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Geometric Generalized Wronskians: Applications in Intermediate Hyperbolicity and Foliation Theory

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Abstract

In this paper, we introduce a sub-family of the usual generalized Wronskians, that we call geometric generalized Wronskians. It is well-known that one can test linear dependance of holomorphic functions (of several variables) via the identical vanishing of generalized Wronskians. We show that such a statement remains valid if one tests the identical vanishing only on geometric generalized Wronskians. It turns out that geometric generalized Wronskians allow to define intrinsic objects on projective varieties polarized with an ample line bundle: in this setting, the lack of existence of global functions is compensated by global sections of powers of the fixed ample line bundle. Geometric generalized Wronskians are precisely defined so that their local evaluations on such global sections globalize up to a positive twist by the ample line bundle. We then give three applications of the construction of geometric generalized Wronskians: one in intermediate hyperbolicity, and two in foliation theory. In intermediate hyperbolicity, we show the algebraic degeneracy of holomorphic maps from C p to a Fermat hypersurface in P N of degree δ\delta > (N + 1)(N -- p): this interpolates between two well-known results, namely for p = 1 (first proved via Nevanlinna theory) and p = N -- 1 (in which case the Fermat hypersurface is of general type). The first application in foliation theory provides a criterion for algebraic integrability of leaves of foliations: our criterion is not optimal in view of current knowledges, but has the advantage of having an elementary proof. Our second application deals with positivity properties of adjoint line bundles of the form K F + L, where K F is the canonical bundle of a regular foliation F on a smooth projective variety X, and where L is an ample line bundle on X.
arXiv:2108.02993v2 [math.AG] 9 Aug 2021
GEOMETRIC GENERALIZED WRONSKIANS. APPLICATIONS
IN INTERMEDIATE HYPERBOLICITY AND FOLIATION
THEORY.
ANTOINE ETESSE
Abstract. In this paper, we introduce a sub-family of the usual generalized
Wronskians, that we call geometric generalized Wronskians. It is well-known
that one can test linear dependance of holomorphic functions (of several vari-
ables) via the identical vanishing of generalized Wronskians. We show that such
a statement remains valid if one tests the identical vanishing only on geometric
generalized Wronskians.
It turns out that geometric generalized Wronskians allow to define intrinsic
objects on projective varieties polarized with an ample line bundle: in this
setting, the lack of existence of global functions is compensated by global sections
of powers of the fixed ample line bundle. Geometric generalized Wronskians are
precisely defined so that their local evaluations on such global sections globalize
up to a positive twist by the ample line bundle.
We then give three applications of the construction of geometric generalized
Wronskians: one in intermediate hyperbolicity, and two in foliation theory. In
intermediate hyperbolicity, we show the algebraic degeneracy of holomorphic
maps from Cpto a Fermat hypersurface in PNof degree δ > (N+ 1)(Np):
this interpolates between two well-known results, namely for p= 1 (first proved
via Nevanlinna theory) and p=N1(in which case the Fermat hypersurface is
of general type). The first application in foliation theory provides a criterion for
algebraic integrability of leaves of foliations: our criterion is not optimal in view
of current knowledges, but has the advantage of having an elementary proof.
Our second application deals with positivity properties of adjoint line bundles
of the form KF+L, where KFis the canonical bundle of a regular foliation F
on a smooth projective variety X, and where Lis an ample line bundle on X.
Introduction
Wronskians first appeared in the litterature with the work of Wronski [HW12]
in 1812: being given f0,...,fm(m+ 1) holomorphic functions on the complex line
C(or any open subset of it), their Wronskian W(f0,...,fm)is the determinant of
the square matrix formed with their multi-derivatives:
W(f0,...,fm):= det
f0· · fm
f
0· · f
m
· ·
· ·
f(m)
0· · f(m)
m
.
This definition, specific to one variable functions, has been widely used in the
litterature since it was first introduced: see e.g [GS12] for a survey of its applica-
tions in various areas of mathematics. (Note that their definition of “generalized
Key words and phrases. Wronskian, hyperbolicity, foliation.
1
Wronskians“ differs from ours). A fundamental property of Wronskians is the
following:
(f0,...,fm)are linearly independant over Cif and only if their Wronskian
W(f0,...,fm)does not vanish identically.
This was pointed out by Peano in [Pea89], and first proved by Bôcher in [Boc16].
It is only later that the definition of Wronskians was extended to the multi-
variables case in [Ost19]. The definition goes as follows (see also Section 1.1). Let
pN1, and let f0, . . . , fmbe (m+ 1) holomorphic functions on Cp(or an open
subset of it). Consider a set of words U={u1, . . . , um}, where each word uiis
written with the alphabet {1,...,p}. Define the following determinant:
WU(f0,...,fm):= det
f0· · fm
u1f0· · u1fm
· ·
· ·
umf0· · umfm
,
where by definition
u=α1(u)+···+αp(u)
∂zα1(u)
1···∂zαp(u)
p
,
with u= 1α1(u)···pαp(u). This is called a pure generalized Wronskian if and only
if the length (ui)of the word uiis less or equal than ifor every 1im.
Ageneralized Wronskian is then simply a linear combination of pure generalized
Wronskians. It was already hinted at in [Ost19] that generalized Wronskians must
statisfy the same fundamental property than Wronskians, namely:
(f0,...,fm)are linearly independant over Cif and only if one of their pure
generalized Wronskians WU(f0,...,fm)does not vanish identically.
This result was proved and used in [Rot55] in the context of Diophantine ap-
proximation, and reproved later in [BCL94] in the context of Nevanlinna Theory
and algebraic dependance of real numbers. Note that each proof provided in the
previously quoted papers uses an induction on the number of variables.
Recently, in respectively [LP+18] and [Bro17], Wronskians in one variable were
used to construct global jet differentials (of 1-germs) on projective varieties: see
Section 1.2.1 for definitions of jet differentials for p-germs, p1, or [Bro17] for
the case p= 1. Their construction reads as follows.
Proposition 0.0.1 ([Bro17], [LP+18]).Let s0,...,smbe global sections of a line
bundle Lon a projective variety X. The Wronskian operator Winduces a global
section W(s0,...,sm)of E1,m, m(m+1)
2XLm+1, defined locally on a trivializing open
set Uof Las follows:
W(s0, . . . , sm)(γ)loc
=W(s0,U γ,...,sm,U γ)(0),
where γ: (C,0) (X, x)is a 1-germ passing through xU, and where the
notation si,U indicates that the section siis written under a trivialization of the
line bundle Lon the open set U.
This is the starting point of this paper, as our main goal is to generalize the pre-
vious proposition in the setting of p-germs, p > 1, using generalized Wronskians.
Page 2
There is however an obstacle in this higher dimensional case: whereas the compat-
ibility condition to define a global object W(s0,...,sm)is automatically satisfied
for the one variable Wronskian W (this gives rise to the positive twist Lm+1), this
is no longer true for generalized Wronskian. This lead us to introduce geometric
generalized Wronskians (see Section 1.2): those are exactly the generalized Wron-
skians satisfying the wanted compatibility condition. The analogue of Proposition
0.0.1 reads as follows (see Section 1.1 for definitions of order and weight, and see
Section 1.2.1 for definitions of relevant jet differentials bundles):
Proposition 0.0.2. Let s0,...,smbe global sections of a line bundle L. Let Wbe
a non-zero geometric generalized Wronskian of order kand weight w. The operator
Winduces a global section W(s0,...,sm)of Ep,k,wXLm+1, defined locally on a
trivializing open set Uof Las follows:
W(s0, . . . , sm)(γ)loc
=W(s0,U γ,...,sm,U γ)(0),
where γ: (Cp,0) (X, x)is a p-germ passing through xU, and where the
notation si,U indicates that the section siis written under a trivialization of the
line bundle Lon the open set U.
