Cohomology of large semiprojective hyperkaehler varieties
In this paper we survey geometric and arithmetic techniques to study the cohomology of semiprojective hyperkähler manifolds including toric hyperkähler varieties, Nakajima quiver varieties and moduli spaces of Higgs bundles on Riemann surfaces. The resulting formulae for their Poincaré polynomials are combinatorial and representation theoretical in nature. In particular we will look at their Betti numbers and will establish some results and expectations on their asymptotic shape.
à Gérard Laumon à l’occasion de son 60éme anniversaire
At the conference “De la géométrie algébrique aux formes automorphes : une conférence en l’honneur de Gérard Laumon” the first author gave a talk, whose subject is well-documented in the survey paper [Ha4]. Here, instead, we will discuss techniques, both geometrical and arithmetic, for obtaining information on the cohomology of semiprojective hyperkähler varieties and we will report on some observations on the asymptotic behaviour of their Betti numbers in certain family of examples.
We call a smooth quasi-projective variety with a -action semiprojective when the fixed point set is projective and for every and as tends to the limit exists.
Varieties with these assumptions were originally studied by Simpson in [Si2, §11] and varieties with similar assumptions were studied by Nakajima in [Na3, §5.1]. The terminology semiprojective in this context appeared in [HS], which concerned semiprojective toric varieties and toric hyperkähler varieties. In particular, a large class of hyperkähler varieties, which arise as a hyperkähler quotient of a vector space by a gauge group, are semiprojective. These include Hilbert schemes of -points on , Nakajima quiver varieties and moduli spaces of Higgs bundles on Riemann surfaces.
It turns out that despite their simple definition we can say quite a lot about the geometry and cohomology of semiprojective varieties. We can construct a Bialinycki-Birula stratification (§1.2), which in §1.3 will give a perfect Morse stratification in the sense of Atiyah–Bott. This way we will be able to deduce that the cohomology of a semiprojective variety is isomorphic with the cohomology of the fixed point set with some cohomological shifts. Also, the opposite Bialinycki-Birula stratification will stratify a projective subvariety of the semiprojective variety, the so-called core, which turns out to be a deformation retract of . This way we can deduce that the cohomology is always pure. Furthermore, we can compactify with a divisor , to get an orbifold . Finally in §1.4 we will look at a version of a weak form of the Hard Lefschetz theorem satisfied by semiprojective varieties.
We will also discuss arithmetic approach to obtain information on the cohomology of our hyperkähler varieties. It turns out that the algebraic symplectic quotient construction of our hyperkähler varieties will enable us to use a technique we call arithmetic harmonic analysis to count the points of our hyperkähler varieties over finite fields. With this technique we can effectively determine the Betti numbers of the toric hyperkähler varieties and Nakajima quiver varieties as well as formulate a conjectural expression for the Betti numbers of the moduli space of Higgs bundles.
To test the range in which the Weak Hard Lefschetz theorem of §1.4 might hold, we will look at the graph of Betti numbers for our varieties when their dimension is very large. The resulting pictures are fairly similar and we observe that asymptotically they seem to converge to the graph of some continuous functions. We will see, for example, the normal, Gumbel and Airy distributions emerging in the limit in our examples. We will conclude the paper with some proofs and heuristics towards establishing such facts.
Acknowledgement We would like to thank Gábor Elek, Stavros Garoufadilis, Sergei Gukov, Jochen Heinloth, Daniel Huybrechts, Andrew Morrison, Antonello Scardicchio, Christoph Sorger, Balázs Szegedy and Balázs Szendrői for discussions related to this paper. The first author was supported by a Royal Society University Research Fellowship (2005-2012) and by the Advanced Grant ”Arithmetic and physics of Higgs moduli spaces” no. 320593 of the European Research Council (2013-2018) during work on this paper. The second author is supported by the NSF grant DMS-1101484 and a Research Scholarship from the Clay Mathematical Institute. He would also like to thank the Mathematical Institute of University of Oxford where this work was started for its hospitality.
