, ζiξ_{1}−

s−1

X

j=2

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

`>2hξ_{`}, ζit^{`})^{j}ξj for 26j 6s−1
and`=s−j+ 1). Observe that |ξ_{1}|^{2} = 1 +|ζ|^{2} and hξ_{j}, ξ_{1}i=hξ_{j}, ξ^{0}_{1}i+hξ_{j}, ζi=hξ_{j}, ζi forj>2,
hence

hξ˜_{s}, ξ_{1}i=hξ_{s}, ζi − hξ_{s}, ζi(1 +|ζ|^{2})−

s−1

X

j=2

jhξs−j+1, ζihξ_{j}, ζi=O(ζ^{2}, ζ^{2},|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(ζ^{2}, ζ^{2},|ζ|^{3}).

Therefore, forε1 = 1, we find

kfk_{ε,β,p} = |ξ_{1}|^{2p}+

k

X

s=2

ε_{s}|ξ_{1}|^{1−sβ}^{s}|ξ˜_{s}|^{β}^{s}2p

!1/2p

= (1 +|ζ|^{2})^{p}+

k

X

s=2

ε^{2}_{s}(1 +|ζ|^{2})^{1−sβ}^{s}|ξ˜_{s}|^{2β}^{s}p

!1/2p

(3.6)