Explicit symmetric differential forms on complete intersection varieties and applications
In this paper we study the cohomology of tensor products of symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. We give several applications. First we prove a non-vanishing result, then we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant. Our main application is the construction of varieties with ample cotangent bundle, providing new results towards a conjecture of Debarre.
- 1 Introduction
- 2 Cohomology of symmetric powers of the cotangent bundle
- 3 Applications
- 4 Varieties with ample cotangent bundle
Varieties with ample cotangent bundle have many interesting properties, however, relatively few examples of such varieties are know (see [Sch85], [Som84], [Deb05], [Bro14]). Debarre conjectured in [Deb05] that: if is a general complete intersection variety in of multidegree high enough and such that then the cotangent bundle of should be ample. The study of this conjecture was the starting point of the present work. In [Bro14] we were able to prove this conjecture when using Voisin’s variational method and inspired by the work of Siu [Siu04] and the work of Diverio Merker and Rousseau [DMR10]. However we were not able to make this strategy work completely in higher dimensions.
The construction of varieties with positive cotangent bundle is closely related to the construction of symmetric differential forms on it. In fact, if one wants to prove that a given variety has ample cotangent bundle, it is natural to start by producing many symmetric differential forms, to be more precise, this means proving that the cotangent bundle is big. In general, this is already a highly non-trivial question and this leads to very interesting considerations. In that direction we would like to mention the recent work of Brunebarbe, Klingler and Totaro [BKT13] as well as the work of Roulleau and Rousseau [R-R14].
However, ampleness is a much more restrictive condition than bigness, in some sense, bigness only requires a quantitative information on the number of symmetric differential forms whereas ampleness requires a more qualitative information on the geometry of the symmetric differential forms. The most natural way to produce symmetric differential forms is to use Riemann-Roch theorem or a variation of it. For instance under the hypothesis of Debarre’s conjecture, one can use Demailly’s holomorphic Morse inequality, in the spirit of Diverio’s work [Div08] and [Div09], to prove that the cotangent bundle is big (see [Bro14]). Nevertheless, this approach doesn’t give much information on the constructed symmetric differential forms besides its existence. One can wonder if it is possible, given a complete intersection variety in to write down explicitly the equation of a symmetric differential form on (if such an object exists).
If is a curve in of genus greater than , this is a very classical exercise. In higher dimensions, very few results towards that problem are known. Brückmann [Bru85] constructed an example of a symmetric differential form on a complete intersection in given by Fermat type equations, and more recently Merker [Mer13] was able to study examples of symmetric differential forms on complete intersection variety in in the spirit of work of Siu and Yeung [S-Y96] (see also [Mer14] for related results for higher order jet differential equations).
The aim of this paper is to develop a cohomological framework which will enable us to describe the space of holomorphic symmetric differential forms on a complete intersection variety in in terms of its defining equations, and to give several applications. The outline of the paper is as follows.
In Section 2 we prove the main technical result of this work. In view of the possible generalizations to higher order jet differential as well as for its own sake, we will not only study the space of symmetric differential forms, but also the more general spaces for a complete intersection variety in Recall the following vanishing result of Brückman and Rackwitz.
Theorem 1.1 (Brückmann-Rackwitz [B-R90]).
Let be a complete intersection of dimension and codimension . Take integers . If then
It is natural to look at what happens in the case in the above theorem. Our result in that direction is the following.
Let , set . Let be a smooth complete intersection variety of codimension , dimension , defined by the ideal , where . Take integers take an integer , let and . Then one has a commutative diagram
Such that all the arrows are injective and such that:
The important thing to note in that result, is that it gives a way of describing the vector space as a sub-vector space of , that this last space is easily described, and that one can precisely determine, in terms of the defining equations of what is the relevant sub-vector space. Therefore this result (and the more general statements in Theorem 2.17 and Theorem 2.24) should be understood as our main theoretical tool to construct symmetric differential forms on complete intersection varieties.
In Section 3 we provide the first applications of Theorem A. First we describe how Theorem A can be used very explicitly in Čech cohomology. Then we illustrate this by treating the case of curves in . After that (Proposition 3.3) we prove that the result of Brückmann and Rackwitz is optimal by providing the following non-vanishing result.
Let , let . Take integers and . Suppose . Then, there exists a smooth complete intersection variety in of codimension , such that
Then, in Corollary 3.7, we provide a new example of a family illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant.
For any , for any , there is a family of varieties over a curve, of relative dimension , and a point such that for generic
This phenomenon has already been studied (see for instance [Bog78], [BDO08] and [R-R14]) and is well known. However, this example shows that invariance fails for any , whereas the other known examples (based on intersection computations) provide the result for large enough.
In Section 4 we provide our main application, which is a special case of Debarre’s conjecture.
Let such that . If be a general complete intersection variety of multidegree , such that , then is ample.
To our knowledge, this is the first higher dimensional result towards Debarre’s conjecture. The proof of this statement does not rely on the variational method neither does it need the Riemann-Roch theorem nor Demailly’s holomorphic Morse inequalities. The idea is to use the results of Section 2 to construct one very particular example of a smooth complete intersection variety in (with prescribed dimension and multidegree) whose cotangent bundle is ample. Then by the openness property of ampleness, we will deduce that the result holds generically. Such an example is produced by considering intersections of deformations of Fermat type hypersurfaces.
Notation and conventions: In this paper, we will be working over he field of complex numbers . Given a smooth projective variety and a vector bundle on , we will denote by the -th symmetric power of , we will denote by the projectivization of rank one quotients of , we will denote the tangent bundle of by and the cotangent bundle of by Moreover we will denote by the canonical projection. Given a line bundle on and an element we will denote the zero locus of by and the base locus of by
Given any , we will denote by the set of homogenous polynomials of degree in variables and by the set of polynomials of degree less or equal to in variables. Given any set and any we will write and
Also, we will say that a property holds for a “general” or a “generic” member of a family if there exists a Zariski open subset such that the property holds for for any .
Acknowledgments. This work originated during the author’s phd thesis under the supervision of Christophe Mourougane. We thank him very warmly for his guidance, his time and all the discussions we had. We also thank Junjiro Noguchi and Yusaku Tiba for listening through many technical details. We thank Joël Merker for his many encouragements and for all the interest he showed in this work. We also thank Lionel Darondeau for motivating discussions and for his suggestions about the presentation of this paper.
2 Cohomology of symmetric powers of the cotangent bundle
2.1 The tilde cotangent bundle
It will be convenient for to use the bundle, studied in particular by Bogomolov and DeOliveira in [BDO08], but also by Debarre in [Deb05]. In some way, the bundle will allow us to work naturally in homogenous coordinates. Let us recall some basic facts about this bundle. Consider with its homogenous coordinates Let be a smooth subvariety. We denote by the Gauss map
where is the embedded tangent space of at , and where denotes the grassmannian of -dimensional linear projective subspace of Let denote the tautological rank vector bundle on Then define
We will refer to this bundle as the tilde cotangent bundle of , and a holomorphic section of will be called a tilde symmetric differential form. Observe that one has a natural identification
Therefore given any homogenous degree polynomial one can define a map
One easily verifies that if is a smooth subvariety and if defines a hypersurface such that is a smooth hypersurface in then the above map fits into the following exact sequence,
We will refer to it as the tilde conormal exact sequence. On the other hand, let be the cone above , and let be the natural projection. Observe that The differential is not invariant under the natural action on because for any , any and any , . We can easily compensate this by a simple twist by as in the following
This yields an exact sequence which we twist and dualize to get
Will refer to it as the Euler exact sequence. Note that the map can be understood very explicitly. Indeed, if we consider the chart , with for any , then Let us mention that in our computations we will often write instead of for simplicity. Those two exact sequences fit together in the following commutative diagram
Observe that can never be ample because it has a trivial quotient. However, Debarre proved in [Deb05] that under the hypothesis of his conjecture, the bundle is ample.
2.2 A preliminary example
The combinatorics needed in the proof of the main results of Section 2 may seem elaborate, but the idea behind it is absolutely elementary. In fact the proofs of the statements in Section 2 are only a repeated us of long exact sequences in cohomology associated to short some exact sequences which are deduced from the restriction exact sequence, the conormal exact sequence, the tilde conormal exact sequence and the Euler exact sequence. But because our purpose is to study tensor produces of symmetric powers of some vector bundle, many indices have to be taken into account, the only purpose of all the notation we will introduce is to synthesis this as smoothly as possible.
Let us illustrate the idea behind this by considering a basic example. Suppose that is a smooth degree hypersurface in defined by some homogenous polynomial Suppose that we want to understand the groups for some and To do so we look at the tilde conormal exact sequence
and take the -th symmetric power and twist it by of it to get the exact sequence
By considering the long exact sequence in cohomology associated to it, we see that the groups can be understood from the groups for and from the applications appearing in the long exact sequence in cohomology. But to understand those groups, we consider the restriction exact sequence
and twist it by to get
Once again, we look at what happens in cohomology, and we see that the groups can be understood from the groups for and the maps appearing in the long exact sequence. But observe that for all from this we get that for all and from this we deduce that for all . Moreover, a more careful study shows that we obtain the following chain of inclusions:
The inclusions appearing in Theorem A are of this type. Now if one wants to describe what is the image of this composed inclusion, one needs to look more carefully at what are exactly the maps between the cohomology groups in the different long exact sequences. For instance the second injection comes the following exact sequence
Hence, To understand similarly is less straightforward, combining the different long exact sequences one obtains the following commutative diagram:
where the vertical arrows are injective. Then, by linear algebra, we obtain that for suitable maps and This example already contains the main idea of the proof of the first part of Theorem A. To study more generally tensor products of symmetric powers of the tilde cotangent bundle is done similarly by considering each factor independently, and to deduce the results concerning the cotangent bundle instead of the tilde cotangent bundle is done in a similar fashion using the Euler exact sequence.
2.3 An exact sequence
In the rest of Section 2, the setting will be the following. Take an integer , let and take . Take non-zero elements , and for any we set . For any let Set and We suppose that is smooth. For simplicity, we will also suppose that is smooth for each
We make this additional smoothness hypothesis here so that we can work without worrying with all the conormal exact sequences between and (and this hypothesis will be satisfied in all our applications). However, a more careful analysis of the proof of the main results shows that the only thing we need to have is the smoothness of each of the s in a neighborhood of , and this follows from the smoothness of .
To simplify our exposition, we introduce more notation. If is a vector bundle on a variety , and if is a -uple of non-negative integers, then we set
If is a -uple of non-negative integers, we set
The following definition gives a convenient framework for our problem.
With the above notation.
A -setting is a -uple where , and for any , for some .
If is as above, we set:
If set . Otherwise, let and set
Take as above. We set:
For any and any , we set:
We will also need a more general definition which will allow us to work simultaneously with and .
A -pair is a couple of -settings and such that
Given a -pair we set:
If for all we set Otherwise, let and set
With the above notation, we set
For any and any , we set
To describe our fundamental exact sequence, we introduce some notion of successors.
Take a -setting with and where for any one denotes . We define -settings and as follows.
If for all then set
If there exists such that , let and let . Then we define
We will need the following generalization to -pairs.
Take a -pair where and . Define and as follows.
If for all set
If there exists such that or set .
Now we come to an elementary, but important, observation.
For any -pair , we have a short exact sequence
Take and We have to consider two cases.
Case 1: Set and so that We have the restriction exact sequence
Since , it suffices to tensor this exact sequence by to obtain the desired exact sequence.
Case 2: there exists such that or . Set . Suppose in a first time that . Recall that . Set also . Set so that , and . By taking the -th symmetric power of the tilde conormal exact sequence when is seen as a hypersurface of and restricting everything to , we obtain
It suffices now to tensor this exact sequence by and to obtain the desired result. If with use the same decomposition on (and the ’s) and we use the usual conormal exact sequence instead of the tilde conormal exact sequence. ∎
2.4 A vanishing lemma
In this section, we prove a vanishing lemma which we will often use later. To be able to give the statement, we need some more notation. Given any -uple of integers , we define to be the number of non-zero terms in
Take a -setting , where for all , Then we set
, where .
, where .
, where and .
We will also need the generalization to -pairs.
Take a -pair where , Then we set
, where .