1 Semiprojective varieties
1.1 Definition and examples
We start with the definition of a semiprojective variety, first considered in [Si2, Theorem 11.2].
Let be a complex quasi-projective algebraic variety with a -action. We call semiprojective when the following two assumptions hold:
The fixed point set is proper.
For every the exists as tends to .
The second condition could be phrased more algebraically as follows: for every we have an equivariant map such that and acts on by multiplication.
First example is a projective variety with a trivial (or any) -action. For a large class of non-projective examples one can take the total space of a vector bundle on a projective variety, which together with the canonical -action will become semiprojective.
A good source of examples arise by taking GIT quotients of linear group actions of reductive groups on vector spaces. Examples include the semiprojective toric varieties of [HS] (even though the definition of semiprojectiveness is different there, but equivalent with ours) and quiver varieties studied by Reineke [Re1].
Semiprojective hyperkähler varieties
In this survey we are interested in semiprojective hyperkähler varieties. Examples arise by taking the algebraic symplectic quotient of a complex symplectic vector space by a symplectic linear action of a reductive group . In practice and arises as the doubling of a representation . If denotes the Lie algebra of , we have the derivative of as . This gives us the moment map
at by the formula
By construction is equivariant with respect to the coadjoint action of on . Taking a character will yield the GIT quotient using the linearization induced by . Sometimes can be chosen generically so that becomes non-singular (and by construction) quasi-projective. We assume this henceforth. By construction of the GIT quotient we have the proper affinization map
to the affine GIT quotient
The -action on given by dilation will commute with the linear action of on it so that the moment map (1.1.1) will be equivariant with respect to this and the weight action of on . This will induce a -action on , such that on the affine GIT quotient it will have a single fixed point corresponding to the origin in . This and the fact that the affinization map (1.1.3) is proper implies, that is semiprojective, provided that is non-singular, which we always assume.
An important special case is when . In this case we can take a square root of the action above by acting only on by dilation and trivially on . This action will also commute with the action of on and will indeed reduce to a -action on the quotient whose square is the -action we considered in the previous paragraph. In particular this new action also makes semiprojective. In fact, it will have an additional property. Namely, the natural symplectic form on will be of homogeneity with respect to the action; in other words, it will satisfy
under this action. This property will be inherited by the quotient . Following [Ha2] we make the following
A semiprojective hyperkähler variety with a symplectic form of homogeneity one as in (1.1.4) is called hyper-compact.
When is a torus, are the toric hyperkähler varieties of [HS]; these always can be arranged to become hyper-compact. When the representation arises from a quiver with a dimension vector is a quiver variety as constructed by Nakajima in [Na2]. When the quiver has no edge loops, one can always arrange that becomes hyper-compact. When the quiver is the tennis-racquet quiver, i.e. two vertices connected with a single edge and with a loop on one of them, and the dimension vector is in the simple vertex and on the looped one, the Nakajima quiver variety becomes isomorphic with the Hilbert scheme of points on . This semiprojective hyperkähler variety however is not hyper-compact as we will see later.
Finally, the following hyper-compact examples originally arose from an infinite dimensional analogue of the above construction. In [Hi1] Hitchin constructs the moduli space of semistable rank degree Higgs bundles on a Riemann surface as an infinite dimensional gauge theoretical quotient. A Higgs bundle is a pair of a rank degree vector bundle on the Riemann surface and . Nitsure [Ni] constructed such moduli spaces in the algebraic geometric category, which are non-singular quasi-projective varieties when . There is a natural action on given by scaling the Higgs field . Hitchin [Hi1] when and Simpson [Si2, Corollary 10.3] in general showed that is semiprojective. A nice argument to see this, is similar for the argument for above. Namely the affinization
turns out to be the famous Hitchin map [Hi2], which by results of Hitchin [Hi1] when and Nitsure [Ni] for general is a proper map. It is also -equivariant wich covers a -action on the affine with a single fixed point. This implies that is indeed semiprojective.
