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ON THE SCHMIDT-KOLCHIN CONJECTURE ON DIFFERENTIALLY
HOMOGENEOUS POLYNOMIALS. APPLICATIONS TO (TWISTED)
JET DIFFERENTIALS ON PROJECTIVE SPACES.
ANTOINE ETESSE
Abstract. The main goal of this paper is to prove the Schmidt–Kolchin conjecture.
This conjecture says the following: the vector space of degree ddifferentially homoge-
neous polynomials in (N+ 1) variables is of dimension (N+ 1)d. Next, we establish a
one-to-one correspondance between differentially homogeneous polynomials in (N+ 1)
variables, and twisted jet differentials on projective spaces. As a by-product of our
study of differentially homogeneous polynomials, we are thus able to understand ex-
plicitly twisted jet differentials on projective spaces.
Contents
Introduction. 1
Organization of the paper. 4
1. Preliminaries. 5
1.1. Representation theory for symmetric groups and general linear groups. 5
1.2. Zero-dimensional systems of partial differential equations with constant
coefficients. 9
2. Structure of differentially homogeneous polynomials. 10
2.1. Families of differentially homogeneous polynomials. 10
2.2. A natural basis for the highest weight vectors of V(k)
d. 14
2.3. An intermediate algebraic problem. 15
2.4. Where one relates the previous problem to the question at hand. 19
2.5. Proof of the Schmidt–Kolchin conjecture. 21
3. Differentially homogeneous polynomials and twisted jet differentials on
projective spaces. 23
3.1. Green–Griffiths vector bundles on complex manifolds. 23
3.2. Differentially homogeneous polynomials and twisted jet differentials on
projective spaces. 24
3.3. On the zeroth cohomology group of twisted Green–Griffiths bundles of
projective spaces. 25
Appendix A. 27
Appendix B. 29
References 31
Introduction.
The goal of this paper is to prove the Schmidt-Kolchin conjecture on differentially
homogeneous polynomials, and give applications to Green–Griffiths bundles of projective
2020 Mathematics Subject Classification. 12H05, 13N15, 14A25, 15A69, 20C15, 32N10, 35E20.
Key words and phrases. Differential polynomials, Schmidt–Kolchin conjecture, Green–Griffiths bun-
dles of projective spaces, Jet differentials of projective spaces.
1
spaces. In order to state this conjecture, let us first set some notations (which are going
to be used throughout the whole paper). We fix a natural number N∈N≥1, and consider
the complex vector space of differential polynomials
V:=C(X(k))k∈N]
in the formal variables X(k):= (X(k)
0, . . . , X(k)
N). An element in this vector space is usually
called a differential polynomial (in the variables X= (X0, . . . , XN)). The upper index
in parenthesis is reminiscent of a formal derivation, for the following reason. Consider
a formal variable T, and for any differential polynomial P∈Vand any polynomial
Q∈C[T], form a new polynomial Q·P∈V[T]by setting
Q·P:=P(QX)(0),(QX )(1) , . . . .
Here, for any k∈N, the symbol (QX)(k)is by definition:
(QX)(k):=
k
X
i=0 k
iQ(k−i)X(i).
Namely, one applies formally the usual Leibnitz rule. There are two natural notions of
homogeneity on V. The first one is the usual one: a differential polynomial P∈Vis
homogeneous of degree dif and only if the following equality holds for any λ∈C:
λ·P=λdP.
The second notion of homogeneity, called differential homogeneity, is defined as follows.
A differential polynomial P∈Vis called differentially homogeneous of degree dif and
only if, for any polynomial Q∈C[T], the following equality holds:
Q·P=QdP.
Accordingly, a differentially homogeneous polynomial of degree dis necessarily homoge-
neous of degree d. Denote by
Vdiff ⊂V
the sub-vector space of differentially homogeneous polynomials.
The vector space Vis naturally filtered by the maximal order of derivation
(0) ⊂V(0) ⊂V(1) ⊂ · · · ⊂ V(k)⊂ · · · ,
where for any k∈N, one defines V(k):=C(X(i))0≤i≤k]. Furthermore, each piece in the
filtration admits a natural and compatible graduation given by the degree of homogeneity
V(k)=M
d∈N
V(k)
d,
where V(k)
dis the set of differential polynomials in V(k)that are homogeneous of degree
d. Both the filtration and the graduation descend to the sub-vector space Vdiff , and we
will denote by
(Vdiff
d)(k)
the sub-vector space of differentially homogeneous polynomials in V(k)
d.
We are now ready to state the so-called Schmidt–Kolchin conjecture:
Conjecture 1 (Schmidt–Kolchin).The dimension of the vector space Vdiff
dof differen-
tially homogeneous polynomials of degree din the variables X= (X0, . . . , XN)is equal
to (N+ 1)d.
Page 2
The origin of this questions goes back to Schmidt’s paper [Sch79], where he proved
that the dimension is at least equal to (N+ 1)d, and suggested that "Perhaps, equality
is true here; no upper bound seems to be known at present". A few years later, Kolchin
proved in [Kol92] that the conjecture holds for N= 1 and d= 1,2,3. Then, in [RS96],
Reinhart and Sit showed that the following equality holds for N= 1:
Vdiff
d= (Vdiff
d)(d−1).
In particular, this implies that the vector space of differentially homogeneous polynomials
of a given degree d∈N≥1in the variables X= (X0, X1)is finite dimensional. Three years
later, Reinhart improved the previous result, and solved the Schmidt–Kolchin conjecture
for N= 1 in [Rei99]. Since then, no progress has been made towards this conjecture. It
is not even known if the vector space Vdiff
dis finite dimensional when N > 1.
In this paper, we solve the Schmidt–Kolchin conjecture, and prove the following (see
also Theorem 2.5.3):
Theorem 0.0.1. The dimension of Vdiff
dis equal to (N+ 1)d. Furthermore, an explicit
basis of this vector space is provided, and one deduces accordingly the equality
Vdiff
d= (Vdiff
d)(d−1).
In a few words, the strategy of the proof is as follows (for more details, see Organization
of the paper). One easily sees (see the beginning of Section 2) that there is a natural
action of the general linear group on Vd, which descends to Vdiff
d. Fixing k∈Na natural
number, this makes (Vdiff
d)(k)into a sub-representation of V(k)
d. There is a natural basis of
the vector space V(k)
d,hw of highest weight vectors of V(k)
d(see Section 1.1.3 for definitions,
and Section 2.2). Therefore, the problem amounts to understanding which suitable linear
combinations of elements in this basis give differentially homogeneous polynomials. The
key part in the proof lies in a suitable algebraic formulation of this problem, and the way
it is tackled (see Section 2.3 and Section 2.4).
Beside the intrinsic interest one may have in such a question, our motivation was
rather geometric. It turns out that there is one-to-one correspondance between differ-
entially homogeneous polynomial in the variables X= (X0, . . . , XN)and (positively)
twisted jet differentials on the projective space PN(see Section 3.1 for definition of jet
differentials on complex manifolds, and see Proposition 3.2.1). As a corollary of our study
on differentially homogeneous polynomials, we obtain the second main theorem of this
paper (see Section 3.1 for notations and definitions):
Theorem 0.0.2. Let d∈Z,k≥0and n≥0be natural numbers. Then:
(1) for any k≥d−1and any n∈N, one has the isomorphism
H0(PN, Ek,nPN(d)) 'H0(PN, Ed−1,n PN(d));
(2) the following equality holds:
∞
X
n=0
dim H0(PN, Ed−1,nPN(d)) = (N+ 1)d;
(3) for any k∈Nand any n > b(1 −1
N+1 )d2
2c, one has the vanishing:
H0(PN, Ek,nPN(d)) = (0).