In view of this, and the fundamental property of generalized Wronskians, the
following question is important:
“Are geometric generalized Wronskians enough to ensure linear dependance of
functions? “
The answer is positive, and it is our first Main Theorem (labeled Theorem 1.4.1
in Section 1.4):
Theorem 0.0.3 (Main Theorem I).Let (f0,...,fm)be (m+1) holomorphic func-
tions on Cp. They are linearly independant if and only there exists a geometric
generalized Wronskian Wsuch that W(f0,...,fm)does not vanish identically.
As a matter of fact, we prove a stronger result: one can test the non-vanishing
condition only on pure geometric generalized Wronskians. Those form a particular
subfamily of pure generalized Wronskians: see Section 1.3 for definitions, and
Section 1.4 for the proof.
In the second part of this paper, we use our construction (detailed in Proposition
0.0.2) and our Main Theorem 0.0.3 in order to study intermediate hyperbolicity
properties of Fermat hypersurfaces. For an introduction to hyperbolic spaces see
e.g. [Kob13], and for intermediate notions of hyperbolicity, namely p-analytic
hyperbolicity see e.g. [Dem97a]. For our purposes, it is enough to know that
intermediate hyperbolicity is concerned with family of entire curves (an entire
curve in a variety Xis a non-constant holomorphic map from the complex line C
to X). Our application in intermediate hyperbolicity (labeled Theorem 2.2.2 in
Section 2) reads as follows:
Theorem 0.0.4 (Main Theorem II).Let Hλ={λ0Xδ
0+···+λNXδ
N= 0} ⊂ PNbe
a Fermat hypersurface of degree δN1in the projective space PN,N1, where
λ= [λ0,...,λN]PN. Let 1pN1, and suppose that δ > (N+ 1)(Np).
Then any non-degenerate holomorphic map
f:CpHλPage 3
is algebraically degenerate, i.e. its image lies in an hypersurface of Hλ.
This result is well-known for the extremal cases. For p=N1, this follows
from the fact that Hλis of general type. For p= 1, this was first proved via
Nevanlinna theory: see e.g [Kob13][p.144-145]. Therefore, our result interpolates
between these two situations.
In the third and last part of this paper, we provide two applications in folia-
tion theory. The first application has starting point the following application of
Proposition 0.0.1, which was pointed out to me by Erwan Rousseau, following an
observation of Jorge Vitorio Pereira (see Section 3 for relevant definitions).
Proposition 0.0.5. Let Xbe a normal projective variety, and let Fbe a foliation
by curves on X. Suppose that Fis not algebraically integrable, i.e. that a general
leaf of Fis not algebraic. Then the canonical bundle of the foliation KFis pseudo-
effective.
Note that pseudo-effectiveness of canonical bundles of foliations was thoroughly
studied in the litterature, and one has in particular the following difficult result:
if the canonical bundle of a foliation on a smooth projective variety is not pseudo-
effective, then the foliation is uniruled (see [CP19]). Since a uniruled foliation
of rank 1is always algebraically integrable, this implies in particular the above
Proposition 0.0.5.
However, in Proposition 0.0.5, one should see the canonical bundle KFas the
cotangent bundle F. This is indeed the right terminology to obtain a result
valid for any foliation, which constitutes a result owing to works of Bost [Bos04],
Bogomolov and McQuillan [BM16], Campana and Paun [CP19], and Druel [Dru18]:
Theorem 0.0.6 ([Bos04], [BM16], [CP19], [Dru18]).Let Xbe a normal projective
variety, and let Fbe a foliation of rank pon X. Suppose that Fis not algebraically
integrable. Then the cotangent bundle of the foliation Fis pseudo-effective.
In this statement, the definition of pseudo-effectiveness is the following: the (re-
flexive) coherent sheaf F:=T
F(TFis the involutive saturated coherent subsheaf
of TX:= Ω
Xdefining the foliation F, see Definition 3.0.1) is pseudo-effective if
and only if for any cN1, there exists m, n N1satisfying m > nc such that
the following holds
H0(X, S[m]FLn)6={0},
where S[m]F:= (SmF)∗∗ is the reflexive hull of SmF, see Definition 3.0.3.
Using geometric generalized Wronskians, we were able to obtain a weak version
of the previous Theorem 0.0.6. Our first application in foliation theory (labeled
Theorem 3.1.2 in Section 3.1) reads then as follows:
Theorem 0.0.7 (Main Theorem III).Let Xbe a normal projective variety, Fa
foliation of rank p1on Xand LXan ample line bundle. Suppose that
Fis not algebraically integrable. Then for any cN1, there exists m, n N1
satisfying m > nc such that the following holds
H0(X, [m]
FLn)6={0}.
Therefore, the conclusion of our Theorem 0.0.7 concerns the non-vanishing of
(reflexive hulls of) twisted tensor products of the cotangent bundle of the foliation,
Page 4
whereas in Theorem 0.0.6, it concerns the non-vanishing of (reflexive hulls of)
twisted symmetric powers of the cotangent bundle of the foliation.
Our second application in foliation theory concerns positivity of adjoint line
bundles of the form
KF+L
on a foliated smooth projective variety (X, F), where Lis an ample line bundle
and Fis a regular foliation on X. A somewhat simplified statement reads as
follows:
Theorem 0.0.8 (Main Theorem IV).Let Xbe a smooth projective variety, Fa
regular foliation of rank p1and LXan ample line bundle. Suppose that
there exists a constant K > 0such that the following inequality holds outside a
countable union of points:
ε(L;x)K.
Then the line bundle KF+(p+1)
KLis nef.
In this statement, the number ε(L;x)denotes the Seshadri constant of Lat x
(see e.g. [Laz04][Section 5]). For instance, if one takes F=T X the trivial foliation,
one obtains informations on the usual adjoint line bundles of the form KX+mL,
where mN1and Lis an ample line bundle. Note however that our result
gives nothing new in this regard: powerful tools such as Kodaïra type vanishing
theorem (see [Laz04][Theorem 4.3.1]) allows to obtain much better conclusions
(see [Laz04][Proposition 5.1.19]). It may however provide indications that some
vanishing results involving the canonical bundle KXmight also work if one replaces
it by the canonical bundle of a foliation F. We refer to Section 3.2 for a more
detailed discussion.
The paper is organized as follows.
Section 1.1 is devoted to recalling the construction of generalized Wronskians as
well as the proof of their fundamental property, namely that they allow to detect
linear independance of holomorphic functions.
Section 1.2 introduces geometric generalized Wronskians, and is twofold. In Sec-
tion 1.2.1, we define jet differentials for p-germs, pN1, on a complex manifold:
these are the right geometric objects to define differential operators acting on p-
germs, e.g. generalized Wronskians of functions. In Section 1.2.2, we then define
geometric generalized Wronskian: these are the generalized Wronskians which,
when evaluated on sections of line bundles, act effectively on p-germs (not every
generalized Wronskian does so!).
Section 1.3 exhibits a particular family of geometric generalized Wronskians,
that we call pure geometric generalized Wronskians.
Section 1.4 is devoted to the proof of our Main Theorem 0.0.3. It is actually
a more precise version since we show the following: pure geometric generalized
Wronskians are enough to read linear independance of holomorphic functions. This
is the main technical part of the paper.
Section 1.5 describes a filtration of geometric generalized Wronskians that will
be used in our first application in folitation theory Theorem 0.0.7.