1.2 Bialinycki-Birula decomposition of semiprojective varieties
Let be a non-singular semiprojective variety. Let be the decomposition of the fixed point set into connected components. Then is finite and are non-singular projective subvarieties of . According to [Dol, Corollary 7.2] we can linearize the action of on a very ample line bundle on . On each then will act on through a homomorphism with weight which we can assume, by suitably changing the linearisation, are always non-negative . We introduce a partial ordering on by setting
Introduce as the set of points for which . Similarly, as above, we can define as the points for which . These are locally closed subsets and Bialynicki-Birula [Bia, Theorem 4.1] proves that both and are subschemes of which are isomorphic to certain affine bundles (so-called -fibrations) over .
It will be convenient to make the following
The core of the semiprojective variety is
By assumption 2 of Definition 1.1.1 we get the Bialinycki-Birula decomposition . This decomposition satisfies that
To see this we note that using the linearisation on the very ample line bundle we can equivariantly embed into some projective space with a linear action. (1.2.2) follows from the corresponding statement for the linear action of on , where it is clear.
It follows from the Hilbert-Mumford criterium for semistability that with respect to our linearisation. Thus we have a geometric quotient , which is proper according to [Si2, Theorem 11.2] and is, in fact, an orbifold as there are no fixed points of on . Using this construction for the semiprojective where acts via the diagonal action (with the standard multiplication action on the second factor) we get
which decomposes as corresponding to points in with non-zero (respectively zero) second component. This thus yields an orbifold compactification of , the algebraic analogue of Lerman’s symplectic cutting [Ler], as studied in [Ha1].
An immediate consequence of this compactification is the following:
The core of a semiprojective variety is proper.
The proper has two Bialinycki-Birula decompositions. One of them is
Thus by property (1.2.2) the core is closed in the proper . The claim follows. ∎
1.3 Cohomology of semiprojective varieties
Generalities on cohomologies of complex algebraic varieties
We denote by the integer and by the rational singular cohomology of a CW complex . is a graded anti commutative ring; while is a graded anticommutative -algebra.
When is a complex algebraic variety there is further structure on its rational cohomology. Motivated by the Frobenius action on the -adic cohomology of a variety defined over an algebraic closure of a finite field Deligne in 1971 [De] introduced mixed Hodge structures on the cohomology of any complex algebraic variety .
Here we only recall the notion of the weight filtration on rational cohomology. It is an increasing filtration:
by -vector spaces . It has many nice properties. For example it is functorial,
for a smooth , and
for a projective variety . We say that the weight filtration on is pure when both (1.3.1) and (1.3.2) holds for every . In particular a smooth projective variety always has pure weight filtration. We will see in Corollary 1.3.2 that semiprojective varieties also have pure weight filtration.
We denote by
the mixed Hodge polynomial. It has two important specializations. The polynomial
is the Poincaré polynomial of , while the specialization
the virtual weight polynomial. In the case when the weight filtration is pure on we have the relationships
In the general case however there is no such relationships.
The case of semiprojective varieties
We now have the following commutative diagram:
Here the top row is the cohomology long-exact sequence of the pair and
is excision followed by the Thom isomorphism theorem, where . The bottom row is the cohomology long-exact sequence of the pair , where again
is the Thom isomorphism. Finally , and denote the corresponding imbeddings.
Clearly is an isomorphism. So if we know that is an isomorphism, so will be by the five lemma. If denotes a minimal element in , then and so . Therefore by induction we get that
is an isomorphism for all satisfying (1.3.5). Thus in particular we have:
The embedding induces an isomorphism .
A smooth semiprojective variety has pure cohomology.
Interestingly our techniques can also be used to prove the purity of the cohomology of certain, typically affine, varieties which are deformations of semiprojective varieties as in the following corollary.
Let be a non-singular complex algebraic variety and a smooth morphism, i.e. a surjective submersion. In addition, let be semiprojective with a action making equivariant covering a linear action of on with positive weight. Then the fibers have isomorphic cohomology supporting pure mixed Hodge structures.
