Differential equations as embedding obstructions and vanishing theorems

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Differential equations as embedding obstructions and vanishing theorems

Damian Brotbek

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Damian Brotbek. Differential equations as embedding obstructions and vanishing theorems. 2011.

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Differential equations as embedding obstructions and vanishing theorems

Damian Brotbek November 22, 2011

Abstract

We generalize a vanishing theorem for the cohomology of symmetric powers of the cotangent bundle of subvarieties of projective space due to Schneider. From this we deduce new vanishing results for Green-Griffiths jet differential bundles, generalizing results of Diverio and Pacienza-Rousseau.

1 Introduction

Given a smooth projective n-dimensional variety X, one can search for the smallest integer N such that there exists an embeddingX ֒→P^N. Obviously,n6N, and, on the other hand, a simple argument shows that N 62n+ 1 (see[Har74]§4). In general, these bounds are optimal, and therefore it seems natural to try to understand what geometric properties of X yield obstructions for projective embeddings. Schneider proved in[Sch92]that the existence of symmetric differential forms onX imposes strong restriction on such embeddings. His result, and its generalization for higher order cohomology groups can be stated as follows.

Theorem 1.1(Schneider [Sch92] Theorems 1.1 and 1.2). Let X⊂P^N be a smooth subvariety of dimension nand codimension c. Let m∈Nanda∈Z. Consider an integer 06j < n−c. Ifa < m−min{j,1}, then

H^j(X, S^mΩX⊗ OX(a)) = 0.

If one regards global sections ofS^kΩX as algebraic differential equations of order1 defined onX, one is naturally led to consider the problem of finding a higher order analogue to this statement. In that direction, Diverio [Div08] established a vanishing theorem for Green-Griffiths jet differential bundles on complete intersection varieties (see section 3 for the definition).

Theorem 1.2(Diverio [Div08] Theorem 7). LetX ⊆P^N be a smooth complete intersection of dimensionn and codimension c. Then

H⁰(X, E_k,m^GGΩX) = 0 for allm>1 and16k <ⁿ_c.

Pacienza and Rousseau[PR08]constructed a generalized version of Green-Griffiths jet differential bundle and they proved a similar result for those more general bundles (see Theorem8.1).

The aim of our work is to generalize, and unify, Schneider’s and Diverio’s result. Our main result is the following (see Theorem 6.3).

Theorem A. Let X ⊆P^N be a smooth variety of dimension n and codimension c=N−n. Consider an integer k>1, integers ℓ1, . . . , ℓk>0, anda∈Z. Ifj < n−k·c anda < ℓ1+· · ·+ℓk−min{j, k}, then

H^j(X, S^ℓ1ΩX⊗ · · · ⊗S^ℓkΩX⊗ OX(a)) = 0.

From this generalization of Schneider’s theorem one can deduce a new vanishing result for jet differentials (see Corollary7.2).

Theorem B. LetX ⊆P^N be a smooth variety of dimensionnand codimensionc. Leta∈Z. Ifj < n−k·c anda < ^m_k −min{j, k} then

H^j(X, E_k,m^GGΩX⊗ OX(a)) = 0.

Therefore the existence of higher order jet differential equation is an obstruction to projective embeddings, analogues to the ones pointed out by Schneider. We also prove an analogous result for Pacienza-Rousseau’s generalized Green-Griffiths jet differential bundles (see Corollary 8.2).

Notation and conventions. We work over the field of complex numbersC. WhenX is a complex manifold, we denote the tangent bundle ofX byT X, and the cotangent bundle ofX byΩX :=T X^∗.

2 The bundle Ωe

It will be convenient for us to work with the bundleΩ, studied in particular by Bogomolov and DeOliveira ine [BDO08], but also by Debarre in[Deb05]. Therefore we recall some basic fact on this bundle. WhenX ⊆P^N is ann-dimensional variety thenΩeX is, roughly speaking, the sheaf of differential forms onXb ⊆C^N+1, the cone overX, which are invariant under theC^∗-action but which do not necessarily satisfy the Euler condition.

A more geometrical approach is to consider the Gauss map γX :X → Gr(n,P^N)

x 7→ T_xX

whereT_xX ⊆P^N is the embedded tangent space ofX atx, and whereGr(n,P^N)denotes the Grassmannian variety of n-dimensional projective subspaces of P^N. Let Sn denote the rank-n tautological bundle on Gr(n,P^N), then set

ΩeX =γ_X^∗S_n^∗⊗ OX(−1).