Page 5
Section 2 details the proof of our application in intermediate hyperbolicity Theo-
rem 0.0.4. We show the algebraic degeneracy of non-degenerate holomorphic maps
from Cp,pN1, to Fermat hypersurfaces in PNof degree δ > (N+ 1)(Np).
Section 3.1 details the proof of our first application in foliation theory Theorem
0.0.7, which gives a criterion for algebraic integrability of leaves of foliations.
Section 3.2 contains our second application in foliation theory Theorem 0.0.8,
where some specific geometric generalized Wronskians are used to study the posi-
tivity of adjoint line bundles of the form KF+mL, where Fis a regular foliation,
Lan ample line bundle, and ma positive integer.
Appendix A gives two formulas repeatedly used in the text: one is a Leibniz rule
to compute partial multi-derivatives of functions, and one is a formula to compute
partial multi-derivatives of composition of maps.
Appendix B details the proof of a vanishing result needed for our application in
intermediate hyperbolicity Theorem 0.0.4.
Conventions. Throughout this paper, a variety is a reduced and irreducible
scheme separated and of finite type over the field of complex numbers C.
1. Geometric Generalized Wronskians
1.1. Generalized Wronskians. In this section, we introduce generalized Wron-
skians as it was defined in e.g. [Rot55]. We prove their fundamental property,
following the scheme of proof given in [BD10]: note that it differs from the one in
given in [Rot55] or [BCL94] as it does not proceed by induction. Every detail is
given since the proof of our first Main Theorem 0.0.3 relies on it.
Let pN1be an integer. Denote Wpthe set of words written in the lexico-
graphic order with the alphabet {1,...,p}. Accordingly, any word u∈ Wpwrites
uniquely
u= 1α1(u)···pαp(u),
where by definition, αi(u)is the number of occurences of the letter iin the word
u. The length of the word uis the number (u) = α1(u) + ···+αp(u). Given a
natural number kN1, denote Wp,kthe set of words of length less or equal
than k, and Wp,k the set of words of length k.
Definition 1.1.1 (Size, order, weight and caracteristic exponent of sets of words).
Let Ube a finite set of words in Wp.
The size mof the set Uis its cardinal, i.e. m=|U|.
The order of Uis the maximal length kof a word belonging to U.
The weight of Uis the integer w(U):=P
u∈U
(u).
The caracteristic exponent β(U)Npof the set Uis defined as follows:
β(U):= (β1,...p),
where βi=P
u∈U
αi(u). In particular, one has that
|β(U)|=w(U).
Definition 1.1.2 (Admissible sets).We say that Uis an admissible set if and
only if there exists an ordering {u1, . . . , um}of the words in Usuch that for every
1im,(ui)i.
Page 6
Remark 1.1.3. The following ordering will be adopted. Start by listing every
word of length 1in U, according to the lexicographic order on {1,...,p}. Then,
list every word of length 2in U, according to the lexicographic order on {1,...,p}2.
Do this until all the words of Uare exhausted.
To any set of words Uof size mis associated the linear functional on Λm+1 OCp
defined as follows:
(WU)x(f0,...,fm):= det
f0· · fm
u1f0· · u1fm
· ·
· ·
umf0· · umfm
(x),
where the fi’s belong to OCp,x (i.e. are germs of holomorphic functions around x),
and where we recall that
u=α1(u)+···+αp(u)
∂zα1(u)
1···∂zαp(u)
p
.
Definition 1.1.4 (Generalized Wronskians).Let mN1be an integer, and let
Ube an admissible set of size m. The functional WUis called a pure generalized
Wronskian, and its size is by definition equal to m.
Denote by W(m)the finite dimensional C-vector space spanned by pure gen-
eralized Wronskians (WU)|U|=m. An element Wof W(m)is called a generalized
Wronskian of size m. Its order is by definition the least order of the admissible
sets appearing in the writing of W.
A generalized Wronskian Wis called unmixed of weight wif and only if the
admissible sets appearing in its writing all have the same weight w.
Remark 1.1.5. By definition, a pure generalized Wronskian is always unmixed.
There is a natural action of the group of biholomorphisms of (Cp,0) on the
vector space W(m)obtained as follows. Let ϕbe a biholomorphism of (Cp,0), let
xCp, and let f0, . . . , fmbe germs of holomorphic functions around x. Define
(ϕ·W)x(f0, . . . , fm):= (WU)0(f0(x+ϕ(·)),...,fm(x+ϕ(·))).
Using the formula Lemma A.0.2 to compute multi-derivatives of compositions of
holomorphic maps, as well as the multi-linearity of the determinant, one easily sees
that the functional ϕ·Wcan be expressed as a linear combination of the functionals
(WU)|U|=m, with constant coefficients depending on the multi-derivatives (evalu-
ated in 0) of the coordinates functions of the biholomorphism ϕ: the functional
ϕ·Wis indeed in W(m), and the action is well-defined. In particular, considering
only the linear biholomorphisms of (Cp,0), one obtains a representation of the
linear group GLp(C). Hence the natural following question:
Question 1. What are the irreducible components of the previous representation?
The fundamental property of generalized Wronskians is the following:
Theorem 1.1.6 ([Rot55]).Let f0,...,fmbe (m+ 1) be holomorphic functions on
Cp. They are linearly independant if and only if there exists an admissible set U
of size msuch that WU(f0,...,fm)6≡ 0.
Page 7
We detail the proof, following [BD10].
Proof. If the holomorphic functions f0,...,fmare linearly dependant, it follows
from the multi-linearity and alternating property of the determinant that, for
every admissible set Uof size m, the function WU(f0,...,fm)vanishes identically.
Thus, the non-trivial part amounts to proving that if (f0,...,fm)is a free family,
then there exists an admissible set Uof size msuch that WU(f0,...,fm)6≡ 0.
It is enough to prove this in the local case, i.e. one can suppose without loss of
generality that f0,...,fmare series in C[[z1,...,zp]]. Put the lexicographic order
on the exponents of the power series, and call the order of a power serie s, denoted
ord(s), the least exponent in the lexicographic order appearing effectively in the
writing of the serie s. We start with the following elementary lemma
Lemma 1.1.7. There exists t0,...,tmC[[z1,...,zp]] with distinct order, and an
invertible matrix AGLm+1(C)such that
(t0,...,tm) = (f0,...,fm)·A.
proof of Lemma 1.1.7. By applying a suitable permutation matrix A, one can be-
forehand suppose that
ord(f0)ord(f1)...ord(fm).
If the series already have distinct orders, there is nothing to prove. Otherwise,
let i0be the least integer isuch that ord(fi) = ord(fi+1). Then, as the family
(fi, fi+1)is free, there exists λCsuch that
>ord(fi+1 λfi)>ord(fi+1) = ord(fi).
Let Abe the matrix of transvection associated to the previous transformation, so
that
(f0,...,fi, fi+1 λfi, fi+2,...,fm) = (f0,...,fm)·A.
Apply now the same reasoning to the new free family of series
(˜
f1,..., ˜
fm) = (f1,...,fi, fi+1 λfi, fi+2,...,fm).
Reorder them, and consider the least integer i1such that the order of two series
is the same (if it exists, otherwise we are done). As i1> i0, an obvious induction
proves the result.
One immediately checks that, since the coefficients of the invertible matrix A
are constants, one has the equality
WU(t0,...,tm) = det(A)WU(f0,...,fm)
for every admissible set Uof size m. Therefore, with Lemma 1.1.7, one can suppose
without loss of generality that the series f0,...,fmhave distinct orders
αi= (α1,i,...,αp,i ).
The key observation is then the following: if for an admissible set Uof size m,
the holomorphic function WU(zα0,...,zαm)is non-zero, then so is WU(f0,...,fm).
(Recall the multi-index notation zα:=zα1
1. . . zαp
p). Indeed, by multi-linearity of
the determinant, one sees that in this situation, the monomial term with least
Page 8
order appearing in the writing of WU(f0,...,fm)comes from WU(zα1,...,zαm);
more specifically, this least order is then equal to
m
X
i=0
αiX
u∈U
α1(u),...,X
u∈U
αp(u).
To conclude the proof, it is now enough to prove that if the p-uples (αi)0im
are distinct, then there exists an admissible set Uof size msuch that
WU(zα0,...,zαm)6≡ 0.
For a proof of this statement, we refer to [BD10]. In Section 1.4, we will actually
prove a stronger statement.
1.2. Geometric Generalized Wronskians. In this section, we first recall the
construction of jet spaces and jet differentials bundles for p-germs into a complex
variety X,pN1. This is the right context to define generalized Wronskians
acting on sections of line bundles on the complex variety X: those generalized
Wronskians are called geometric. Their definition and properties is the object of
the second part of this section.
1.2.1. Jet spaces and jet differentials bundles for p-germs. Let Xbe a complex
manifold of dimension N, and let pN1. For kN1, define the bundle Jp,k X
of k-jets of p-germs of holomorphic maps γ: (Cp,0) Xon the complex manifold
Xas follows. Consider an atlas (Ui, ϕi)iIof X, and for iI, consider the (trivial)
bundle on Uiwhose fiber over xUiis the C-vector space of dimension N×|Wp,k|
u(ϕiγ)(0)u∈Wp,kγ: (Cp,0) (X, x)holomorphic p-germ .
Glue these trivial bundles Ui×CN×|Wp,k|via (the maps naturally induced by) the
transition maps ϕjϕ1
ito obtain (up to isomorphism) the bundle Jp,kX. The
general formula to change charts involves higher order derivatives of the transition
maps as soon as k > 1, and in particular, it does not preserve the structure of
vector space of the fibers: Jp,k Xis not a vector bundle for k > 1. For sake of
notation, denote p,kγthe element in (Jp,k X)xdefined by the holomorphic p-germ
γ: (Cp,0) (X, x).
Example 1.2.1. In the case where k= 1,Jp,1Xis pcopies of the tangent bundle
T X . In the case where p= 1, one recovers jet spaces of curves.
The torus (C)pacts on the fibers of Jp,k Xas follows
λ·u(ϕiγ)u∈Wp,k
=λα1(u)
1. . . λαp(u)
pu(ϕiγ)u∈Wp,k
,
where λ= (λ1,...,λp)(C)p. Indeed, the element λ= (λ1,...,λp)(C)p
induces a diagonal automorphism of Cp
Dλ: (z1,...,zp)7→ (λ1z1,...,λpzp),
and one has the equality
λ·p,kγ= (u(ϕiγDλ))u∈Wp,k.
Page 9
Therefore, the action commutes with a change of chart, and is thus well defined
on the bundle Jp,kX. More generally, the group of biholomorphisms of (Cp,0) acts
on Jp,kXby setting
ϕ·p,kγ:=p,k(γϕ),
where ϕis a biholomorphism of (Cp,0) and γ: (Cp,0) Xis a p-germ. Since one
considers derivatives up to the order k, the algebraic description of the previous
action is encoded in the following complex Lie group
Gp,k :=nX
1(u)k
a1,u·zu,..., X
1(u)k
ap,u·zuai,j 1i,jpGLp(C)o,
in which the composition law is given by reducing modulo every monomial of
degree strictly greater than k, and where zu:=zα1(u)
1···zαp(u)
p.
Define the vector bundle Ep,k,βXof partial jet differentials (of p-germs) of order
kand multi-degrees β= (β1,...,βp)Npas follows. Construct the bundle
whose fiber over xXis the vector space of complex valued polynomials Q
of muti-degrees βon the fiber (Jp,k X)x, i.e. for any λCpand any p-germ
γ: (Cp,0) (X, x), the polynomial Qsatisfies the equality
Q(λ·p,kγ) = λβQ(p,kγ),
where we recall the usual multi-indexes notation λβ=λβ1
1···λβp
p. The formula
to compute multi-derivatives of compositions of maps Lemma A.0.2 allows one to
see that the structure of vector space of the fibers is preserved under a change of
chart: the bundle Ep,k,βXis indeed a vector bundle.
Let wN1, and gather the multi-indexes βNpof same weight |β|=win
order to define the vector bundle of partial jet differentials of order kand weight
w
Ep,k,w X:=M
|β|=w
Ep,k,βX.
The group of biholomorhisms of (Cp,0) acts in a natural fashion on Ep,k,w Xby
acting on Jp,kX:
(ϕ·Q)(p,kγ):=Q(ϕ·p,k γ),
where Qis a partial jet differentials of order kand weight w, and ϕis a biholo-
morphism of (Cp,0). Indeed, observe that one has the following equality for any
λCand any ϕbiholomorphism of (Cp,0):
(λIpϕ)·p,kγ= (ϕλIp)·p,kγ,
which follows for instance from the formula Lemma A.0.2. It then readily implies
that if QEp,k,w X, then so does ϕ·Qfor any ϕbiholomorphism of (Cp,0).
Remark 1.2.2. Note that Ep,k,βXis not stable under the action of biholomor-
phisms, unless β= (0, . . . , 0).
One can define particular subbundles of Ep,k,wXby imposing a condition under
the action of the biholomorphisms of (Cp,0). Here, we introduce the following:
Definition 1.2.3 (Linear and invariant subbundles).The linear subbbundle of
Ep,k,w Xis defined as follows:
E1
p,k,w X:={QEp,k,wX| ∀ ϕbiholomorphism of (Cp,0), ϕ ·Q= d ϕ(0) ·Q}.
Page 10
This is the subbundle on which the action of the biholomorphisms of (Cp,0) reduces
to the action of the linear group GLp(C). The bundle E1
p,k,w Xhas itself a particular
subbundle, the invariant subbundle of Ep,k,wX, defined as follows:
Einv
p,k,w X:={QEp,k,wX| ∀ ϕbiholomorphism of (Cp,0), ϕ·Q= det(d ϕ(0))w
pQ}.
Note that this bundle is the zero-bundle if wis not divisible by p.
Example 1.2.4. In the case where p= 1, one recovers the usual invariant jet
differentials bundle. Note that in this situation, one has the equality
Einv
1,k,w X=E1
1,k,w X.
There is a natural filtration of the vector bundles Ep,k,βXobtained as follows.
Locally, an element of Ep,k,βXis a linear combination of monomial terms of the
following form
N
Y
i=1
(u1γ1)τ1···(uNγN)τN
where γ= (γ1,...,γN) : (Cp,0) Xis a p-germ (written in a fixed trivialization,
with N= dim(X)), and where the following is satisfied:
1jN, ℓ(uj)kand 1ip,
N
X
j=1
τj×αi(uj) = βi.
Fix u∈ Wp,k a word of length k, and define the total degree deguwith respect to
the word uas follows:
degu N
Y
i=1
(u1γ1)τ1···(uNγN)τN!:=X
i, ui=u
τi.
It is extended to any element Pof Ep,k,βXby taking the maximum degree of the
monomials appearing in the writing of P. The degree degudoes not depend on the
trivialization, as one can readily check by changing charts and using the formula
Lemma A.0.2 to compute multi-derivatives of compositions of maps. It is thus
well-defined on Ep,k,βXand it induces the following filtration Fuof Ep,k,βX:
Fu(Ep,k,βX) : 0 Fu,0(Ep,k,βX) · · · ⊂ Fu,|β|
k(Ep,k,βX) = Ep,k,βX
where Fu,d(Ep,k,βX)is the subvector bundle of elements of degree degud. Note
that for any 0d≤ ⌈|β|
k, one has the isomorphism
Fu,d(Ep,k,βX)/Fu,d1(Ep,k,βX)SdXFu,0(Ep,k, ˜
β(u,d))
where ˜
β(u, d):=βd×(α1(u),...,αp(u)). Inductively, this allows to obtain a
filtration of Ep,k,βXwhose graded terms are as follows:
O
u∈Wp,k
Sd(u)X
O···O
O
u∈Wp,1
Sd(u)X
,
where the integers d(u)satisfy the following equality for any 1jp
X
u∈Wp,k
d(u)αj(u) = βj.
Page 11
Remark 1.2.5. Let wbe an integer, and consider all the p-uples βof weight
|β|=w. Putting the previous filtrations together, one obtains a filtration for the
vector bundle Ep,k,wX.
1.2.2. Geometric generalized Wronskians. Fix mN1, and consider the following
subvector space Wg(m)of W(m):
Wg(m):={W∈ W(m)| ∀g, f0,...,fm∈ OCp, W (gf0,...,gfm) = gm+1W(f0,...,fm)}.
Definition 1.2.6 (Geometric generalized Wronskians).Ageometric generalized
Wronskian is an element of Wg(m), for some mN1.
Observe that the vector space of geometric generalized Wronskians of size mis
stable under the action of the biholomorphisms of (Cp,0). Hence the analogue of
Question 1:
Question 2. What are the irreducible components of the representation (of
GLp(C))Wg(m)?
Let Xbe a complex variety, equipped with a line bundle LX. By very
definition of geometric generalized Wronskians and partial jet differentials bundles,
one has the following proposition:
Proposition 1.2.7. Let s0,...,smbe global sections of the line bundle L. Let W
be an unmixed geometric generalized Wronskian of size m, order kand weight w.
The operator Winduces a global section W(s0,...,sm)of Ep,k,w XLm+1, defined
locally on a trivializing open set Uof Las follows:
W(s0,...,sm)(γ)loc
=W0(s0,U γ,...,sm,U γ),
where γ: (Cp,0) (X, x)is a p-germ passing through xU, and where the
notation si,U indicates that the section siis written under the trivialization of the
line bundle Lon the open set U.
Proof. Compute that, for any λ= (λ1,...,λp)(C)pand any full set Uof size
mand order k, one has the following equality
Dλ·WU=λβ(U)WU,
where we recall that Dλ= Diag(λ1,...,λp). This implies in particular that for
any p-germ γ: (Cp,0) (U, x), and any λ(C)pone has the following equality:
(WU)0s0,U (γDλ),...,sm,U (γDλ)=λβ(U)(WU)0s0,U γ,...,sm,U γ.
As (WU)0s0,U γ, . . . , sm,U γclearly writes as a polynomial in the coordinates
of p,kγ, this implies that WU(s0,U ,...,sm,U )defines a section of Ep,k,β(U)U.
Now, by hypothesis, the geometric generalized Wronskian Wwrites
W=X
U,w(U)=w
λUWU,
where λUC, and the above shows that W(s0,U ,...,sm,U )defines a section of
Ep,k,w U=M
|β|=w
Ep,k,βU.
Page 12
The very definition of geometric generalized Wronskians allows then to glue the
local sections W(s0,U , . . . , sm,U )into a global section W(s0,...,sm)of Ep,k,w X
Lm+1.
1.3. Pure geometric generalized Wronskians.
Definition 1.3.1 (Pure geometric generalized Wronskians).Let Ube an admissi-
ble set. One says that Uis a full set if and only if the set Usatisfies the following
property:
if a word ubelongs to U, then so does every one of its subwords.
In this situation, the generalized Wronskian WUis called a pure geometric gener-
alized Wronskian.
In order to justify this terminology, we show that pure geometric generalized
Wronskian are indeed geometric generalized Wronskians. To this end, one uses a
Leibniz rule for the operators u, given in Lemma A.0.1.
Proposition 1.3.2. A pure geometric generalized Wronskian in the sense of Defi-
nition 1.3.1 is a geometric generalized Wronskian in the sense of Definition 1.2.6.
Proof. Let Ube a full set of size m, and let g, f0,...,fmbe elements of OCp. One
must show the following equality:
WU(gf0,...,gfm) = gm+1WU(f0,...,fm).
Denote, for ua word belonging to U,Lu(resp. L
u) the line
Lu:= (uf0,...,∂ufm) (resp. L
u:= (u(gf0),...,∂u(gfm))) .
By convention, set L= (f0,...,fm)(resp. L
= (gf0,...,gfm)). Ob-
serve that those lines are the lines of the matrix defining WU(f0,...,fm)(resp.
WU(gf0,...,gfm)).
One proceeds by describing operations on the lines yielding the wanted result.
For sake of notation, one keeps the same notation for the lines after having per-
formed an operation on them: e.g. if one transforms L
u, then L
ustill denotes the
transformed line. Start by taking ua word of length 1in the full set U, and make
the operation on the lines
L
uL
uug·L,
which transforms L
uinto g·Luby the classic Leibniz rule. Do this for all the
elements of size 1in U. Take now ua word of length 2in U, and make the
following operation on the lines
L
uL
uX
u1·u2=u
u16=
1
gCu1,u2u1g·gLu2,
where Cu1,u2are constants defined in Lemma A.0.1. This is a well defined opera-
tion, keeping in mind the effect of the previous transformations and the fact that
Uis a full set. This operation transforms the line L
uinto the line g·Luby the
formula Lemma A.0.1. One performs this for all words of length 2in U. One keeps
Page 13
doing these operations until all the words in the full set Uare exhausted. The key
features are on the one hand formula Lemma A.0.1, which gives the equality
u(gf )g∂uf=X
u1·u2=u
u16=
Cu1,u2u1g·u2f,
and on the other hand the fact that in the sum on the right, the words v2ap-
pearing belong to U(since they are subwords of u) and are of length strictly less
than u: the previous operations made on the lines allow then to implement the
right operation to transform L
uinto g·Lu. To conclude, the multi-linearity and
alternating property of the determinant yields the equality
WU(gf1,...,gfm) = gm+1WU(f0,...,fm),
which is what we wanted to prove.
As a corollary of the proof of Proposition 1.2.7, one then has the following:
Proposition 1.3.3. Let s0,...,smbe global sections of the line bundle L. Let U
be a full set of size mand order k, with caracteristic exponent β=β(U). Then
WU(s0,...,sm)is in fact a global section of Ep,k,βXLm+1.
Note that in the case where p= 1, one easily shows that W(s0,...,sm)defines
in fact a section of Einv
1,k, m(m+1)
2
XLm+1: see [Bro17]. This is no longer true for
p > 1, unless for very specific Wronskians.
Example 1.3.4. If Uis the full set of order kcontaining every word of length i
for 1ik, i.e. U=Wp,k, then one checks that WU(s0,...,sm)defines an
invariant section: see Lemma 3.2.1.
1.4. Pure geometric generalized Wronskians and linear dependance. The
goal of this section is to prove that, in the statement of Theorem 1.1.6, one can
replace “generalized Wronskians“ with “pure geometric generalized Wronskians“:
Theorem 1.4.1. Let f0,...,fmbe (m+ 1) be m+ 1 holomorphic functions on Cp.
They are linearly independant if and only if there exists a full set Uof size msuch
that WU(f0,...,fm)6≡ 0.
Following verbatim the scheme of proof of Theorem 1.1.6 (as well as the nota-
tions), recall that it is enough to prove the following: if the p-uples (αi)0imare
distinct, then there exists a full set Uof size msuch that
WU(zα0,...,zαm)6≡ 0.
In the first part of this section, we introduce geometric Vandermondes as well as
relevant algebraic sets associated to them. Provided a suitable description of these
sets, we end the first subsection with the proof of Theorem 1.4.1. The second part
of this section, longer and more technical, is devoted to the proof of the description
of these algebraic sets.
Page 14
1.4.1. Geometric Vandermondes. Let pN1, and let mN1. Denote Fp,m the
set of full sets of size m, whose elements are words in Wp. For ua word in Wp,
denote
Xu(m):= (Y
k
xuk,0,...,Y
k
xuk,m),
where the product runs implicitely on the letters of the word u=u1u2..., and
where (xk,ℓ)1kp
0m
are indeterminates. By convention, one sets for the empty word:
X(m) = (1,...,1
|{z }
×(m+1)
).
Using the Kronecker product, one has the equality:
Xu(m) = Xα1(u)
1(m). . . Xαp(u)
p(m).
Let U Fp,m . Denote MU(resp. ˜
MU) the following square matrix
MU=
X(m)
Xu1(m)
·
·
Xum(m)
resp. ˜
MU=
Xu1(m1)
·
·
Xum(m1)
,
where {u1, . . . , um}is the listing of the words in Uobtained by applying the algo-
rithm of Remark 1.1.3.
Definition 1.4.2 (Geometric Vandermondes).The polynomial VU:= det MU
(resp. ˜
VU:= det ˜
MU) is called a geometric Vandermonde (resp. an homogeneous
geometric Vandermonde).
Define the following algebraic sets:
Vp,m =Z(VU| U Fp,m)(Cp)m+1 ;˜
Vp,m =Z(˜
VU| U Fp,m)(Cp)m.
Denote, for 0im,
Ci(p) =
x1,i
·
·
xp,i
.
For a full set U Fp,m, the polynomial VU(resp. ˜
VU) can be seen as depending
on the variables (C1(p),...,Cm(p)) (resp. (C1(p),...,Cm1(p))). As the index p
is already specified in the nature of set U, we will write
VU=VU(C1,...,Cm)resp. ˜
VU=˜
VU(C1,...,Cm1).
Note that this polynomial can also be seen as depending on the variables
(X1(m),...,Xp(m)) (resp. (X1(m1),...,Xp(m1))), so that we will also
sometimes write
VU=VU(X1,...,Xp)resp. ˜
VU=˜
VU(X1,...,Xp).
(Once again we drop the index m, as it is implicit). We aim at proving the following
description of the sets Vp,m and ˜
Vp,m.
Page 15
Theorem 1.4.3. Let p, m N1. The following holds:
(C0,...,Cm)(Cp)m+1 belongs to Vp,m if and only if there exists two distinct
indexes i6=jsuch that Ci=Cj.
Theorem 1.4.4. Let p, m N1. The following holds:
(C0,...,Cm1)(Cp)mbelongs to ˜
Vp,m if and only if there exists an index isuch
that Ci= 0, or two disctinct indexes i6=jsuch that Ci=Cj.
Before proceeding to the (simultaneous) proof of the two previous theorems, let
us finish the proof of Theorem 1.4.1.
Proof of Theorem 1.4.1. Suppose that the p-uples (αi)0imare distinct. We must
show that there exists a full set of size msuch that
WU(zα0,...,zαm)6≡ 0.
By Proposition 1.3.2, the functional WUis a geometric generalized Wronskian. By
multiplying each monomial zαiby the same monomial zα, one obtains therefore
the equality
WU(zα+α0,...,zα+αm) = z(m+1)αWU(zα0,...,zαm).
Therefore, up to replacing the p-uples (αi)0imby the p-uples (α+αi)0im, one
can always suppose that the αiare large enough so that for any word uappearing
in a full set Uof size m,uzαiis not identically zero.
To conclude, the key observation is the following equality for any full set Uof
size m
WU(zα0,...,zαm)(1) = VU(α0,...,αm),
which is shown using appropriate operations on the lines (in a fashion very similar
to the proof of Proposition 1.3.2) . The result then follows immediately from
Theorem 1.4.3.
1.4.2. Proof of Theorem 1.4.3 and 1.4.4. Let us denote Hp,m (resp. ˜
Hp,m) the
property that Theorem 1.4.3 (resp. Theorem 1.4.4) holds for the values p, m. We
first prove the following elementary lemma which relates these two properties, and
follows from definitions:
Lemma 1.4.5. Let p, m N1. If Hp,m holds, then so does ˜
Hp,m.
Proof. Let (C0,...,Cm1)be in ˜
Vn,m, and denote 0the zero column in Cp. Observe
that
MU(C0,...,Cm1,0) = ˜
MU(C0,...,Cm1),
so that VU(C0, C1,...,Cm1,0) = 0 for every U Fp,m . Since Hp,m holds, this
implies that two of these columns are equal, which gives exactly that either there
exists 0im1such that Ci= 0 or 1i < j m1such that Ci=Cj.
Accordingly, if Theorem 1.4.3 holds, so does Theorem 1.4.4. Before diving into
the proof Theorem 1.4.3, we start with the following lemma:
Page 16
Lemma 1.4.6. For any full set U Fp,m, and any column
C=
λ1
·
·
λp
,
where the λi’s are indeterminates, the following equality holds:
VU(C0,...,Cm) = VU(C0+C,...,Cm+C).
Proof. Consider the surjective map
π1:U → U =π1(U),u7→ u= 2α1(u)· · · · · pαp(u),
whose fibers form a partition of U. Pick an element uin U, and observe that
since Uis a full set, there exists k10such that
π1
1({u}) = {u,1·u,...,1k1·u}.
Consider the lines Xu, X1·u,...,X1k1·u. Denote Q=
p
Q
j=2
Tαj(u)
jZ[T2,...,Tn],
and observe that by definition
X1i·u=xi
1,0Q(x2,0,...,xp,0),...,xi
1,mQ(x2,m ,...,xp,m).
Make then successively the following operations on the lines:
X1k1·uX1k1·u+k1
1λ1X1k11·u+···+λk1
1Xu
X1k11·uX1k11·u+k11
1λ1X1k12·u+···+λk11
1Xu
• ·
• ·
X1·uX1·u+λ1Xu,
after which the line X1i·uhas been replaced by
(x1,0+λ1)iQ(x2,0,...,xp,0),...,(x1,m +λ1)iQ(x2,m,...,xp,m).
By doing this operation for every fiber, observe that one simply replaces, in the
arguments of VU, the line X1by the line X1+(λ1,...,λ1) = (x1,1+λ1,...,x1,m+λ1).
Therefore, the following equality is proved:
VU(X1+ (λ1,...,λ1), X2,...,Xp) = VU(X1, X2,...,Xp).
By considering the other projections
πj:u7→ u= 1α1(u)···(j1)αj1(u)·(j+ 1)αj+1(u)···pαp(u)
and doing the same reasoning, one finishes the proof of the lemma.
We now turn to the proof of Theorem 1.4.3 (giving also Theorem 1.4.4 by Lemma
1.4.5).
proof of theorem 1.4.3. Proceed by induction on (m+p)1, where mN0and
pN1. One first proves the result for the extremal cases, i.e. for m= 0,pN1
arbitrary, as well as for p= 1,mN0arbitrary. If m= 0, there is nothing to
prove since the condition is empty, and the algebraic set Vp,1is the whole set. If
Page 17
p= 1, the only full set in W1of size mis the set U={1,12,...,1m}, and the
polynomial associated is
VU=
1 1 · · 1
x1,0x1,1· · x1,m
· · · · ·
· · · · ·
xm
1,0xm
1,1· · xm
1,m
,
which is the usual Vandermonde determinant: the result is accordingly well-known.
Suppose the result granted for any mN1,pN2with m+pr3. The
goal is prove it for every mN1,pN2with m+p= (r+ 1).
Let (C0,...,Cm)be a zero of Vp,m. By considering only the full sets Ubelonging
to Fp1,m Fp,m, one deduces the following by induction hypothesis:
0i6=jm, ˆ
Ci=ˆ
Cj,
where given a column Cof size p, the column ˆ
Cis the column of size p1obtained
by suppressing the last entry of C. Without loss of generality, suppose that i= 0
and j= 1. By Lemma 1.4.6, one obtains that (0, C1C0,...,CmC0)is also a
zero of Vp,m. Compute then that for every U Fp,m, the following equality holds:
VU(0, C1C0,...,CmC0) = ˜
VU(C1C0,...,CmC0).
Accordingly, one is reduced to showing that ˜
Hn,m holds, in a particular case:
indeed, the column C1C0is zero everywhere except possibly at the last line
(since ˆ
C1=ˆ
C0). Consider accordingly the following algebraic set:
X:=n(xp,1, C2, . . . , Cm)C×(Cp)m1|
0
·
·
0
xp,1
|{z }
:=C1
, C2,...,Cm˜
Vp,mo.
One must show that if (xp,1, C2,...,Cm)I, then one of the following holds:
()(1im, Ci= 0) or (1i < j m, Ci=Cj).
Denote π:X(Cp)m1the projection onto the second factor, and observe
that π(X∩ {xp,16= 0})is included in the algebraic set
Y:={(C2,...,Cm)|(ˆ
C2,..., ˆ
Cm)˜
Vp1,m1}.
Indeed, there is an inclusion
a:Fp1,m1֒Fp,m Page 18
obtained by sending U Fp1,m1to a(U):={p} ∪ U Fp,m . One computes that
˜
Va(U)(
0
·
·
0
xp,1
, C2,...,Cm) = xp,1˜
VU(ˆ
C2,..., ˆ
Cm),
so that the inclusion follows.
Denote for 2im
Yi={(C2,...,Cm)Y|ˆ
Ci= 0},
as well as for 2i < j m
Yi,j ={(C2,...,Cm)Y|ˆ
Ci=ˆ
Cj}.
By induction hypothesis, the closed subsets Yiand Yi,j form the irreducible com-
ponents of Y. Consider
Oi=Yi∩ {xp,i 6= 0} ∩ { ˆ
Ck6=ˆ
C2k < ℓ m}
as well as
Oi,j =Yi,j ∩{xp,i 6=xp,j }∩{x1,i ···xp1,i 6= 0}∩{ ˆ
Ck6=ˆ
C2k < ℓ m, (k, ℓ)6= (i, j )}.
Those are dense open subsets of Yiand Yi,j respectively.
One first shows that for 2im, the map
πi:π1(Yi)Yi
is generically 21. More precisely, the following holds for any (C2,...,Cm)Oi:
(1) π1{(C2, . . . , Cm)}={(0, C2,...,Cm)(xp,i, C2,...,Cm)}.
It is straightforward to check that the fiber contains at least these two elements,
so that one must show that there are no others. Consider the injection
b:Fp1,m2֒Fp,m
obtained by sending U Fp1,m2to b(U) = {p, p2} ∪ U. Consider the matrix
˜
Mb(U)(
0
·
·
0
xp,1
, C2,...,Ci1,
0
·
·
0
xp,i
, Ci+1,...,Cm).
Page 19
By developping its determinant according to the two lines corresponding to the
words pand p2, one finds, up to a sign, the following equality:
˜
Vb(U)(
0
·
·
0
xp,1
, C2,...,Ci1,
0
·
·
0
xp,i
, Ci+1,...,Cm)
=
xp,1xp,i
x2
p,1x2
p,i
˜
VU(ˆ
C2,..., ˆ
Ci1,ˆ
Ci+1,..., ˆ
Cm).
On the one hand, by definition of Oiand by induction hypothesis, there exists
at least one full set U Fp1,m2such that ˜
VU(ˆ
C2,..., ˆ
Ci1,ˆ
Ci+1,..., ˆ
Cm)6= 0.
On the other hand, by definition of Oi, one has xp,i 6= 0. The equality (1) follows
immediately.
Second, one shows that for 2i < j m
πi,j :π1(Yi,j)Yi,j
is generically 11. More precisely, the following holds for any (C2,...,Cm)in
Oi,j :
(2) π1{(C2,...,Cm)}={(0, C2,...,Cm)}.
Once again, it is straightforward to see that the fiber contains at least this element,
so that one must show that it is the only one. By definition of Oi,j , and by induction
hypothesis, one can find a full set Uin Fp1,m2such that
˜
VU(ˆ
C2,..., ˆ
Ci1,ˆ
Ci+1,..., ˆ
Cm)6= 0.
Let 1k(p1) be a letter appearing in U, and consider the full set Uin Fp,m
defined as follows:
U={p, k ·p} ∪ U .
Consider the matrix
˜
MU(
0
·
·
0
xp,1
, C2,...,Cm),
and apply the following operation on the columns
ColiColiColj,
where Colis the th column of the previous matrix. Since ˆ
Ci=ˆ
Cj, all the
elements of the new column are zero, except for the elements corresponding to the
words pand k·p, which are respectively xp,i xp,j and xk,i (xp,i xp,j ). Develop
now the determinant of the new matrix (which is equal to the determinant of the
Page 20
previous one) according to the lines corresponding to the words pand k·pto find
that
˜
VU(
0
·
·
0
xp,1
, C2,...,Cm)
=
xp,1xp,i xp,j
0 (xp,i xp,j )xk,i
˜
VU(ˆ
C2,..., ˆ
Ci1,ˆ
Ci+1,..., ˆ
Cm).
Since by definition of Oi,j,xk,i 6= 0 and xp,i 6=xp,j , the very choice of Uimplies
the equality (2).
Now, conclude the proof as follows. Let x= (xp,1, C2,...,Cm)Xand let
y=π(x) = (C2,...,Cm). If xp,1= 0, then ()holds: suppose therefore that
xp,16= 0, so that ylies in Y. Suppose first that ybelongs to Yifor some 2im,
and that its fiber with respect to πis finite. If xp,i = 0, then Ci= 0, so that ()
holds. Suppose therefore that xp,i 6= 0. Accordingly, the fiber π1({y})must be
equal to
{(0, C2,...,Cm),(xp,i, C2,...,Cm)}
as these two elements lies in it, and one knows that the cardinal of the fiber
cannot exceed 2(since πiis generically 21). Therefore, one necessarily has that
xp,1=xp,i, so that C1=Ci(recall that ˆ
Ci= 0 by definition of Yi): the property
()holds.
Second, suppose that ybelongs to Yi,j for some 2i < j m, and that its
fiber with respect to πis finite. Then its fiber must necessarily be equal to
{(0, C2,...,Cm)},
as it contains at most one element (since πi,j is generically 11), and this element
obviously lies in it. In this case, one deduce that C1= 0, so that ()holds.
Finally, suppose that yhas infinite fiber. Define an injection
c:Fp,m1Fp,m
as follows: for U Fp,m1, let 0kmbe the greatest integer such that pkis
in U; define then c(U) = {pk+1} ∪ U . Now, observe that by developping according
to the first column the determinant of the matrix
˜
Mc(U)(
0
·
·
0
xp,1
, C2,...,Cm),
one obtains a polynomial in xp,1whose leading coefficient is ˜
VU(C2,...,Cm). For
y= (C2,...,Cm)such that the fiber π1({y})is not finite, the previous polynomial
has therefore an infinite number of roots (i.e. all the elements in the fiber) so that
it must be zero. It implies that for every U ∈ Fp,m1, one has the following:
˜
VU(C2,...,Cm) = 0.
Page 21
By induction hypothesis, there exists either 2imsuch that Ci= 0, or
2i < j msuch that Ci=Cj, and thus ()holds. This finishes the proof.
1.5. Filtration of geometric generalized Wronskians & representation
theory. Recall that for any mN1, the group of biholomorphisms of (Cp,0),
and in particular the linear group GLp(C), acts on the finite-dimensional C-vector
space Wg(m). In this section, we establish a filtration of this representation, and
we discuss an elementary property in representation theory: those will be needed
for our first application in foliation theory (see Section 3.1).
Definition 1.5.1 (Caracteristic sequence).Let Ube an admissible set of words of
size m, order kand weight w. Define the caracteristic sequence of Uas the k-uple
of positive integers
n(U):= (n1,...,nk),
where niis the number of words of length ibelonging to U. Note that m=
n1+n2+···+nkand w(U) = n1+ 2n2+···knk.
Let mN1, and let n= (n1,...,nk)Nk
1with kN1. The k-uple of
positive integers nis called a caracteristic sequence of order kof the integer mif
and only if it is the caracteristic sequence of an admissible set (of size mand order
k). The weight of the caracteristic sequence nis by definition
w(n) = n1+ 2n2+···knk,
i.e. it is the weight of any of the possible admissible sets associated to the carac-
teristic sequence n.
One orders such caracteristic sequences n= (n1,...,nk)
by increasing order of their order k;
at fixed order k, by decreasing order in the lexicographic order of Nk.
Namely, n<nif and only if, either the order of nis strictly less than the order
of n, or, if their order is equal to k, if and only if nis greater than nin the
lexicographic order of Nk. This defines a total order on caracteristic sequences.
Fix n= (n1,...,nk)a caracteristic sequence of m, and define
Wn
g(m):={W∈ Wg(m)|W=X
Uadmissible,
n(U)n
λUWU, λUC}.
Observe that Wn
g(m)is stable under the action of the biholomorphisms of (Cp,0)
(thus in particular under the action of the linear group GLp(C)), which follows
immediately from the formula Lemma A.0.2 to compute multi-derivatives of com-
positions of maps. This gives a filtration of the representation Wg(m)by sub-
representations of Gp,m. We will also denote
W<n
g(m):={W∈ Wg(m)|W=X
Uadmissible,
n(U)<n
λUWU, λUC}.
Remark 1.5.2. The previous filtration is also valid if one replaces Wg(m)by
W(m).
Page 22
Let Ube an admissible set of size mand caracteristic exponent β. The gener-
alized Wronskian WUis a weight vector of weight β(in the usual sense in repre-
sentation theory), i.e. one has the following equality
(3) Diag(λ1,...,λp)·WU=λβ1
1···λβp
pWU
for any λ1,...,λpin C.
Suppose that Vis an irreducible representation of GLp(C)appearing in
Wn
g(m), and suppose that this representation does not come from the ones of
W<n
g(m). One knows that for any λCand any W∈ V, one has the egality
Diag(λ, . . . , λ)·W=λrW
for some integer rN1. In particular, this implies that every element of Vis an
unmixed geometric generalized Wronskians of weight r. Since a pure generalized
Wronskian WU, with Uan admissible set of caracteristic sequence n(U) = n, must
appear in the writing of at least one element W∈ V (recall that we assumed that
V 6⊂ W<n
g(m)), one deduces from (3) that necessarily
r=w(n) = n1+ 2n2+···=β1+β2+··· .
Then, by standard facts in representation theory, one deduces that there exists a
partition λof w(n)such that
V SλCp.
This observation will be used in the course of the proof of Theorem 3.1.2.
2. Application in intermediate hyperbolicity
In this section, we start by recalling a fundamental vanishing theorem of Siu,
which asserts that on a projective manifold X, every entire curve must satisfy the
differential equation defined by a global jet differentials vanishing along an ample.
We then state a similar statement concerning non-degenerate holomorphic maps
CpX: every such map must satisfy the partial differential equation defined by
a global partial jet differentials vanishing along an ample. We give the proof of
this statement in Appendix B, following the approach of the original proof, but
using Nevanlinna theory in several variables (instead of one variable).
Then, as an application of this vanishing theorem and our construction of global
partial jet differentials via Wronskians, we prove Theorem 0.0.4, i.e. we show
the algebraic degeneracy of non-degenerate holomorphic maps from Cpto Fermat
hypersurfaces of degree δin projective spaces PN, as soon as δ > (N+ 1)(Np).
2.1. A vanishing theorem. Recall the following fundamental vanishing theorem,
due to Siu:
Theorem 2.1.1 (Siu).Let Xbe a projective manifold equipped with an ample line
bundle L, and suppose that there exists a (non-zero) global section
PH0(X, E1,k,w XL1)
where k, w N1. Then any entire curve f:CXsatisfies the differential
equation
P(f) = Pf(1,kf)0.
Page 23
Recall that the conclusion of this theorem means that for any zC, the evalu-
ation of Pf(z)on the 1-germ
f(z+·): (C,0) (X, f (z))
is zero. This theorem was first proved by Siu using methods from Nevanlinna
theory: see [Dem97b] for a short exposition of the proof. Later, Demailly proved
this vanishing result first for invariant jet differentials bundles Einv
1,k,w , using the
Demailly-Semple tower and the Ahlfors-Schwarz lemma. Then, with a clever ar-
gument, he obtained the general statement for every jet differentials bundles (see
[Dem20][Chapter 7])). In the setting of p-germs, Siu’s scheme of proof can be
carried over verbatim, with some Nevanlinna theoretic considerations in several
variables. In Appendix B, we prove the following generalization of Theorem 2.1.1:
Theorem 2.1.2. Let Xbe a projective manifold equipped with an ample line bundle
L, and suppose that there exists a (non-zero) global section
PH0(X, Ep,k,w XL1),
where p, k, w N1. Then any non-degenerate map f:CpXsatisfies the
partial differential equation
P(f) = Pf(p,k f)0.
Remark 2.1.3. The Demailly-Semple tower and the Ahlfors Schwarz lemma can
be generalized in the setting of p-germs, p > 1, and Demailly’s scheme of proof
leads to a vanishing result (see [PR11]). However, for p > 1, the bridge between
the generalized Demailly-Semple tower and the invariant p-jet differentials is not
completely clear (see Problem 2.9 in [PR11]). Furthermore, while the gap between
invariant and non-invariant jet bundles is small for p= 1 (see [Dem20]), this is no
longer true for p > 1.
2.2. Holomorphic maps from Cpto a Fermat hypersurface. Recall the fol-
lowing standard definition:
Definition 2.2.1 (Fermat hypersuface).A Fermat hypersurface Fof degree δ1
inside the projective space PN,N1, is any hypersurface defined by an equation
of the following form
λ0Xδ
0+···+λNXδ
N= 0
where (λ0,...,λN)is a non-zero vector in CN+1.
The goal of this section is to prove the following theorem:
Theorem 2.2.2. Let HPNbe a Fermat hypersurface of degree δ. If
δ > (N+ 1)(Np),
then any non-degenerate holomorphic map f:CpHis algebraically degenerate.
More precisely, there exists a Fermat hypersurface H(PN1(PNof degre