The proof can be found in [HLV1, Appendix B]. It proceeds by proving that the embedding of every fiber of induces an isomorphism
which implies the statement in light of Corollary 1.3.2. This is clear for as is itself semiprojective and it shares the same core with . The proof of (1.3.7) for is more difficult and is using a version of the compactification technique as in (1.2.3) and Ehresmann’s theorem for proper smooth maps; in particular the proof is not algebraic. ∎
In fact Simpson’s [Si2] main example for a semiprojective variety was , the moduli space of stable rank degree -connections on the curve which comes with satisfying the conditions of Corollary 1.3.3. Here is our moduli space of Higgs bundles while is the moduli space of certain holomorphic connections. The Corollary 1.3.3 then shows that have isomorphic and pure cohomology. This argument was used in [HT, Theorem 6.2] and [Ha3, Theorem 2.2] in connection with topological mirror symmetry.
Another crucial use of this Corollary 1.3.3 is in our arithmetic harmonic analysis technique explained in §2. We will be able to compute the virtual weight polynomial of an affine symplectic quotient, and to deduce that it gives the Poincaré polynomial we will put in a family satisfying the conditions of Corollary 1.3.3.
The core is a deformation retract of the smooth semiprojective variety .
First we note that replacing cohomology with homology in the proof of Theorem 1.3.1 yields that that induced by the embedding is also an isomorphism. By the homology long exact sequence this is equivalent with
We also claim that induces an isomorphism on the fundamental group (from whose notation we omitted the base-point for simplicity). This follows by induction similarly as in the proof of Theorem 1.3.1. First note by [Bia, Theorem 4.1] that retracts to thus have isomorphic fundamental group. Then by induction we assume is an isomorphism for an index set satisfying (1.3.5). Take minimal and cover with open sets and a small tubular neighborhood of , small in the sense that it is disjoint from the proper ( is the core of the semiprojective ; thus proper by Theorem 1.2.2). This implies that is a tubular neighborhood of . Then we have two commutative diagrams:
where the maps are all induced by the embedding of the indicated varieties in each other. The four vertical arrows are all isomorphisms. The last one because of the induction hypothesis. The second one as both and retract to . Finally, the first and the third because these spaces all retract to .
Using the diagrams (1.3.9) and the Seifert-van Kampen theorem applied to both the open covering
we see that
By induction we get the desired
In particular, the homotopy long exact sequence of the pair implies that as well as that is a quotient of and so abelian. From this and (1.3.8) the relative Hurewitz theorem [Whi, Theorem IV.7.3] implies for every , thus
is an isomorphism. Therefore and are weakly homotopy equivalent, and as varieties they are CW complexes and so by Whitehead’s theorem [Whi, Theorem V.3.5] is a homotopy equivalence. ∎
The Bialinycki-Birula decomposition of a semiprojective variety is perfect. In particular .
This follows from studying the top long-exact sequence of (1.3.6) considered with rational coefficients. Here we assume the same situation as there:
This is a sequence of Mixed Hodge structures, and the weights are pure according to Corollary 1.3.2 in the cohomology of the semiprojective varieties and , and in by the Thom isomorphism. Therefore the connecting morphism must be trivial. Therefore the long exact sequence splits, the stratification is perfect, and the formula for Poincaré polynomials follow by induction. ∎
1.4 Weak Hard Lefschetz
Fix a very ample line bundle on a smooth semiprojective variety and let . Then we have
Theorem 1.4.1 (Weak Hard Lefschetz).
Let be a semiprojective variety with core . Assume is equidimensional of pure dimension Then the Hard Lefschetz map
is injective for .
It follows from Corollaries 1.3.6 and 1.3.2 that the core has pure cohomology. Then the result follows from [BE, Theorem 2.2] as we have assumed is equidimensional. Their argument goes by first showing that the natural map is injective, and then concludes by using [BBD, Theorem 5.4.10] for the Hard Lefschetz theorem for . ∎
An immediate consequence of the injectivity of (1.4.1) for are the inequalities
for the Betti numbers of the smooth semiprojective variety. As a consequence both sequences of odd and even Betti numbers grow until and satisfy .