Observe thatΩeP(V) =V ⊗ OP(V)(−1). The properties of ΩeX we will use are summarized in the following commutative diagramm.

0 0

 y

 y N_X/^∗ PN N_X/^∗ PN

 y

 y 0 −−−−→ ΩPN

|X −−−−→ ΩePN

|X −−−−→ OX −−−−→ 0

 y

 y

0 −−−−→ ΩX −−−−→ ΩeX −−−−→ OX −−−−→ 0

 y

 y

0 0

We refer to[BDO08]and[Deb05]for some details. In this paper the exact sequence 0→N_X/^∗ P^N →ΩeP^N

|X →ΩeX→0

will be called thetilde conormal exact sequence. And the exact sequence 0→ΩX→ΩeX → OX→0 will be called theEuler exact sequence.

3 The Green-Griffiths jet differential bundles E_k,m^GGΩX

We recall very briefly the definitions of Green-Griffiths jet differential bundles. We refer to [Dem97] for more information, and also to[Mer10]where many details are carried out explicitly. LetX be a projective variety of dimensionn. For allk>1, we denote byJkX→^pk X the holomorphic bundle of k-jets of germs of holomorphic curvesf : (C,0)→X, that is JkX:={f : (C,0)→X}/∼, where two germsf, g: (C,0)→X are equivalent(f ∼g)if and only iff^(j)(0) =g^(j)(0)for all06j6k. The projection is then simply defined by

JkX →^pk X f 7→ f(0).

Those spaces naturally have the structure of holomorphic fiber bundles overX, but unlessk= 1they are not vector bundles. Moreover, for anyx∈X,JkXx∼= (Cⁿ)^k. To see this, take local coordinates(z1, . . . , zn) aroundx∈X; we can then writef = (f1, . . . , fn)and thek-jet will then be entirely determined, by Taylor’s formula, by

(f₁^′(0), . . . , f_n^′(0), f₁^′′(0), . . . , f_n^′′(0), . . . , f₁^(k)(0), . . . , f_n^(k)(0)).

With the above notation, for each k >1, there is a natural C^∗-action on JkX. Namely if λ∈C^∗ and f : (C,0)→(X, x)then

λ·f : (C,0) → (X, x) t 7→ f(λt).

This action is easily expressed fiberwise. Namely, if one considers local coordinates aroundx=f(0) =λ·f(0) thenf is represented by(f^′(0), f^′′(0), . . . , f^(k)(0))and

λ·

f^′(0), f^′′(0), . . . , f^(k)(0)

=

λf^′(0), λ²f^′′(0), . . . , λ^kf^(k)(0) .

Then the Green-Griffiths jet differentials are defined as follows. Fiberwise we consider E_k,m^GGΩX,x:=n

Q∈C[f^′, . . . , f^(k)]/ Q λ·

f^′, . . . , f^(k)

=λ^mQ

f^′, . . . , f^(k)o .

One can make this even more explicit: whenx∈X is fixed, one can consider coordinates (f₁^′, . . . , f_n^′, . . . , f₁^(k), . . . , f_n^(k))

onJkXx. Then an elementQ∈E_k,m^GGΩX,x is exactly a polynomial in the variables(f_i^(j))of the form

Q= X

I1,...,Ik∈Nⁿ

|I1|+2|I2|+···+k|I_k|=m

aI1,...,Ik(f^′)^I1· · ·(f^(k))^Ik

where we use the standard multi-index notation for I = (i1, . . . , in) and 1 6 j 6 k we set f^(j)I

:=

f₁^(j)i1

· · · fn^(j)

in

. It turns out that these fibers can be arranged into a vector bundle overX.

The bundleE_k,m^GGΩX admits a natural filtration. We briefly recall its construction; however, we will not go into details, in particular we don’t justify why everything is well-defined.

Fix local coordinates aroundx∈X as above. For eachp∈Nand for each16s6kdefine F_s^p=F_s^p(E_k,m^GGΩX,x) =

Q∈E_k,m^GGΩX,x involving only monomials(f^′)^I1· · ·(f^(k))^Ik with|I1|+ 2|I2|+· · ·+s|Is|>p

.