It should be noted that the previous Theorem 0.0.2 does not encapsulate all the in-
formation we have on twisted jet differentials on projective spaces. Indeed, since dif-
ferentially homogeneous polynomials are explicitly understood, so are these twisted jet
differentials (see Proposition 3.2.1). Whereas, being given a differentially homogeneous
Page 3
in the canonical basis of Vdiff (see Section 2.1.1), it is easy to determine which global
section it defines (see Proposition 3.3.1), it seems to be quite difficult to count the num-
ber of differentially homogeneous polynomials falling into a same class. Such a purely
combinatoric problem might be the object of a future work, allowing to obtain closed
formulas for the dimension of the space of global sections of Green–Griffiths bundles of
projective spaces.
To finish this introductory part, we would like to mention another motivation. As
any algebraic geometer knows, to any graded ring Sis associated the scheme Proj(S),
which is of particular interest if Sis furthermore finitely generated as a S0algebra. It is
therefore very natural to study, for any k∈N, the (bigger and bigger as kgrows) scheme
Proj M
d≥0
(Vdiff
d)(k),
along with its closed subschemes. This is the object of an ongoing work.
Organization of the paper. The paper is organized as follows.
Section 1 is devoted to preliminaries.
In Section 1.1, we recall some classic facts on representation theory of symmetric
groups and general linear groups, allowing along the way to fix notations.
In Section 1.2, we recall a correspondance between systems of zero-dimensional partial
differential equations with constant coefficients on the one hand, and algebraic varieties
on the other hand. Such a correspondance allows to evaluate the number of independent
solutions of a system of PDE’s via methods coming from intersection theory.
Section 2 is devoted to the proof of the Schmidt–Kolchin conjecture.
In Section 2.1, following [Rei99], we introduce, for each integer d≥0, a natural sub-
family Vdiff
d⊂Vdiff
dof differentially homogeneous polynomials of degree d, which is shown
to be (N+1)d-dimensional.1The main result of this Section 2.1 is the following: we show
that Vdiff
dis actually a sub-representation of Vdiff
dfor the natural action of the general
linear group.2As a crucial corollary, one deduces a lower bound on the dimension of
λ-highest weight vectors in Vdiff
d, for λ`da partition of dwith at most (N+ 1) parts:
(1) dim Vdiff
hw(λ)≥fλ.
In Section 2.2, we fix k∈N≥0an integer, and provide a natural basis for the vector
space
V(k)
d,hw
of highest weight vectors in the finite-dimensional representation V(k)
d.
In Section 2.3, we study a purely algebraic problem, which consists in the evaluation
of the dimension of the space of vectors lying simultaneously in the kernel of a family
of particular endomorphisms of a fixed finite-dimensional vector space. We feel that
one of the main originality of this paper lies in the way we tackled this problem, as
we crucially used the correspondance between systems of PDE’s and algebraic varieties,
briefly described above.
In Section 2.4, we provide a set of equations characterizing highest weight vectors of
V(k)
dthat are differentially homogeneous. The technical part of this Section 2.4 amounts
1As indicated in the Introduction, the lower bound
dim(Vdiff
d)≥(N+ 1)d
was already proved by Schmidt in [Sch79]. However, he did not prove this result by exhibiting an explicit
free family of differentially homogeneous polynomials.
2Believing in the Schmidt–Kolchin conjecture, this is the least one could expect.
Page 4
to relate this set of equations to the previous algebraic problem. At this stage, the small
miracle is that the dependency on Nand k(almost) vanishes. The outcome of this
Section 2.4 is that we have an upper bound for the number of independent λ-highest
weight vector in (Vdiff
d)(k)(where λ`dis partition with at most (N+ 1) parts), which
depends only on d.3
In Section 2.5, we finish the proof of the Schmidt–Kolchin conjecture, and to this end,
the lower bound (1) turns out to be crucial.
Section 3 is devoted to establishing the correspondance between differentially homoge-
neous polynomials and twisted jet differentials on projective spaces, and infer applications
from the previous study.
In Section 3.1, we recall the construction of Green–Griffiths bundles on complex man-
ifolds.
In Section 3.2, we establish the correspondance between differentially homogeneous
polynomials, and twisted jet differentials on projective spaces.
In Section 3.3, we study the 0th cohomology group (i.e. the vector space of global
sections) of twisted Green–Griffiths bundles via the acquired knowledge on differentially
homogeneous polynomials.
Acknowledgements. I would like thank Gleb Pogudin, as I learnt from him the termi-
nology of differential homogeneity, as well as the Schmidt–Kolchin conjecture. It turns
out that I had used the defining property of differential homogeneity in a previous paper,
and called it geometric, out of ignorance. This is precisely the geometric implications of
the Schmidt–Kolchin conjecture that motivated me to tackle it.
I also would like to thank the Institut de Mathématiques de Toulouse (IMT) for pro-
viding a very comfortable and stimulating working environment. And of course, I am
very grateful to the CIMI LabEx for giving me the opportunity to work here.
1. Preliminaries.
In this Section 1, we gather a few results that will be used during our study of differen-
tially homogeneous polynomials. These results come from representation theory on the
one hand (Section 1.1), and systems of partial differential equations on the other hand
(Section 1.2).
1.1. Representation theory for symmetric groups and general linear groups.
The references for this section are [Ful97], [FH91] and [Boe70].
1.1.1. Partitions, Young tableaux, and some combinatorial identities. Recall that a par-
tition λof a natural number d≥1is simply a finite, decreasing sequence of positive
natural numbers whose sum equals to d. To any partition λwith sparts
λ= (λ1≥ · · · ≥ λs>0)
is associated a Young diagram of shape λ, which is a collection of cells arranged in left-
justified rows: the first row contains λ1cells, the second λ2, and so on (cf. Figure 1 for
an example).
Fix n∈Na natural number, and Ta Young diagram of shape λ. A Young tableau
filled with the set {1, . . . , n}is, by definition, a filling Tof the boxes of the Young diagram
Twith the numbers {1, . . . , n}. Furthermore, if one imposes that
•browsing from left to right a line of the diagram, the sequence of numbers is
non-decreasing;
3Accordingly, at this stage, one has already proved that Vdiff
dis finite dimensional.
Page 5
(λ1= 5)
(λ2= 3)
(λ3= 3)
(λ4= 1)
Figure 1. Young diagram of the partition λ= (5,3,3,1).
•browsing from top to bottom a column of the diagram, the sequence of numbers
is increasing.
then such a Young tableau is called a semi-standard Young tableau of shape λ, filled
with the set {1, . . . , n}. In the case where n=d, and where one imposes furthermore
that each number 1≤k≤doccurs exactly once in the tableau, then the tableau Tis
called a standard tableau of shape λ.
One is naturally lead to consider the following (combinatorial) numbers:
•fλ:=Tstandard Young tableau of shape λ;
•dλ(n):=Tsemi-standard Young tableau of shape λ, filled with {1, . . . , n},
Furthermore, for a fixed n-uple a= (a1, . . . , an)∈Nn, with |a|=|λ|=d, define the
so-called Kostka numbers:
Kλ,a:=Tsemi-standard Young tableau of shape λ, filled with a11’s ,. . .,ann’s.
Note that, from the very definition of Kostka numbers, the following equality holds:
(2) dλ(n) = X
|a|=d
Kλ,a.
The Robinson–Schensted–Knuth correspondance (see e.g. [Ful97][I.4]) allows to obtain
the following identities:
Proposition 1.1.1. The following equalities hold for any d≥1and any n≥1:
•d! = P
λ`d
f2
λ;
•nd=P
λ`d
fλdλ(n).
Here, the symbol λ`dmeans that λis a partition of the natural number d.
1.1.2. Representations of symmetric groups. Let d∈N≥1, and denote by Σdthe sym-
metric group of the set {1,2, . . . , d}. It is well-known (see e.g. [FH91][Lectures 1 & 2])
that there are as many irreducible representations of Σdas there are conjugacy classes
in Σd. Note that the conjugacy classes are in one-to-one correspondance with partitions
of d.
Denote by C[Σd]the regular represention of Σd, i.e.
C[Σd]:=M
σ∈Σd
C·σ.
Note that C[Σd]is tautologically a representation of the symmetric group Σd. Note also
that any representation (M, ρ : Σd→GLC(M)) of the symmetric group Σdhas a natural
structure of (left) C[Σd]-module defined as follows. For m∈Mand (λσ∈C)σ∈Σd, set:
(X
σ∈Σd
λσσ)·m:=X
σ∈Σd
λσρ(σ)(m).
Every irreducible representations of Σdappears in C[Σd](this is actually a general fact
for the regular representation of any finite group: see e.g. [FH91][Lecture 2]). One has
Page 6
furthermore canonical projections onto irreducible factors, which are defined as follows.
Let λ`dbe a partition of d, and let Tbe a standard Young tableau with shape λ. To
the standard Young tableau Tis associated the following two subgroups of Σd:
•the subgroup R(T)of permutations that preserve the rows of T;
•the subgroup C(T)of permutations that preserves the columns of T.
The so-called Young symmetrizer associated to the tableau Tis the vector in C[Σd]
defined as follows:
cT:=X
q∈C(T)
(q)q
| {z }
:=bT
×X
p∈R(T)
p
| {z }
:=aT
,
where (·)is the signature. One has the following theorem (see e.g. [Boe70][IV. 3]):
Theorem 1.1.2. The following holds.
•The multiplication map
pT:C[Σd]−→ C[Σd]
v7−→ v×cT
is an almost-projection4onto an irreducible representation of Σd.
•For two different standard tableaux Tand T0with same shape λ`d, the irre-
ducible representations Im(pT)and Im(pT0)are isomorphic. The abstract irre-
ducible representation is usually denoted Sλ.
•For two standard tableaux Tand T0with different shape, the irreducible represen-
tations Im(pT)and Im(pT0)are non-isomorphic.
•The representation C[Σd]splits as follows into direct sum of irreducible represen-
tations:
C[Σd] = M
T
Im(pT),
where Truns over standard tableaux with dboxes.
1.1.3. Representations of general linear groups. Let n∈N≥1, and consider the general
linear group GLn(C), along with the subgroups D⊂B⊂GLn(C)where
•Dis the subgroup of diagonal matrixes;
•Bis the Borel subgroup of upper triangular matrixes.
Fix Ea finite-dimensional polynomial5representation of GLn(C). A weight vector with
weight λ= (λ1, . . . , λn)∈Nnis a vector e∈Esuch that, for every x1, . . . , xn∈C, the
following equality holds:
Diag(x1, . . . , xn)·e= (xλ1
1· · · xλn
n)e.
Aλ-highest weight vector is a weight vector of weight λwhich is left invariant, up to
homothety, by B. Namely, the following equality holds:
B·e=C∗e.
General theory of representations (see e.g [Ful97][II.8] and [FH91][Lectures 14 & 15])
tells that finite-dimensional polynomial representations are uniquely determined by their
highest weight vectors. More precisely, one has that
•every representation splits as a direct sum of irreducible representations;
•a representation is irreducible if and only if it has a unique highest weight vector,
up to a multiplicative factor;
4By definition, this means that pT◦pT=λpTfor some non-zero scalar λ.
5This means that the map ρ: GLn(C)→GL(E)is polynomial, i.e. after choosing a basis for E, the
dim(E)2coordinates functions of ρare polynomial in the n2variables of GLn(C).
Page 7
•two irreducible representations are isomorphic if and only if their highest weight
vector has same weight.
Furthermore, in the case of the general linear group, possible highest weight vectors
λ∈Nnare in one-to-one correspondance with partitions with at most nparts. Said
otherwise, to any λ∈Nnsatisfying
λ1≥λ2≥ · · · ≥ λn≥0
is associated a unique irreducible representation of GLn(C), and vice-versa. This unique
representation is denoted SλCn, and called λth Schur power of Cn. For later use, let us
record the following classic proposition (see e.g. [Ful97][II.8.3]):
Proposition 1.1.3. The dimension of the representation SλCnis equal to dλ(n).
To any finite-dimensional polynomial representation Eis associated its character,
denoted χE, which is by definition the symmetric polynomial in the variables x=
(x1, . . . , xn)
χE(x):= Trace(Diag(x)).
In the case where E=SλCn, the character χEis called the λth Schur polynomial (in
the variables x= (x1, . . . , xn)), and is denoted by sλ. General theory on symmetric
polynomials (see e.g. [Ful97][I.2.2]) shows that Schur polynomials form a basis of sym-
metric polynomials. Accordingly, one deduces that a representation Eof the general
linear group is uniquely determined by its character χE.
There is a natural interaction between representations of symmetric groups and general
linear groups given as follows. Let E= (Cn)⊗d. On the one hand, it is a natural
representation of the general linear group GLn(C). On the other hand, it has a natural
structure of right C[Σd]-module given by permutations of factors of tensor products.
Namely, for σ∈Σdand v1, . . . , vd∈Cn, one sets:
(v1⊗ · · · ⊗ vd)·σ:=vσ(1) ⊗ · · · ⊗ vσ(d).
The simple but important observation is that these actions are compatible, in the sense
that they commute with each other:
A·((v1⊗ · · · ⊗ vd)·σ)=(A·(v1⊗ · · · ⊗ vd)) ·σ.
Here, Ais a matrix in GLn(C),σis an element in Σd, and v1, . . . , vdare vectors in Cn.
For any representation Mof Σd, seen as a left C[Σd]-module, denote by
E(M):=E⊗C[Σd]M
the tensor product over C[Σd]between Eand M. In particular, one has tautologically
the equality
E(C[Σd]) = E.
Following what was seen in the previous Section 1.1.2, one is naturally lead to consider,
for Ta standard Young tableau with dboxes, the linear map
E(pT): E−→ E
v7−→ v⊗C[Σd]cT.
Using in particular Theorem 1.1.2, one can show the following (see e.g. [Boe70][V.4] and
[Ful97][II.8]):
Theorem 1.1.4. The following holds.
Page 8
•The linear map
E(pT): E−→ E
v7−→ v⊗C[Σd]cT
is an almost-projection onto an irreducible representation of GLn(C).
•For two different standard tableaux Tand T0with same shape λ`d, the ir-
reducible representations Im(E(pT)) and Im(E(pT0)) are isomorphic to the λth
Schur power SλCn'E(Sλ).
•For two standard tableaux Tand T0with different shape, the irreducible represen-
tations Im(E(pT)) and Im(E(pT0)) are non-isomorphic.
•The representation E= (Cn)⊗dsplits as follows into direct sum of irreducible
representations:
E=M
T
Im(E(pT)),
where Truns over standard tableaux with dboxes.
1.2. Zero-dimensional systems of partial differential equations with constant
coefficients. Fix n∈N≥1a natural number, and denote by O(Cn)the set of entire
functions on Cn(i.e. the set of holomorphic maps in nvariables defined everywhere).
Consider
A:=C[∂1, . . . , ∂n]
the polynomial ring in the formal variables ∂1, . . . , ∂n. The set of holomorphic maps
O(Cn)has a natural structure of A-module defined as follows on monomials
(∂α1
1· · · ∂αn
n)·P:=∂α1+···+αnP
(∂z1)α1· · · (∂zn)αn,
and extended by linearity. To any ideal I⊂Ais associated a system S(I)of partial
differential equations (PDE’s):
S(I):={P∈ O(Cn)|E·P≡0∀E∈I}.
Note that since Ais Noetherian (by Hilbert’s basis theorem), S(I)is defined by a finite
number of partial differential equations with constant coefficients.
Suppose that the ideal Iis zero-dimensional, i.e. the affine variety Spec(A
I)is finite.
In this case, one can relate the dimension (as a C-vector space) of the space of solutions
S(I)to the algebraic structure of the quotient space A
I. Indeed, one of the main result
that initiated the theory of systems of PDE’s with constant coefficients is the following
theorem, proved in [Pal70][VII] and [Ehr70]:
Theorem 1.2.1. The A-module OCnis injective, i.e. the functor
HomA(·,OCn)
is exact.
Further properties of the A-module OCnwere obtained by Oberst in [Obe90], where
he proved that OCnis a large injective cogenerator: see loc.cit for definitions and details.
As a by-product, this allowed Oberst to prove the following result in [Obe96], crucial to
us:
Theorem 1.2.2. Let I⊂Abe zero-dimensional ideal. Then one has the following
equality:
dimCS(I) = length(A
I).
We refer for instance to [Eis95][2.4] for the notion of length of an Artinian module.
Page 9
2. Structure of differentially homogeneous polynomials.
There is a natural left action (by change of variables) of the general linear group
GLN+1(C)on the vector space Vof differential polynomials in (N+ 1) variables defined
as follows. Consider A∈GLN+1(C),P∈V, and set:
A·P:=P(AX)(0),(AX )(1) , . . . ).
This action respects the natural filtration and grading on Vdefined in the Introduction.
Furthermore, a straightforward but crucial observation is that the set Vdiff of differen-
tially homogeneous polynomials forms a sub-representation of V. As a matter fact, these
two facts are so important that we record them in a lemma:
Lemma 2.0.1. The following two facts hold:
•The natural action of the general linear group on Vpreserves the filtration and
the grading;
•The vector space Vdiff forms a sub-representation of V.
The goal of this Section 2 is to study the structure of differentially homogeneous
polynomials, and eventually prove the Schmidt–Kolchin conjecture.
2.1. Families of differentially homogeneous polynomials.
2.1.1. Construction of differentially homogeneous polynomials. Fix d≥1a natural num-
ber. For any d-uple n= (n1, . . . , nd)∈ {0, . . . , N}dand any family of polynomials in
one variable R:= (R1, . . . , Rd)∈C[t]⊕d, one can define a differentially homogeneous
polynomial (denoted Wronsk(R1Xn1, . . . , RdXnd)) of degree das follows. Consider the
following element in M1,d(V[t])
Ln(R):= (R1(t)Xn1, . . . ,Rd(t)Xnd).
For a natural number r≥0, denote by Ln(R)(r)the line obtained by replacing, for
1≤k≤d, the kth entry of Ln(R)(i.e. Rk(t)Xnk) by
r
X
j=0 r
jRk(t)(r−j)X(j)
nk.
Form then the d×dsquare matrix
Wn(R):=
Ln(R)
L(1)
n(R)
·
·
L(d−1)
n(R)
,
and consider its determinant. One has the following proposition:
Proposition 2.1.1. The determinant det(Wn(R))|t=0 is a differentially homogeneous
polynomial of degree d, denoted by Wronsk(R1Xn1, . . . , RdXnd).
Proof. Let Q∈C[T]. From the definition of the action of Qon V(defined in the
Introduction) and from the very definition of L(r)
n(R), observe that the following equality
holds
Q·L(r)
n(R)=(QLn(R))(r).
Therefore, one deduces that the polynomial
Q·det(Wn(R)) ∈V[t, T ]
Page 10
is equal to the determinant of the d×dsquare matrix whose rth line is equal to
(QLn(R))(r). Using elementary operations on the lines as well as the anti-linearity
of the determinant, one deduces that
Q·det(Wn(R)) = Qddet(Wn(R)).
This shows that each coefficient in tof det(Wn(R)) is indeed differentially homogeneous
of degree d, and so does the evaluation at t= 0.
As observed by Reinhart in [Rei99], one can extract a free family of (N+ 1)ddifferen-
tially homogeneous polynomials as follows6:
Proposition 2.1.2. Consider the following data P:
•A(N+ 1)-uple m= (m0, . . . , mN)∈NN+1 such that |m|=d. Let
i1<· · · < ir
be the ordered set of indexes isuch that mi6= 0.
•For any 1≤`≤r, an increasing sequence of mi`integers satisfying:
0≤αi`,1<· · · < αi`,mi`< mi1+· · · +mi`.
To any such data P, denote by WPthe following differentially homogeneous polynomial
of degree d
WP:= Wronsk(tαi1,1Xi1, . . . , tαi1,mi1Xi1, . . . , tαir,1Xir, . . . , tαir,mirXir).
Then, the family (WP)Pforms a free family of (N+ 1)ddifferentially homogeneous
polynomials of degree d(and order less or equal than (d−1)).
Proof. First, one easily computes that there are indeed (N+ 1)dsuch data P. Indeed,
for a fixed (N+ 1)-uple mas in the statement, there are:
d
mird−mir
mir−1· · · d−(mir+· · · +mi2)
mi1=d
mi1, . . . , mir
ways of creating sequences of integers satisfying the second item in the statement (one
has used the usual notation for multinomial coefficients). The counting of the data P
follows from Newton multinomial formula.
To see that the constructed differentially homogeneous polynomials of degree dare
independent, proceed as follows. Put the following order on the variables (X(k)
i):
X(d−1)
0>· · · > X(0)
0> X(d−1)
1>· · · > X(0)
N.
This induces a total order on the monomials by first comparing the term of higher order
in X(·)
0, then the degree in that factor, and so on. Now, the key observation is that
monomials of least order in the polynomials WP’s are pairwise distinct, and respectively
equal to the product of the entries on the diagonal of the defining matrix. This shows
the linear independency, and finishes the proof of the proposition.
Example 2.1.3. Take N= 2,d= 6, and consider Wronsk(X0, tX0, tX1, t3X1, X2, t4X2).
This differentially homogeneous polynomial is given by the determinant of the following
6In his paper, the detailed construction is different from the one presented here, but they amount to
the same. I take this opportunity to thank Gleb Pogudin, as he indicated me the construction presented
here.
Page 11
matrix:
X(0)
00 0 0 X(0)
20
X(1)
0X(0)
0X(0)
10X(1)
20
X(2)
02X(1)
02X(1)
10X(2)
20
X(3)
03X(2)
03X(2)
16X(0)
1X(3)
20
X(4)
04X(3)
04X(3)
124X(1)
1X(4)
224X(0)
2
X(5)
05X(4)
05X(4)
160X(2)
1X(5)
2120X(1)
2
.
Denote by
Vdiff
d:= SpanC((WP)P)
the (N+ 1)d-dimensional C-vector space spanned by the family given in the previous
Proposition 2.1.2. The differential polynomials (WP)Pwill be referred to as the canonical
basis of Vdiff
d.
According to the Schmidt–Kolchin conjecture, the vector space Vdiff
dshould be the
whole vector space of differentially homogeneous polynomials of degree d. In particular,
by Lemma 2.0.1, this should be a representation of the general linear group GLN+1(C).
Proving that it is indeed the case is the object of the next section.
Remark 2.1.4. There are other natural families of differentially homogeneous polyno-
mials that one can construct. For instance, one can consider, for θ∈C∗, the following
family:
nWronsk Xn1,(θ+t)Xn2,...,(θ+t)d−1Xnd)|n= (n1, . . . , nd)∈ {0, . . . , N }do.
Whereas one immediately sees that the vector space spanned by the previous family
defines a sub-representation of Vdiff
d, it is not obvious at all to determine its dimension.
We conjecture that, for a generic value of θ, this is (N+ 1)d-dimensional, but we were
not able to prove it.
2.1.2. Invariance under the action of the general linear group. The sought invariance of
Vdiff
dunder the action of GLN+1(C)will follow from more general considerations that we
develop in Appendix A. We chose to put this part in an appendix because, in itself, it
has little to do with differentially homogeneous polynomials. It is rather a more general
statement about families of determinants of a certain shape.
For now, let us state a corollary of the main statement in Appendix A, which will be
useful for our purposes. Let us fix Y1, . . . , Ydformal variables, along with their formal
derivatives Y(k)
1, . . . , Y (k)
d, with ka natural number. We consider the following family of
differentially homogeneous polynomials of degree d:
Fd:=Wronsk(tα1Y1, . . . , tαdYd)0≤αi≤d−1,
as well as the following subfamily:
˜
Fd:=Wronsk(tα1Y1, . . . , tαdYd)0≤αi≤i−1.
Remark 2.1.5. Note that the canonical basis of Vdiff
ddefined in the previous Section
2.1.1 is obtained from the family ˜
Fdby suitable substitutions Yi=Xni,ni∈ {0, . . . , N}.
Remark 2.1.6. If one the αi’s in the previous definition is taken larger or equal than d,
then it is easily seen that the resulting differential polynomial is identically zero.
As a corollary of Proposition A.0.5 and remark A.0.2, one deduces immediately the
following:
Page 12
Proposition 2.1.7. The C-vector space spanned by ˜
Fdis the same than the one spanned
by Fd.
Proof. See Appendix A.
This proposition implies in turn the following:
Theorem 2.1.8. The C-vector space Vdiff
dis a sub-representation of Vdiff
d(for the natural
action of the general linear group GLN+1(C)).
Proof. Consider a data Pas described in Proposition 2.1.2, and the differential polyno-
mial WPassociated to it:
WP= Wronsk(tαi1,1Xi1, . . . , tαi1,mi1Xi1, . . . , tαir,1Xir, . . . , tαir,mirXir).
Denoting
α= (αi1,1, . . . , αir,mr)∈Nd,
observe that, by construction, one has the inequality αi≤i−1for any 1≤i≤d.
Accordingly, the differentially homogeneous polynomial WPis nothing but
Wronsk(tα1Y1, . . . , tαdYd)
after the evaluations
Y1=. . . =Ymi1=Xi1, Ymi1+1 =· · · =Ymi1+mi2=Xi2, . . . .
Reciprocally, being given
Wronsk(tα1Y1, . . . , tαdYd)
with αi≤i−1for any 1≤i≤d−1, this yields, up to sign, a differential polynomial in
the canonical basis of Vdiff
dafter evaluations of the form
Y1=. . . =Ymi1=Xi1, Ymi1+1 =· · · =Ymi1+mi2=Xi2,...,
provided that the following holds:
•α1, . . . , αm1are pairwise distinct;
•αm1+1, . . . , αm1+m2are pairwise distinct;
•etc.
If these conditions are not satisfied, then the evaluation yields trivially zero by anti-
linearity of the determinant.
Now, if one lets an invertible matrix A∈GLN+1(C)act on WP, then, by expanding
out, one finds that A·WPwrites as a linear combination of terms of the form
W:= Wronsk(tβ1Y1, . . . , tβdYd)|Y0=Xn0,...,Yd=Xnd
for some 0≤βi≤d−1and 0≤n0≤ · · · ≤ nd−1≤N. By Proposition 2.1.7, one can
suppose that such a term actually satisfies the inequalities
βi≤i−1
for any 1≤i≤d−1. By the above, the differential polynomial Wis either zero, or an
element in Vdiff
d. This proves that Vdiff
dis left stable under the action of GLN+1(C). It
is therefore a sub-representation of Vdiff
d, so that the proof is complete.
For our purposes, the following corollary will be of particular importance:
Proposition 2.1.9. For any partition λ`dwith at most (N+ 1) parts, the following
inequality holds:
dim Vdiff
hw(λ)(d−1) ≥fλ,
where one recalls that Vdiff
hw(λ)is the vector space of λ-highest weight vectors in Vdiff
d.
Page 13
Proof. Now that one knows that Vdiff
dis a (finite-dimensional) representation, one can
compute its character. Since the canonical basis of Vdiff
dis made of weight vectors, one
computes immediately that:
χVdiff
d
(x0, . . . , xN) = X
m∈NN+1
|m|=dd
m0, . . . , mNxm0
0· · · xmN
N
= (x0+· · · +xN)d.
Using for instance Theorem 1.1.4, one sees that the symmetric polynomial (x0+· · ·+xN)d
writes:
(x0+· · · +xN)d=X
λ`d
fλsλ.
Note that, if λhas more than (N+ 1) parts, then sλ= 0. Since a representation
is uniquely determined by its character, one deduces that Vdiff
dhas fλindependent λ-
highest weight vectors. The proposition now follows from the straightforward observation
that Vdiff
d⊂(Vdiff
d)(d−1).
2.2. A natural basis for the highest weight vectors of V(k)
d.Note that there is a
tautological isomorphism of representation
V(k)
d'M
a∈Nk+1
|a|=d
Sa0CN+1 ⊗ · · · ⊗ SakCN+1.
Elementary plethysm using the so-called Pieri’s formula (see e.g. [Ful97][I.2.2 & II.8.3])
tells that V(k)
ddecomposes into irreducible representations as follows:
(3) V(k)
d'M
λ`dM
a∈Nk+1
|a|=d
Sλ(CN+1)⊕Kλ,a.
An easy corollary of the decomposition (3) is the following:
Lemma 2.2.1. The number of irreducible representations of V(k)
dis equal to
X
λ`d
dim SλCk+1.
Proof. This follows from the equality (3), and the following identity:
X
a∈Nk+1
|a|=d
Kλ,a= dim SλCk+1
(see the equality (2) and Proposition 1.1.3).
In order to define a basis of highest weight vectors inside V(k)
d, let us first introduce
some notations. To any sequence of natural numbers i= (i0, . . . , ir), with 0≤r≤N,
define
Di:= det
X(i0)
0· · X(i0)
r
· ·
· ·
X(ir)
0· · X(ir)
r
.
Page 14
To any Young tableau Tof shape λ= (λ0≥ · · · ≥ λs)with at most (N+ 1) rows, filled
with the numbers {0, . . . , k}, define
DT:=DT(·,1) × · · · × DT(·,λ0).
Here, for 1≤i≤λ0, the symbol T(., i)represents the sequence of integers read off (from
top to bottom) from the ith column of the Young tableau T.
Now, for λ`da partition of d, consider the following sub-vector space of V(k)
d:
D(k)
λ:= SpanCDT|T Young tableau of shape λ, filled with {0, . . . , k}
One has the following important lemma:
Proposition 2.2.2. The following holds:
(1) Let λ`dbe a partition of the integer dwith at most (N+ 1) parts. The family
(DT)T
where Truns over semi-standard Young tableaux of shape λfilled with {0, . . . , k}
(i) is a free family of λ-highest weight vectors in V(k)
d;
(ii) spans the vector space D(k)
λ, which is canonically isomorphic to SλCk+1.
(2) The direct sum M
λ`d
D(k)
λ
spans all the highest weight vectors of V(k)
d(note that if λhas more than (N+ 1)
parts, then D(k)
λis zero).
Proof. Let λbe a partition with at most (N+ 1) parts, and let Tbe a Young tableau
of shape λfilled with the numbers {0, . . . , k}. A straightforward application of the anti-
linearity of the determinant shows that DTis indeed a λ-highest weight vector in V(k)
d.
The fact that the family
DTTsemi-standard Young tableaux of shape λfilled with {0, . . . , k }
is a free family that spans the vector space D(k)
λis in particular the content of [Ful97][II.
8.1 Corollary of Theorem 1]. This very result also shows that D(k)
λis canonically isomor-
phic to SλCk+1.
The last part of the statement follows immediately from the previous Lemma 2.2.1: one
has indeed exhibited as many independent highest weight vectors as there are irreducible
representations in V(k)
d.
Now that we have a basis of highest weight vectors in V(k)
d, the goal is the following:
evaluate how many independent linear combinations of elements in this basis provide
differentially homogeneous polynomials. Tackling this question is the object of the next
two Sections 2.3 and 2.4.
2.3. An intermediate algebraic problem. Let us start this Section 2.3 with a simple
observation. A differentially homogeneous polynomial Pof degree dmust satisfy the
following property: for any complex number α∈C, the following equality holds
P(α+T)X, (α+T)X(1), . . . |T=0 =αdPX, X(1), . . . .
This equality is equivalent to saying that, if one makes the substitution
(4) X(i)←→ αX(i)+iX(i−1)
Page 15
for any i∈N, then the polynomial Pbecomes αdP. One is therefore naturally lead to
study differential polynomials satisfying such a property.
The goal of this Section 2.3 is to study a purely algebraic problem related to transfor-
mations of the type (4). We will then relate this algebraic problem to our situation of
interest in the next Section 2.4.
2.3.1. Bounding the dimension of the kernel of a family of nilpotent endomorphisms of
(Ck+1)⊗d.Consider the following nilpotent endomorphism of Ck+1:
J:=
010· · 0
002· · 0
· · · · · ·
· · · · · ·
· · · · · k
0 0 · · · 0
.
Fix v∈Ck+1 a vector such that Jkv6= 0. It induces a natural basis of E:= (Ck+1)⊗d
given by: Jα1v⊗ · · · ⊗ Jαdv
| {z }
:=Jαv0≤α1,...,αd≤k.
For any 1≤`≤d, define an endomorphism J(`)of Einduced by Jas follows. Denote
by (e1,...,ed)the canonical Z-basis of Zd, and set:
J(`)(Jαv):=X
1≤i16=···6=i`≤d
Jα+ei1+···+ei`v.
Equivalently, if v1, . . . , vdare vectors in Ck+1, the endomorphism J(`)is defined as follows
on v1⊗ · · · ⊗ vd:
J(`)(v1⊗ · · · ⊗ vd):=X
1≤i16=···6=i`≤d
(⊗
d
Y
i=1
)(Jδi,{i1,...,i`}vi).
Here, for any set I⊂N, one sets
δ·,I :N→ {0,1}
to be the function that is equal to 1if i∈I, and zero otherwise.
The problematic is the following: we are looking for tensors in E= (Ck+1)⊗dthat
are in the kernel of the endomorphisms J(`)for any 1≤`≤d. This problem can be
reformulated as follows. Consider the isomorphism of vector spaces that identifies
Jαv←→ Xk−α1
1
(k−α1)! · · · Xk−αd
d
(k−αd)! ∈C[X1, . . . , Xd].
The key observation is that, under this identification, the endomorphism J(`)is nothing
but the endomorphism induced by the partial differential equation:
X
1≤i16=···6=i`≤d
∂`
∂Xi1· · · ∂Xi`
.
With this reformulation, it becomes relevant to study solutions in O(Cd)of the fol-
lowing system of PDE’s with constant coefficients:
(S): X
1≤i16=···6=i`≤d
∂`
∂Xi1· · · ∂Xi`1≤`≤d.
Page 16
Note that by classic considerations on symmetric polynomials, the above system is equiv-
alent to the following system of PDE’s:
(S0): d
X
i=1
∂`
(∂Xi)`1≤`≤d.
Following Section 1.2, we are lead to study the ideal in C[X1, . . . , Xd]
I:= (S1, . . . ,Sd),
where, for r∈N≥1,Sris the rth Newton polynomial in dvariables, namely:
Sr:=
d
X
i=1
Xr
i.
One has then the following lemma:
Lemma 2.3.1. The affine variety V(I)consists of the origin
V(I) = {0},
and one has the equality:
length C[X1, . . . , Xd]
I=d!.
Proof. The fact that the affine variety V(I)consists only of the origin is classic. If
(a1, . . . , ad)∈V(I), then one has the equality in C[Z]
Zd=
d
Y
i=1
(Z−ai),
so that necessarily a1=· · · =ad= 0.
Let m0= (X1, . . . , Xd)be the maximal ideal at the origin, and consider the Artinian
ring M:=C[X1,...,Xd]
I. One easily sees that
length(M) = length(Mm0),
where Mm0is the localized module at m0. Consider now the following trick. Add one
variable Tto the polynomial ring C[X1, . . . , Xd], and denote by
˜
I= (S1, . . . , Sd)⊂C[X1, . . . , Xd, T ]
the homogeneous ideal in C[X1, . . . , Xd, T ]induced by S1, . . . , Sd∈C[X1, . . . , Xd]. The
homogeneous variety Proj( ˜
M), where ˜
M:=C[X1,...,Xd,T]
˜
I, consists then of the single point
∞:= [0 : . . . : 0 : 1] ∈Pd.
Denoting m∞the ideal sheaf of the closed point ∞, elementary intersection theory (see
e.g. [Ful98][Proposition 8.4, and discussion below]) allows to show that
length(M) = length(Mm0) = length( ˜
Mm∞) = OPd(1) · OPd(2) · · ·· · OPd(d)
=d!.
This finishes the proof of the lemma.
Now, Theorem 1.2.2 readily implies the following proposition:
Proposition 2.3.2. One has the following estimate:
dim \
1≤`≤d
Ker J(`)≤d!.
Page 17
Proof. On the one hand, one has seen that elements in T1≤`≤dKer J(`)embeds as polyno-
mial solutions of the system of PDE’s (S). On the other hand, Theorem 1.2.2 combined
with Lemma 2.3.1 shows that the system (S)admits exactly d!independent solutions in
OCd⊃C[X1, . . . , Xd]. This proves the proposition.
Remark 2.3.3. Note that the upper bound depends only on d, and not on k.
2.3.2. Equivariance of the endomorphisms J(`)with respect to the action of the symmetric
group Σd.We keep the notations introduced in the previous Section 2.3.1. Recall that
there is a natural right action of symmetric group Σdon E= (Ck+1)⊗dobtained by
permuting the factors:
(v1⊗ · · · ⊗ vd)·σ:=vσ(1) ⊗ · · · ⊗ vσ(d).
It makes Einto a right C[Σd]-module. A key observation is that the endomorphisms
(J(`))1≤`≤dcommutes with the action of Σd:
Lemma 2.3.4. For any σ∈Σd, any 1≤`≤dand any v1⊗ · · · ⊗ vd, the following
equality holds: J(`)(v1⊗ · · · ⊗ vd)·σ=J(`)(v1⊗ · · · ⊗ vd)·σ
Proof. On the one hand, compute that:
J(`)(v1⊗ · · · ⊗ vd)·σ=X
1≤i16=···6=i`≤d
(⊗
d
Y
i=1
)(Jδi,{i1,...,i`}vi)·σ
=X
1≤i16=···6=i`≤d
(⊗
d
Y
i=1
)(Jδσ(i),{i1,...,i`}vσ(i)).
On the other hand, compute that:
J(`)(v1⊗ · · · ⊗ vd)·σ=X
1≤i16=···6=i`≤d
(⊗
d
Y
i=1
)(Jδi,{i1,...,i`}vσ(i))
=X
1≤i16=···6=i`≤d
(⊗
d
Y
i=1
)(Jδi,{σ−1(i1),...,σ−1(i`)}vσ(i))
=X
1≤i16=···6=i`≤d
(⊗
d
Y
i=1
)(Jδσ(i),{i1,...,i`}vσ(i)).
This proves the lemma.
By the above Lemma 2.3.4, one deduces that for any 1≤`≤d, and any stan-
dard Young tableau Twith dboxes, the endomorphisms J(`)commute with the almost-
projections E(pT)(see Section 1.1.3 for notations). In particular, these endomorphisms
stabilize Im(E(pT)). One has then the following elementary result:
Lemma 2.3.5. The dimension of
\
1≤`≤d
Ker J(`)
|Im(E(pT))
is independent of the standard Young tableau of shape λ`d.
Proof. Let Tand T0be two standard tableaux of shape λ`d. There exists a unique
permutation σ∈Σdsuch that
σ·T=T0.
Page 18
Denote by E(σ)the isomorphism
E(σ): E(C[Σd]) −→ E(C[Σd])
v7−→ v⊗C[Σd]σ.
Observe that the following equality holds in C[Σd]:
cT0=σ×cT×σ−1.
This implies in turn the following equality
E(pT0) = E(σ−1)◦E(pT)◦E(σ).
In particular, if v∈Im E(pT)∩Ker J(`)for some 1≤`≤d, then
E(σ−1)(v)∈Im E(pT0)∩Ker J(`).
This allows to show that E(σ−1)realizes a bijection between T1≤`≤dKer J(`)
|Im(E(pT)) and
T1≤`≤dKer J(`)
|Im(E(pT0)). This finishes the proof of the lemma.
For a partition λ`d, denote by Tcan(λ)the canonical standard Young tableau of shape
λ, which is defined as follows:
•the first row of the diagram is filled with 1,2, . . . , λ1;
•the second row of the diagram is filled with λ1+ 1, . . . , λ1+λ2;
•. . .
From Proposition 2.3.2 and Lemma 2.3.5, one deduces the following:
Proposition 2.3.6. The following inequality holds:
X
λ`d
fλ×dim \
1≤`≤d
Ker J(`)
|Im(E(pTcan(λ)))≤d!.
Proof. By Theorem 1.1.4, one has the direct sum decomposition
E=M
T
Im(E(pT)),
where Truns over standard tableaux with dboxes. The statement now follows immedi-
ately from Proposition 2.3.2 and Lemma 2.3.5.
2.4. Where one relates the previous problem to the question at hand. Let us
now relate the study carried over in the previous Section 2.3 to the Schmidt–Kolchin
conjecture. We fix λ`da partition of dwith at most (N+ 1) parts. Following Section
2.2 and the very beginning Section 2.3, we wish to understand polynomials Pin D(k)
λ
(see Lemma 2.2.2 for notations) which satisfy the following identity
(5) P(αX, αX (1) +X(0), αX(2) + 2X(1), . . . ) = αdP(X, X (1), X (2), . . . )
for any α∈C. This can be restated as follows. Denote by
F:=C·X⊕C·X(1) ⊕ · · · ⊕ C·X(k)'Ck+1.
Observe that the linear group GL(F)'GLk+1(C)acts naturally on D(k)
λby change of
variable7, i.e. for P∈ D(k)
λand A∈GL(F):
A·P:=P(AX, AX(1), . . . , AX(k)).
The equality (5) is then equivalent to the following equality (see Section 2.3 for the
definition of the matrix J):
(6) (αId +J)·P=αdP.
7Note that this action has nothing to do with the usual linear action on V.
Page 19
Note that for an arbitrary P∈ D(k)
λ, the polynomial
(αId +J)·P
is a polynomial of degree din α, with leading term P(X, X (1), X(2), . . . ). Therefore, in
order to obtain (5), we must require the vanishing of all the coefficients in front of the
terms αi,i<d.
For any Young tableau Twith dboxes, denote
eT:= (⊗Y
(i,j)∈T
)X(T(i,j)) ∈E:=F⊗d,
where the tensor product is taken over the elements in the tableau T, read in the usual
fashion, i.e. from left to right and top to bottom. Let Tcan(λ)be the canonical standard
tableau with shape λ. For sake of notations, let us denote cλ:=cTcan(λ). We have the
following crucial proposition:
Proposition 2.4.1. The linear map
e: D(k)
λ−→ Im(E(pTcan(λ))) ⊂E=F⊗d
DT7−→ eT⊗C[Σd]cλ!
is well-defined, and is an isomorphism8.
Proof. It essentially follows from [Ful97][II.8.1 Lemma 3 & Theorem 1] and [Ful97][II.7.4
Proposition 4]. Details are provided in Appendix B.
Remark 2.4.2. This proposition is essential, because it allows to almost completely get
rid of the dependency on N: the sole dependency on Nfor the space on the right lies on
the constraint on the partition λ(it must not have more than (N+ 1) parts).
Consider the following natural surjective linear map
π:E−→ D(k)
λ
eT7−→ DT,
and observe that one has the following commutative diagram:
(7) E
E(pTcan(λ))
//
π
Im(E(pTcan(λ)))
D(k)
λ
e
88
.
A simple but important observation is that the action of GL(F)on D(k)
λcommutes with
the projection π:
Lemma 2.4.3. For any v∈Eand any A∈GL(F), the following equality holds:
π(A·v) = A·π(v).
Proof. This follows from [Ful97][II.8.1 Exercices 3 & 4].
As an immediate corollary, the action of GL(F)commutes with the isomorphism e:
Lemma 2.4.4. For any P∈ D(k)
λand any A∈GL(F), the following equality holds:
e(A·P) = A·e(P).
8The assumption that the partition has at most (N+ 1) parts is important: this map would be the
zero-map otherwise.
Page 20
Proof. By linearity, it suffices to prove the equality for P=DT, where Tis a Young
tableau of shape λfilled with {0, . . . , k}. By Lemma 2.4.3, compute that
e(A·DT) = e(A·π(eT)) = e(π(A·eT)).
By commutativity of the diagram (7), one has:
e(π(A·eT)) = pTcan(λ)(A·eT) = A·pTcan (λ)(eT) = A·e(DT).
This shows the result.
Now, the key observation to relate our problem to what we did in Section 2.3 is the
following:
Lemma 2.4.5. For any v∈E, and any α∈C, the following equality holds:
(αId +J)·v=J(d)(v) + αJ(d−1)(v) + · · · +αd−1J(1) (v) + αdv.
Proof. It suffices to check the equality for v=eT, where Tis a Young tableau of shape
λfilled with {0, . . . , k}. This is then a straightforward computation to show the sought
equality.
Denote ∼
V(k)
hw(λ):=P∈ D(k)
λ|(αId +J)·P=αdP∀α∈C.
As a simple corollary of the previous lemmas, we obtain the following important propo-
sition:
Proposition 2.4.6. One has the following isomorphism of vector spaces:
∼
V(k)
hw(λ)'\
1≤`≤d
Ker J(`)
|Im(E(pTcan(λ))).
Proof. Since eis an equivariant isomorphism (by Proposition 2.4.1 and Lemma 2.4.4),
there is an isomorphism
∼
V(k)
hw(λ)' {v∈Im(E(pTcan(λ))) |(αId +J)·v=αdv}.
Since the endomorphisms J(`)commute with E(pTcan(λ))(by Lemma 2.3.4), Lemma 2.4.5
implies that the set on the right is nothing but
\
1≤`≤d
Ker J(`)
|Im(E(pTcan(λ))).
This finishes the proof of the proposition.
We would like to emphasize again that, in the above statement, the dependency on N
lies only in the partition λ`d(namely, it must not have more than (N+ 1) parts). We
now have all the tools to finish the proof of the Schmidt–Kolchin conjecture
2.5. Proof of the Schmidt–Kolchin conjecture. Fix d∈N≥1a natural number, and
fix k≥d−1.
Remark 2.5.1. One chooses to take k≥d−1simply because the canonical basis of
Vdiff
dlies in (Vdiff
d)(d−1).
Page 21
For the moment, suppose that the natural number Nis greater or equal than d−1.
One knows by the beginning of Section 2.3 that for any partition λ`dwith at most
(N+ 1) parts, the following inclusion holds:
(8) V(k)
hw(λ)⊂
∼
V(k)
hw(λ)=P∈ D(k)
λ|(αId +J)·P=αdP∀α∈C.
Note that, since N≥d−1, any partition λ`dhas at most N+1 parts. From Proposition
2.1.9, one therefore deduces the inequality
dim
∼
V(k)
hw(λ)≥fλ
for any partition λ`d. By Proposition 2.4.6, this is the same as the following inequality:
(9) dim \
1≤`≤d
Ker J(`)
|Im(E(pTcan(λ)))≥fλ.
Now, using Proposition 2.3.6 and the above inequality (9), one obtains the following
string of inequalities:
X
λ`d
f2
λ≤X
λ`d
fλ×dim \
1≤`≤d
Ker J(`)
|Im(E(pTcan(λ)))≤d!.
By Proposition 1.1.1, one has the equality
X
λ`d
f2
λ=d!.
Therefore, one deduces that, for any λ`d, the following equality holds:
(10) dim \
1≤`≤d
Ker J(`)
|Im(E(pTcan(λ)))=fλ.
Remark 2.5.2. Note that one went in the opposite direction than in Section 2.4. Namely,
one has used informations on differentially homogeneous polynomials (obtained in Section
2.1) to deduce properties on the algebraic problem of Section 2.3. This is precisely why
Proposition 2.1.9 is so crucial.
Return now to the case where N∈N≥1is arbitrary. For any partition λwith at most
(N+ 1) parts, Proposition 2.1.9 combined with the inclusion (8) and the equality (10)
forces the equality
dim V(k)
hw(λ)=fλ.
By standard considerations in representation theory recalled in Section 1.1.3, one deduces
the equality
dim (Vdiff
d)(k)=X
λ`d
fλ×dim SλCN+1.
Note that if λhas more than (N+ 1) parts, then SλCN+1 = (0). By Propositions 1.1.3
and 1.1.1, this implies in turn the equality:
dim (Vdiff
d)(k)= (N+ 1)d.
As this holds for any k≥(d−1), this shows that Vdiff
d= (Vdiff
d)(d−1),and that
dim(Vdiff
d)=(N+ 1)d.
This finishes the proof of the Schmidt–Kolchin conjecture:
Theorem 2.5.3. The following equality holds:
Vdiff
d=Vdiff
d= SpanC(WP)P.
In particular, Vdiff
d= (Vdiff
d)(d−1) and dim Vdiff
d= (N+ 1)d.
Page 22
3. Differentially homogeneous polynomials and twisted jet
differentials on projective spaces.
3.1. Green–Griffiths vector bundles on complex manifolds. The reference for
this Section 3.1 is [Dem97]. The goal is to quickly recall the definition of the so-called
Green–Griffiths vector bundles, whose global sections are called jet differentials.
Let Xbe a complex manifold of dimension N. For k∈N≥1, define the bundle JkX
of k-jets of 1-germs of holomorphic maps γ: (C,0) →Xon the complex manifold Xas
follows. Consider an atlas (Ui, ϕi)i∈Iof X, and for i∈I, consider the (trivial) bundle
on Uiwhose fiber over x∈Uiis the C-vector space of dimension N×k
di
dzi(ϕi◦γ)(0)1≤i≤kγ: (C,0) →(X, x)holomorphic 1-germ .
Glue these trivial bundles Ui×CN×kvia (the maps naturally induced by) the transition
maps ϕj◦ϕ−1
ito obtain (up to isomorphism) the bundle JkX. 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: JkXis
not a vector bundle for k > 1. For sake of notation, denote dkγthe element in (JkX)x
defined by the holomorphic 1-germ γ: (C,0) →(X, x).
Example 3.1.1. In the case where k= 1,J1Xis nothing but the (holomorphic) tangent
bundle T X .
The torus C∗acts on the fibers of JkXas follows
λ·di
dzi(ϕi◦γ)(0)1≤i≤k=λidi
dzi(ϕi◦γ)(0)1≤i≤k,
where λ∈C∗. It is indeed straightforward to see that this action commutes with a
change of chart, and is thus well defined on the bundle JkX. More generally, the group
of biholomorphisms of (C,0) acts on JkXby setting
ψ·dkγ:= dk(γ◦ψ),
where ψis a biholomorphism of (C,0) and γ: (C,0) →Xa holomorphic 1-germ.
Define the vector bundle Ek,nXof jet differentials (of 1-germs) of order k≥1and
degree n≥1as follows. Construct the bundle whose fiber over x∈Xis the vector space
of complex valued polynomials Qof degree non the fiber (JkX)x, i.e. for any λ∈Cand
any 1-germ γ: (C,0) →(X, x), the polynomial Qsatisfies the equality
Q(λ·dkγ) = λnQ(dkγ).
The formula to compute multi-derivatives of compositions of maps allows to see that the
structure of vector space of the fibers is preserved under a change of chart. The bundle
Ek,nXis thus a vector bundle, and is usually called a Green–Griffiths vector bundle (of
order kand degree n). A global section
P∈H0(X, Ek,n X)
is usually called a jet differential (of order kand degree n). If the complex manifold
is projective, and polarized by an ample line bundle OX(1), one is naturally lead to
consider twisted vector bundles of the form Ek,nX(d):=Ek,nX⊗Ld, with d∈Z. A
global section of such a twisted Green–Griffiths bundle is called accordingly a twisted jet
differential.
Page 23
For reasons that will become transparent in the next Section 3.2, we will rather consider
the following direct sums of vector bundles for k∈N≥0:
Ek,∞X:=
∞
M
n=0
Ek,nX.
Here, by convention, one has set
Ek,0X:=OX
for any k∈N≥0. For sake of consistency, a global section of Ld, with d∈N, will be
considered to be a twisted jet differential of order 0.
3.2. Differentially homogeneous polynomials and twisted jet differentials on
projective spaces. The link between differentially homogeneous polynomials and
twisted global sections of Green–Griffiths bundles of projective spaces is given by the
following proposition:
Proposition 3.2.1. For any k≥1and any d∈N, there is a natural isomorphism of
vector spaces
H0(PN, Ek,∞PN(d)) '(Vdiff
d)(k).
Proof. Let W∈(Vdiff
d)(k). On each standard affine open set Ui:={Xi6= 0}, define
Wi:=W(
∧i
X(0),...,
∧i
X(k)).
Here, one has set for 0≤`≤k
∧i
X(`):=X0
Xi(`),...,XN
Xi(`),
where the upper index in parenthesis stands for the usual formula to differentiate, e.g.:
X0
Xi(1) =X(1)
0Xi−X0X(1)
i
X2
i
.
On the corresponding trivialization of Ek,∞PN
|Ui, the differential polynomial Wi(in the
variables (X`
Xi)0≤`≤N) induces in a natural fashion a section of Ek,∞PN
|Ui. Now, the fact
that Wis differentially homogeneous of degree dimplies that the following equality holds:
Xd
iWi=Xd
jWj.
In particular, this shows that the local sections Wiglue to a global section Wof
Ek,∞PN(d).
Reciprocally, let W ∈ H0(PN, Ek,∞PN(d)) be a global section of Ek,∞PN(d). On
each trivializing open set Ui, the restricted section Wi:=W|Uidefines a differential
polynomial in the variables ∧i
X= (X`
Xi)0≤`≤N. Consider
W:=Xd
iWi,
and note that the rational function (rational in the variables X(0), polynomial in the
variables X(1), X(2), . . .) obtained is independent of 0≤i≤N. In particular, this shows
that Wis in fact a differential polynomial. Let then Q∈C[T], and compute that
W((QX)(0),...,(QX )(k)) = Q(T)dXd
iWi∧i
X(0),...,
∧i
X(k)
=Q(T)dW.
Therefore, Wis differentially homogeneous of degree d.
Both maps are clearly inverse to each other, hence the proposition.
Page 24
As a straightforward corollary of the previous proposition, one has the following im-
portant result:
Theorem 3.2.2. There is a one-to-one correspondance between twisted jet differential
on PN, and differentially homogeneous polynomials in the variables X= (X0, . . . , XN).
Proof. In view of the previous Proposition 3.2.1, it suffices to show that for any d < 0,
and any k, n ∈N, there is no global section of