Possibly the analogous result to (1.4.1) holds when is not equidimensional and is the smallest dimension of the irreducible components of . It was proved in the case of smooth semiprojective toric varieties in [HS]. There however it was used that the components of the core are smooth; but conceivably this can be avoided.
Of course a general semiprojective toric variety could have a non-equidimensional core (as it corresponds to the complex of bounded faces of a non-compact convex polyhedron). However, we do not know of an example of a semiprojective hyperkähler variety whose core is not equidimensional.
When the semiprojective variety is hyper-compact (Definition 1.1.2) one finds that is Lagrangian. In other words, and hence as . Examples include toric hyperkähler manifolds, Nakajima quiver varieties (from quivers without edge-loops) and the moduli space of Higgs bundles. The fact that the nilpotent cone, which agrees with the core of , is Lagrangian was first observed by Laumon [Lau]. Retrospectively, this can also be considered as a consequence of the completely integrability of the Hitchin system [Hi2]. In the hyper-compact case Theorem 1.4.1 appeared as [Ha2, Corollary 4.3].
However, when the quiver contains an edge loop the Nakajima quiver varieties are not hyper-compact. Examples include and more generally the ADHM spaces . Nevertheless, in these cases we know by [Br] and respectively [EL] and [Ba] that the cores are irreducible and in particular equidimensional of dimension and respectively.
We do not know if equidimensionality or even irreducibility of the core of Nakajima quiver varieties for quivers with edge loops holds in general.
In the case of smooth projective toric varieties , the Hard Lefschetz theorem, together with the fact that generates , famously [St1] gives a complete characterization of possible Poincaré polynomials of smooth projective toric varieties, and in turn the face vectors of rational simple complex polytopes.
Thus it is interesting to ask the following:
For semiprojective hyperkähler varieties is there a stronger form of the Weak Hard Lefschetz theorem or the inequalities (1.4.2)? In particular how do the Betti numbers of semiprojective hyperkähler varieties behave after ?
This question was the original motivation to look at the Betti numbers of examples of large semiprojective hyperkähler varieties to find how the Betti numbers behave after the critical dimension .
It turns out that partly due to an arithmetic harmonic analysis technique to evaluate such Betti numbers we have now efficient formulas to compute Poincaré polynomials. This allows us to investigate numerically the shape of Betti numbers of large seimprojective hyperkähler manifolds in several examples. We explain this arithmetic technique and the resulting combinatorial formulas for the Poincaré polynomials in the next section.
2 Arithmetic harmonic analysis on symplectic quotients: the microscopic picture
In the previous section we collected results on the cohomology of a general semiprojective variety . In this section we show that when arises as symplectic quotient of a vector space, we can use “arithmetic harmonic analysis” to count points on over a finite field, and in turn to compute Betti numbers. Counting points of varieties over finite fields is what we call microscopic approach to study Betti numbers of complex algebraic varieties.
2.1 Katz’s theorem
From Katz’s [HV, Appendix] we recall the definition
that a complex algebraic variety is strongly-polynomial
count. This means that there is polynomial
and a spreading out
We then have the following theorem of Katz from [HV, Theorem 6.1, Appendix]:
Theorem 2.1.1 (Katz).
Assume that is strongly-polynomial count with counting polynomial . Then
This result gives the Betti numbers of a strongly polynomial count variety , when additionally it has a pure cohomology. In that case (1.3.4) will compute the Poincaré polynomial from the virtual weight polynomial. This will be the case for many of our semiprojective varieties, where we will be able to use an effective technique to find the count polynomial . This technique from [Ha5, Ha6] we explain in the next section.
2.2 Arithmetic harmonic analysis
We work in the setup of §1.1.1 but change coefficients from to a finite field . For simplicity we denote with the same letters the corresponding objects over . We define the function at as
Let and fix a non-trivial additive character . The number of solutions of the equation over the finite field equals:
Thus in order to count the points of we only need to determine the function as defined in (2.2.1) and compute its Fourier transform as in (2.2.2). In turn we assume that and we use this to count the points of the affine GIT quotients , in cases when acts freely on , when the number of points on is just . In our cases considered below this quantity will turn out to be a polynomial in , yielding by (2.1.1) a formula for the virtual weight polynomials of affine GIT quotient .
