Compact tori associated to hyperkähler manifolds of Kummer type
1. Introduction
1.1. Background and motivation
Let be a hyperkähler manifold, i.e. (for us) a simply connected compact Kähler manifold carrying a holomorphic symplectic form whose cohomology class spans . The KugaSatake construction [KS67, Del72] associates to a compact complex torus , and an inclusion of Hodge structures . The definition of is transcendental: one constructs a weight H.S. out of the weight H.S. on . If is projective with ample line bundle , the KugaSatake construction applied to the primitive cohomology produces an abelian variety and an injective homomorphism of H.S.’s
(1.1.1) 
One might wonder whether it is possible to relate the geometry of and that of , or of . A famous instance of such a relation is provided by Deligne’s proof of the Weil conjectures for (projective) surfaces starting from the validity of the Weil conjectures for abelian varieties [Del72]. In this respect we notice that if is projective, the Hodge conjecture predicts the existence of a KugaSatake algebraic cycle on realizing the homomorphism of H.S.’s in (1.1.1).
There are very few families of hyperkähler manifolds for which one has a geometric description of the corresponding KugaSatake varieties and a proof of existence of a KugaSatake algebraic cycle: Kummer surfaces [Mor85] and surfaces obtained as minimal desingularization of the double cover of a plane ramified over 6 lines [Par88].
The present paper grew out of the desire to understand the KugaSatake torus associated to hyperkähler manifolds of Kummer type, i.e. deformations of the dimensional generalized Kummer manifold associated to an abelian surface (for ).
Among known examples of hyperkähler manifolds, those of Kummer type are distinguished by the fact that they have non zero odd cohomology. Let be such a manifold. Then , and hence there is an associated 4 dimensional intermediate Jacobian . Most of our paper is actually concerned with . Our starting point is the proof that there is an analogue of the key cohomological property of the KugaSatake torus (see (1.1.1)) valid with replacing the KugaSatake torus. From this it follows that if is projective with polarization , then is isogenous to . Thus is a smaller dimensional version of the KugaSatake torus. Moreover, it is easier to relate geometrically to than it is to relate it to (or ), e.g. via the AbelJacobi map.
We will give an explicit recipe that produces the weight H.S. on in terms of the weight H.S. on .
One fact that we discovered is that if is projective, then is an abelian fourfold of Weil type. More precisely, as varies in a complete family of polarized hyperkählers of Kummer type with fixed discrete invariants, the corresponding polarized intermediate Jacobians sweep out a complete family of polarized abelian fourfolds of Weil type with fixed discrete invariants. Notice that the number of moduli for both families is equal to . This result suggests that we will be able to describe explicitly locally complete families of projective hyperkählers of Kummer type starting from the locally complete families of abelian fourfolds of Weil type which are known (see [Sch88]). In this respect, we notice that several locally complete families of projective hyperkählers have been explicitly described, but the varieties in those familes are all of type (deformations of the Hilbert scheme of length subschemes of a surface).
There is a series of papers related to the present work. The first one is [vG00]. Following the proof of Theorem 9.2 of that paper, one shows that the Kuga Satake of a polarized HK of Kummer type is the fourth power of an abelian fourfold of Weil type. Since is isogenous to , it follows that is of Weil type. However we would like to stress that we have precise results on the integral Hodge structure on , not only up to isogeny. Another paper related to this work is [Lom01]. Lastly, the recent preprint [Mar18] is strictly related to our work.
1.2. Main results
Let be a hyperkähler manifold of dimension at least , deformation equivalent to a generalized Kummer variety (following established terminology, we say that is of Kummer type). Then , see [Göt94], and of course . Thus
(1.2.1) 
is a dimensional compact complex torus. If is projective, and is an ample line bundle on , then is an abelian fold (all of is primitive because ), and we let be the polarization defined by .
Recall that, given a HK manifold , there is a class which corresponds to the BeauvilleBogomolovFujiki quadratic form of (see LABEL:subsecdefgenkum for details). Now assume that is of Kummer type, of dimension . Then is an integral class (see LABEL:dfnqubarra). Let
(1.2.2) 
be the composition of the map
and the map defined by cup product followed by integration.
Theorem 1.1.
Let be a HK manifold of Kummer type, of dimension .

The map is surjective, and hence its transpose defines an inclusion of integral Hodge structures
(1.2.3) 
The set
(1.2.4) is a smooth quadric hypersurface in .

The projectivization of is a maximal linear subspace of .
If is a HK manifold of Kummer type, let be the irreducible component of the variety parametrizing maximal dimensional linear subspaces of containing (this definition makes sense by LABEL:thmprimoteor). We recall that , where is one of the two spinor representations of . Recall also that is dimensional. Since has an integral structure, so does . There is a unimodular integral quadratic form on (unique up to multiplication by ) such that is the set of zeroes of . Moreover, if is a family of HK manifolds of Kummer type, the flat connection on induces a flat connection on the fibration with fiber over . Next, we make following
Key observation 1.2.
Let be the map in (1.2.2). Then is equal the one dimensional subspace .
In fact is contained in because is a morphism of Hodge structures, and equality follows from surjectivity of . Notice that Item (3) of LABEL:thmprimoteor follows from the LABEL:kobchiave.
The result below is motivated by the LABEL:kobchiave.
Theorem 1.3.
Let be a HK manifold of Kummer type, of dimension . There exists a codimension subspace defined over such that the following hold:

Given a dimensional vector subspace , the subspace has dimension if and only if is a linear subspace of parametrized by a point . If this is the case, then .

There exist an isomorphism defined over , invariant up to sign under monodromy, and a choice of “sign” for , such that the restriction of to is equal to the dual of the BBF quadratic form.
Item (1) of LABEL:thmsecondoteor amounts to an explicit description of the weight Hodge structure on in terms of the weight Hodge structure on .
The result below was first proved by Mongardi by other methods. We will show that it is a simple consequence of LABEL:thmsecondoteor.
Corollary 1.4 (Mongardi [Mon16]).
Let be a HK of Kummer type. Let be a monodormy operator. Then either acts trivially on the discriminant group (here is embedded into by the BBF quadratic form) and it has determinant , or it acts as multiplication by on the discriminant group and it has determinant .
Below is our last main result.
Theorem 1.5.
Let be a hyperkähler variety of Kummer type, of dimension , and let be an ample line bundle on . Then is of Weil type, with an inclusion
where is the value of the BeauvilleBogomolovFujiki (BBF) quadratic form on . By varying , one gets a complete (up to isogeny) family of dimensional abelian varieties of Weil type with associated field , and trivial determinant. Moreover, tha KugaSatake variety is isogenous to .
1.3. Organization of the paper
Most of LABEL:secgenkum is devoted to the proof of results on the cohomology of HK’s of Kummer type. After recalling the definition of generalized Kummers, and establishing basic notation, we compute the constants which enter into the formula for certain integrals on a HK of Kummer type (see LABEL:prpbellaform). In LABEL:subsecintcohom we describe explicitly the integral 3rd cohomolgy group of a generalized Kummer. In dimension 4 this was done by Kapfer and Menet [KM16]. We extend their result to arbitrary dimension by adapting arguments of Totaro [Tot16]. In LABEL:subsecstruttura we show that, by invariance under the monodromy group of compact complex tori, the map in (1.2.2) for dimensional HK’s of Kummer type has a “shape” which depends on an apriori unknown . In LABEL:subsechilbring, LABEL:subsecgrancul and LABEL:subsecgranculdue we compute the first two entries of (the third entry will be determined up to sign in LABEL:subsectwoone). Most of the effort goes in a painful computation of the cup product of certain cohomology classes on a generalized Kummer. In order to do this we rely on the explicit description of the cohomology ring of Hilbert schemes of smooth projective surfaces with trivial canonical bundle given by Lehn and Sorger [LS03]. The last subsection of LABEL:secgenkum contains the proof of LABEL:thmprimoteor.
In LABEL:secalglin we prove LABEL:thmsecondoteor and LABEL:crlmonlim. Actually we discuss an “abstract” map which has the same shape as , depending on a choice of with no vanishing entry. In such a setup, we have a way of explicitly associating to a weight H.S. of type a weight H.S. If the weight H.S. is polarized, then the weight H.S. is also polarized.
In the short LABEL:secpolandmon we compute the elementary divisors of the natural polarization of for a polarized HK fourfold .
LABEL:sectipoweil is devoted to the proof of LABEL:thmterzoteor. Actually we prove, more generally, that the polarized weight H.S.’s constructed in LABEL:secalglin (depending on a with no vanishing entry) are of Weil type.
1.4. Conventions
We work over : projective varieties will be complex projective varieties.
Throughout the present paper, is an abelian surface.
Notation: in dealing with cohomology, we omit to mention the ring of coefficients when we consider complex coefficients.
Let be a lattice. The divisibility of a non zero is the positive generator of ; we denote it by .
Let . The double factorial of is equal to
(1.4.1) 
It is convenient to set and .
Acknowledgements
After giving a talk in Oberwolfach on some of the results of this paper [O’G17], I had various stimulating conversations related to this work. I am grateful to Francois Charles for sharing an unpublished paper of his on the KugaSatake construction [Cha]. I am indebted to Eyal Markman, who pointed out the importance of triality  this motivated me to formulate LABEL:thmsecondoteor in terms of spinor representations. Thanks go to Eyal also for sending me his preprint [Mar18] while I was finishing writing the present paper.
It is a pleasure to thank Ruggero Bandiera for proving LABEL:lmmideban.
2. Generalized Kummers and their cohomology
2.1. Hilbert schemes parametrizing subschemes of finite length
Let be a smooth projective surface. Let be the Hilbert scheme parametrizing subschemes of length , and let be the symmetric product of copies of . Given a point , we let
The HilbertChow map associates to the cycle . Let be the prime divisor parametrizing non reduced schemes. The divisor class of is divisible by . We let be characterized by
(2.1.1) 
(In order to simplify notation, we will omit whenever there is no ambiguity.)
Let be a (commutative) ring. Given , let be characterized by the formula
(2.1.2) 
where is the quotient map and is the th projection. Let
(2.1.3) 
For let
(2.1.4) 
Then is irreducible of (complex) dimension . Let be the map sending to the support of , and let be the projection. We let
(2.1.5) 
where means Poincaré dual.
2.2. Generalized Kummers
Let be an abelian surface. Let be the summation map (in the group ). The th generalized Kummer variety is
Beauville [Bea85] proved that is a hyperkähler variety of dimension . Let
(2.2.1) 
and
(2.2.2) 
Now suppose that . We have a direct sum decomposition
(2.2.3) 
Moreover, the map for is a homomorphisms of integral Hodge structures. The BeauvilleBogomolovFujiki bilinear form is given by
(2.2.4) 
and the normalized Fujiki constant of equals , i.e.
(2.2.5) 
Remark 2.1.
Let be a complex vector space, equipped with a bilinear symmetric form . Let us say that two permutations are equivalent if we have equality of multilinear symmetric functions
(2.2.6) 
Let be a set of representatives for equivalence classes, and let be the multilinear symmetric function defined by
(2.2.7) 
Then is the polarization of the homogeneous polynomial , i.e. . In particular, Equation (2.2.5) is equivalent to the equation
(2.2.8) 
Now let be a dimensional hyperkähler manifold, of Kummer type. The bilinear form defines an isomorphism . The inverse defines an element in , whose image by the cupproduct map is a class in that we denote by , or if there is no danger of misunderstanding. An explicit expression for is obtained as follows. Let be a standard basis of , i.e.
(2.2.9) 
(Notice that each is a hyperbolic plane.) Then
(2.2.10) 
Before proving a result on products of , we need an identity whose proof was kindly provided by Ruggero Bandiera.
Lemma 2.2 (Ruggero Bandiera).
Let and be natural numbers. Then
(2.2.11) 
Proof.
For fixed natural numbers the left and right hand sides of (2.2.11) are polynomials in (of degree ), that we denote and respectively. In particular and makes sense for any , not only for an integer greater than . One proves that by induction on arguing as follows. First , because they are both equal to the constant polynomial . A straightforward computation shows that
and hence by the inductive hypothesis the difference operators of and of are equal. Since
it follows that . ∎
Proposition 2.3.
Let be a dimensional hyperkähler manifold, of Kummer type. Then for all
Proof.
By a theorem of Fujiki [Fuj87], there exists a rational number (independent of ) such that
(2.2.12) 
In order to determine , it suffices to compute the left hand side of (2.2.12) for one and one such that . We will do the computation for and . Let
Thus
A straightforward computation shows that
With some manipulations, it follows that
Thus it remains to show that
The above equality follows at once from the case of LABEL:lmmideban. ∎
Definition 2.4.
If is a dimensional hyperkähler manifold of Kummer type, let .
The point of the above definition is that (by (2.2.10)).
2.3. On the integral cohomology of generalized Kummers
We will prove the following two results.
Proposition 2.5 (Contained in [Km16] for .).
Let . Then is divisible by in .
Remark 2.6.
Let . By LABEL:prpaccatre, there is a well defined
Theorem 2.7 (Proposition 6.2 in [Km16] for .).
The map
(2.3.1) 
is an isomorphism of integral Hodge structures.
We recall that is naturally stratified, with strata indexed by partitions of . The stratification of defines a stratification of via pull back by the HilbertChow map. Let be a partition of , where . The stratum is equal to the set of such that , where the points are pairwise distinct. Each stratum is irreducible, and
(2.3.2) 
Since is a stratification, the dimension formula (2.3.2) shows that
(2.3.3) 
are open (dense) subsets of .
Lemma 2.8.
The restriction map is an isomorphism.
Proof.
The complement of the open has (complex) codimension ; it follows by a standard argument that the map is an isomorphism. Thus it suffices to prove that the map is an isomorphism. (Notice that .) If , then , and hence we are done. From now on we assume that . A piece of the long exact sequence of cohomology with coefficients for the couple reads
By excision and Thom’s isomorphism, , hence is injective. On the other hand, excision and Thom’s isomorphism give that
where and are the fundamental classes of and respectively. In order to finish the proof it suffices to show that is injective, i.e. that are independent over (if this is to be interpreted as stating that is not a torsion class). We may assume that is a special abelian surface; we will assume that , where are elliptic curves. Given , we let be the embedding as the slice . Let be a generic very ample divisor. Thus meets the curve in a finite non empty set. Choose , and let be pairwise distinct points of . Let be defined by
If , choose distinct , and let be pairwise distinct points of . Let be a degree map. We let be the defined by , i.e. . Let be defined by
Both and are projective, and have (pure) dimension . Thus we may evaluate the classes and on and . Now notice that meets in a finite non empy set, and that meets in a finite non empty set, and it does not meet . It follows that the matrix describing the evaluation of the classes , on and is non degenerate. (If , this is to be interpreted as stating that the evaluation of the class on is non zero). This proves that is injective. ∎
Proof of LABEL:prpaccatre.
It suffices to prove that is divisible by . Let be the open dense subset defined in (2.3.3); by LABEL:lmmrestisom it suffices to prove that is divisible by in . Let be the reduction modulo of ; thus is the reduction modulo of . We must show that
(2.3.4) 
A piece of the long exact sequence of cohomology with coefficients for the couple reads
(2.3.5) 
Thom’s isomorphism gives an identification
(2.3.6) 
Let be the composition of (the restriction of HilbertChow) and the projection . Then
(2.3.7) 
(The above equation makes sense by (2.3.6)). By Lemma 3.1 in [Tot16] (Totaro’s Lemma is stated for , but the same proof gives the statement in general), , and hence (2.3.4) holds. ∎
Proof of LABEL:thmaccatre.
The map in (2.3.1) is a morphism of Hodge structures, integral by LABEL:prpaccatre, hence we are left with the task of proving that it defines an isomorphism between and . We proceed as in the proof of Proposition 6.2 in [KM16].
Let be an oriented basis of , i.e. such that is the orientation class, and let be the dual basis. Since we have the perfect pairing
(2.3.8) 
we may view each as an element of . Let be generic smooth oriented manifolds representing the Poincaré duals of , and let be generic smooth oriented manifolds representing the Poincaré duals of . Choose generic (distinct) points . Let be the smooth oriented manifolds