This gives a filtration

{0}=F_s^m+1⊆F_s^m⊆ · · · ⊆F_s¹⊆F_s⁰=E_k,m^GGΩX. We consider the associated graded terms

Gr^p_s= Gr^p_s(E_k,m^GGΩX) :=F_s^p/F_s^p+1. Observe that

Gr^p_k−1 ∼=

Qinvolving only monomials(f^′)^I1· · ·(f^(k))^Ik with|I1|+ 2|I2|+· · ·+ (k−1)|Ik−1|=p

∼=

Qinvolving only monomials(f^′)^I1· · ·(f^(k))^Ik withk|Ik|=m−p

.

Therefore Gr^p_k−1 6= 0 if and only if there is an integer ℓk ∈ N such that p= m−kℓk. Whenever this is satisfied, if one looks closely at the coordinate changes, one observe that

Gr^p_k−1= Gr^m−kℓ_k−1 ^k∼=E_{k−1,m−kℓ}^GG _kΩX⊗S^ℓkΩX. And therefore,

Gr^•_k−1= M

1≤p≤m

Gr^p_k−1∼= M

06ℓk6⌊^m_k⌋

E_{k−1,m−kℓ}^GG _kΩX⊗S^ℓkΩX.

Combining all those filtrations, we find inductively a filtration F^• on E_k,m^GGΩX such that the associated graded bundle is

Gr^•(E_k,m^GGΩX) = M

ℓ1+2ℓ2+···+kℓk=m

S^ℓ1ΩX⊗ · · · ⊗S^ℓkΩX.

4 Pacienza-Rousseau generalized jet differential bundles

The Green-Griffiths jet differential bundles were constructed to study entire mapsf :C→X. Pacienza and Rousseau[PR08]generalized this construction to study more generally holomorphic mapsf :C^p→X. For each p≥ 1 they constructed bundles E_p,k,m^GG ΩX generalizing E_k,m^GGΩX = E_1,k,m^GG ΩX. Their construction is very similar to the one described above, however there are some unexpected difficulties appearing. We just really briefly recall the definitions. For the details we refer to [PR08].

Fixp≥1. Consider the spaceJk,pX ofk-jets of germs of holomorphic mapsf : (C^p,0)→X. As above, this space comes with a natural(C^∗)^p-action. Namely, for f : (C^p,0)→X andλ= (λ1, . . . , λp)∈(C^∗)^p we define

λ·f : (C^p,0) → X

(t1, . . . , tp) 7→ f(λ1t1, . . . , λptp).

The bundlesE_p,k,m^GG ΩX are defined as follows: for eachx∈X we consider

E_p,k,m^GG ΩX,x :=

Q(f^′, . . . , f^(k))/ Q(λ·(f^′, . . . , f^(k)) =λ^m₁ · · ·λ^m_pQ(f^′, . . . , f^(k)) for allλ= (λ1, . . . , λp)∈(C^∗)^p

.

Those spaces can be arranged into vector bundles. The only thing we will need concerning these bundles is thatE_p,k,m^GG ΩX admits a filtration whose graded terms are

O

α∈I1

S^q1αΩX· · ·O

α∈Ik

S^qαkΩX,

where

Xk

ℓ=1

X

α∈Iℓ

q_α^ℓα= (m, . . . , m),

and where forℓ∈N,Iℓ:={α= (α1, . . . , αp)∈N^p/ α1+· · ·+αp=ℓ}. See Remark 2.6 in[PR08].

5 A cohomological lemma

During the proof of our main result, we will need an elementary cohomological lemma.

Lemma 5.1. Let X be a projective variety. LetGbe a vector bundle on X. Letk>1. Suppose we have k long exact sequences of vector bundles onX:

0→E₁^ℓ→E₁^ℓ−1→ · · · →E₁¹→E₁⁰→F1→0 0→E₂^ℓ→E₂^ℓ−1→ · · · →E₂¹→E₂⁰→F2→0

...

0→E^ℓ_k→E_k^ℓ−1→ · · · →E_k¹→E_k⁰→Fk→0.

Fix an integerq. Suppose that

H^j(X, E₁ⁱ¹⊗ · · · ⊗E_k^ik⊗G) = 0 f or all j6q+i1+· · ·+ik. Then

H^j(X, F1⊗ · · · ⊗Fk⊗G) = 0 f or all j6q.

Remark 5.2. The casek= 1appears already in Schneider’s article [Sch92].

Proof. The proof is straightforward, but we write it down for the sake of completeness. This is just an induction onk.

Let us start whenk= 1. We proceed by induction onℓ.

Whenℓ= 0, this is obvious. Whenℓ= 1, we just have one short exact sequence 0→E₁¹→E₁⁰→F1→0.

Tensoring it byG and looking at the associated long exact sequence in cohomology gives the result. Now whenℓ>2, we cut the long exact sequence into two pieces, and tensor everything byGto obtain

0→E₁^ℓ⊗G→E^ℓ−1₁ ⊗G→ K⊗G →0

0→ K⊗G →E₁^ℓ−2⊗G→ · · · →E₁¹⊗G→E₁⁰⊗G→F1⊗G→0.

Apply the global section functor to the first exact sequence to obtainH^j(X, K⊗G) = 0for allj6q+ℓ−1, and then apply the induction hypothesis.

Now we let k>2. Suppose the result holds for any family of kexact sequences. Take one more exact sequence

0→E_k+1^ℓ →E_k+1^ℓ−1→ · · · →E_k+1¹ →E_k+1⁰ →Fk+1→0. (1) Suppose thatH^j(X, E₁ⁱ¹⊗ · · · ⊗E_k+1^ik+1⊗G) = 0for allj6q+i1+· · ·+ik+1.

Tensoring the exact sequence (1) byF1⊗ · · · ⊗Fk⊗G, we obtain

0→Ee_k+1^ℓ →Ee^ℓ−1_k+1→ · · · →Ee_k+1¹ →Ee_k+1⁰ →F1⊗ · · · ⊗Fk+1⊗G→0,

whereEe_k+1ⁱ :=Eⁱ_k+1⊗F1⊗ · · · ⊗Fk⊗G. Therefore, to prove thatH^j(X, F1⊗ · · · ⊗Fk+1⊗G) = 0for all j6q, it suffices to prove that for any06i6ℓ,

H^j(X,Ee_k+1ⁱ ) = 0

for allj6q+i. To do so, fix06i6ℓ. Consider theklong exact sequences

0→E1^ℓ⊗Ek+1ⁱ →E₁^ℓ−1⊗Ek+1ⁱ → · · · →E1¹⊗Ek+1ⁱ →E1⁰⊗Ek+1ⁱ →F1⊗Ek+1ⁱ →0 0→E₂^ℓ→E₂^ℓ−1→ · · · →E₂¹→E₂⁰→F2→0

...

0→E_k^ℓ→E_k^ℓ−1→ · · · →E_k¹→E_k⁰→Fk →0.

By hypothesis,H^j(X, E₁ⁱ¹⊗E_k+1ⁱ ⊗E₂ⁱ²⊗ · · · ⊗E_k^ik⊗G) = 0for allj6q+i1+· · ·+ik+i. Therefore by the induction hypothesis, we obtain thatH^j(X,Ee_k+1ⁱ ) =H^j(X, E_k+1ⁱ ⊗F1⊗ · · · ⊗Fk⊗G) = 0for allj6q+i.

This concludes the proof.

Similarly, we obtain the following.

Lemma 5.3. Let X be a projective variety. Let k>1. Suppose we have k long exact sequences of vector bundles onX,

0→F1→E₁⁰→E₁¹ · · · →E₁^ℓ−1→E₁^ℓ→0 0→F2→E₂⁰→E₂¹ · · · →E₂^ℓ−1→E₂^ℓ→0

...

0→Fk→E_k⁰→E_k¹ · · · →E_k^ℓ−1→E_k^ℓ →0.

Fix an integerq. Suppose that

H^j(X, E₁ⁱ¹⊗ · · · ⊗E_k^ik⊗G) = 0 f or all j6q−i1− · · · −ik. Then

H^j(X, F1⊗ · · · ⊗Fk⊗G) = 0 f or all j6q.

6 Main results

To prove his result, Schneider considered a Kozsul resolution of S^kΩX. Then, applying his cohomological lemma with Le Potier’s vanishing theorem [LP75], he was able to conclude. Here we follow the same idea in our more general setting. However, we will not be able to apply directly the vanishing theorem of Le Potier, we have to replace it with a more general vanishing theorem. The following theorem appears first in a work of Ein and Lasarsfeld [EL93], but this result follows from Le Potier’s theorem thanks to an idea of Manivel.

We refer to [EL93] and [Man97] for more details.

Theorem 6.1(Ein-Lazarsfeld/Manivel). LetXbe a smooth projective variety of dimensionn. LetE1, . . . , Er

be vector bundles onX, of ranks e1, . . . , er, and let A be an ample line bundle onX. Assume that eachEi

is generated by global sections, and fixa1, . . . , ar>1. Then

H^k(X, KX⊗Λ^aiE1⊗ · · · ⊗Λ^arEr⊗A) = 0 for k >(e1−a1) +· · ·+ (er−ar).

The caser= 1is the original result of Le Potier [LP75].

Before we continue we recall a well-known fact. Consider a subvariety X ⊆P^N, and let NX/PN denote the normal bundle toX inP^N. ThenNX/PN⊗ OX(−1)is globally generated. This follows directly from the normal exact sequence and the Euler exact sequence. As a matter of notation, we fix an(N+ 1)-dimensional complex vector space such thatP^N =P(V).

Putting all this together, we can prove the following.

Theorem 6.2. Let X ⊆P^N be a smooth variety of dimension nand codimension c=N−n. Consider an integerk>1, integers ℓ1, . . . , ℓk>0, anda∈Z. Ifj < n−k·c anda < ℓ1+· · ·+ℓk, then

H^j(X, S^ℓ1ΩeX⊗ · · · ⊗S^ℓkΩeX⊗ OX(a)) = 0.

Proof. LetN:=NX/PN. Recall the tilde conormal exact sequence 0→N^∗→ΩeP^N

|X →ΩeX→0.

Taking different symmetric powers we obtainkexact sequences 0→Λ^ℓ1N^∗→Λ^ℓ1−1N^∗⊗ΩeP^N

|X → · · · →N^∗⊗S^ℓ1−1ΩeP^N

|X →S^ℓ1ΩeP^N

|X →S^ℓ1ΩeX →0

0→Λ^ℓ2N^∗→Λ^ℓ2−1N^∗⊗ΩeP^N

|X → · · · →N^∗⊗S^ℓ2−1ΩeP^N

|X →S^ℓ2ΩeP^N

|X →S^ℓ2ΩeX →0

... 0→Λ^ℓkN^∗→Λ^ℓk−1N^∗⊗ΩePN

|X → · · · →N^∗⊗S^ℓk−1ΩePN

|X →S^ℓkΩePN

|X →S^ℓkΩeX →0

For16p6k, we let

E_pⁱ = ΛⁱN^∗⊗S^ℓp−iΩeP^N

|X = ΛⁱN^∗⊗S^ℓp−iV ⊗ OX(i−ℓp).

By Lemma 5.1 to prove that

H^j(X, S^ℓ1ΩeX⊗ · · · ⊗S^ℓkΩeX⊗ OX(a)) = 0 forj < n−k·c, it suffices to prove that

H^j(X, E₁ⁱ¹⊗ · · · ⊗E_k^ik⊗ OX(a)) = 0 forj < n−k·c+i1+· · ·+ik. We now prove this fact. We have:

H^j(X, E₁ⁱ¹⊗ · · · ⊗Eⁱ_k^k⊗ OX(a)) =H^j(X,Λⁱ¹N^∗⊗S^ℓ1−i1ΩeP^N

|X ⊗ · · · ⊗Λ^ikN^∗⊗S^ℓk−iΩeP^N

|X⊗ OX(a))

= H^j(X,Λⁱ¹N^∗⊗ · · · ⊗Λ^ikN^∗⊗S^ℓ1−i1V ⊗ · · · ⊗S^ℓk−ikV ⊗ OX(i1−ℓ1+· · ·+ik−ℓk+a))

= S^ℓ1−i1V ⊗ · · · ⊗S^ℓk−ikV ⊗H^j(X,Λⁱ¹(N^∗(1))⊗ · · · ⊗Λ^ik(N^∗(1))⊗ OX(a−ℓ1− · · · −ℓk)) On the other hand using Serre duality, we obtain

H^j(X,Λⁱ¹(N^∗(1))⊗ · · · ⊗Λ^ik(N^∗(1))⊗ OX(a−ℓ1− · · · −ℓk)

= H^n−j(X,Λⁱ¹(N(−1))⊗ · · · ⊗Λ^ik(N(−1))⊗ OX(ℓ1+· · ·+ℓk−a)⊗KX)

SinceOX(ℓ1+· · ·+ℓk−a)is ample andN(−1) is a rank-cglobally generated vector bundle, we can apply Theorem 6.1 to see that this last cohomology group vanishes if

n−j > k·c−i1− · · · −ik

or equivalently, if

j < n−k·c+i1+· · ·+ik. This is exactly what was needed to conclude the proof.

We are in a position to deduce the announced generalization of Schneider’s theorem.

Theorem 6.3. Let X ⊆P^N be a smooth variety of dimension nand codimension c=N−n. Consider an integerk>1andkintegersℓ1, . . . , ℓk>0, anda∈Z. Ifj < n−k·c anda < ℓ1+· · ·+ℓk−min{j, k}, then

H^j(X, S^ℓ1ΩX⊗ · · · ⊗S^ℓkΩX⊗ OX(a)) = 0.

Remark 6.4. The particular casek= 1is precisely Schneider’s result.

Proof. Consider thekshort exact sequences gotten from the Euler exact sequence 0→S^ℓ1ΩX→ S^ℓ1ΩeX →S^ℓ1−1ΩeX →0

0→S^ℓ2ΩX→ S^ℓ2ΩeX →S^ℓ2−1ΩeX →0 ...

0→S^ℓkΩX→ S^ℓkΩeX →S^ℓk−1ΩeX →0.

Fix j < n−k·c. By Lemma 5.3 it is now sufficient to check that for any 0 6 i1, . . . , ik 6 1, for all 06i6j−i1− · · · −ik and for alla < ℓ1+· · ·+ℓk−min{j, k}

Hⁱ(X, S^ℓ1−i1ΩeX⊗ · · · ⊗S^ℓk−ikΩeX⊗ OX(a)) = 0.

This follows from Theorem6.2as soon as one observes that under these conditions,i1+· · ·+ik6min{j, k}.

7 Application to Green-Griffiths jet differentials

Now we apply those vanishing results to Green-Griffiths jet differential bundle. The idea of converting a vanishing result for symmetric differential form bundles into a vanishing result for Green-Griffiths jet differential bundles is due to Diverio. In [Div08], Diverio uses a vanishing theorem due to Bruckmann and Rackwitz (see [BR90]) concerning symmetric differential forms on complete intersection varieties. We proceed along the lines of [Div08].

We start by recalling a cohomological lemma (see [Div08]).

Lemma 7.1. Let E→X be a holomorphic vector bundle with a filtration{0}=Er⊂ · · · ⊂E1 ⊂E0 =E.

If H^q(X,Gr^•E) = 0, thenH^q(X, E) = 0.

Combining this lemma with Theorem 6.3 one derives our result.

Corollary 7.2. LetX⊆P^N be a smooth variety of dimensionnand codimensionc. Leta∈Z. Ifj < n−k·c anda < ^m_k −min{j, k}, then

H^j(X, E_k,m^GGΩX⊗ OX(a)) = 0.

Proof. Recall that the Green-Griffiths bundle admits a filtration whose graded bundle is given by Gr^•E_k,m^GGΩX = M

ℓ1+2ℓ2+···+kℓk=m

S^ℓ1ΩX⊗S^ℓ2ΩX⊗ · · · ⊗S^ℓkΩX.

By twisting everything byOX(a), we obtain a filtration forE_k,m^GGΩX⊗ OX(a)whose graded bundle is given by

Gr^•E_k,m^GGΩX⊗ OX(a) = M

ℓ1+2ℓ2+···+kℓk=m

S^ℓ1ΩX⊗S^ℓ2ΩX⊗ · · · ⊗S^ℓkΩX⊗ OX(a).

By Lemma7.1, we only have to prove that

H^j(X, S^ℓ1ΩX⊗S^ℓ2ΩX⊗ · · · ⊗S^ℓkΩX⊗ OX(a)) = 0