Finally we can connect the affine GIT quotient to the GIT quotients with generic linarization as in §1.1.1 by considering a non-singular semiprojective variety with a projection with generic fiber the affine GIT quotient when and the GIT quotient with linearization . Now Corollary 1.3.3 can be applied to show that and have isomorphic pure cohomology, and so our computation by Fourier transform above gives the Poincaré polynomial of our semiprojective varieties, which arise as finite dimensional linear symplectic quotients.
2.3 Betti numbers of semiprojective hyperkähler varieties
Toric hyparkähler varieties
Let be a rational hyperplane arrangement. In this case the toric hyperkähler variety arises as linear symplectic quotient, with generic linearization, induced by a torus action constructed from as in [HS, §6]. The variety is an orbifold and is non-singular when is unimodular. In the unimodular case it was first constructed in [BD] by differential geometric means.
As explained in [Ha5] the above arithmetic harmonic analysis can be used to compute the Betti numbers of the semiprojective ; we get
where the Betti numbers are the -numbers of the hyperplane arrangement ; a combinatorial quantity. In the unimodal case (2.3.1) was first determined in [BD] and in the general case it was proved in [HS].
As explained in [HS, §8] one can construct the so-called cographic arrangement from any graph . Then is just the Nakajima quiver variety of §2.3.3 below. In this case the -polynomial of (2.3.1) can be computed from the Tutte polynomial as follows:
Here the Tutte polynomial of a graph is a two variable polynomial invariant, universal with respect to contraction-deletion of edges. It can be defined explicitly as follows
where denotes the number of connected components of the subgraph with edge set and the same set of vertices as . Note that the exponent equals , the first Betti number of .
We will only consider the external activity polynomial of obtained by specializing to . For connected, we have
where the sum is over all connected subgraphs with vertex set . (This polynomial is related to the reliability polynomial of by a simple change of variables, hence the choice of name.) A remarkable theorem of Tutte guarantees that , and hence also , has non-negative (integer) coefficients.
Twisted ADHM spaces and Hilbert scheme of points on the affine plane
Here , where is an -dimensional vector space
with , and .
The space is empty when (the trace of a commutator is always zero), diffeomorphic with the Hilbert scheme of -points on , when , and is the twisted version of the ADHM space [ADHM] of Yang-Mills instantons of charge on (c.f. [Na3]). As explained in [Ha5, Theorem 2] the arithmetic Fourier transform technique of §2 yields the following generating function for the Poincaré polynomials of (originally due to [NY, Corollary 3.10]):
In particular when this gives for the generating function of Poincaré polynomials of Hilbert schemes of points on
Göttsche’s formula from [Go], which by Euler’s formula reduces to
where is the number of parts in the partition of ; this was the original computation of Ellingsrud-Stromme in [ES].
Nakajima quiver varieties and
Here we recall the definition of the affine version of Nakajima’s quiver varieties [Na2]. Let be a quiver, i.e. an oriented graph on a finite set with a finite set of oriented (perhaps multiple and loop) edges. To each vertex of the graph we associate two finite dimensional vector spaces and . We call the dimension vector, where and . To this data we associate the grand vector space:
the group and its Lie algebra
and the natural representation
The action is from both left and right on the first term, and from the left on the second.
As explained in [Ha5] and [Ha6] the arithmetic harmonic analysis technique of §2 translates to the formula (2.3.9) below. We first introduce some notation on partitions following [Mac]. We let be the set of partitions of . For two partitions and we define . Writing we let be the number of parts in . For any we have .
Let be a quiver, with and , with possibly multiple edges and loops. Fix a dimension vector . The Poincaré polynomials of the corresponding Nakajima quiver varieties are given by the generating function: