Source: https://www.groundai.com/project/cohomology-of-large-semiprojective-hyperkaehler-varieties/
Timestamp: 2019-04-25 00:42:33+00:00

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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.
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 X a smooth quasi-projective variety with a C×-action semiprojective when the fixed point set XC× is projective and for every x∈X and as λ∈C× tends to 0 the limit limλ→0λx 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 n-points on C2, 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 XC× with some cohomological shifts. Also, the opposite Bialinycki-Birula stratification will stratify a projective subvariety C⊂X of the semiprojective variety, the so-called core, which turns out to be a deformation retract of X. This way we can deduce that the cohomology H∗(X;C) is always pure. Furthermore, we can compactify ¯¯¯¯¯X=X∐Z with a divisor Z, to get an orbifold ¯¯¯¯¯X. 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.
We start with the definition of a semiprojective variety, first considered in [Si2, Theorem 11.2] .
The fixed point set XC× is proper.
For every x∈X the limλ→0λx exists as λ∈C× tends to 0.
The second condition could be phrased more algebraically as follows: for every x∈X we have an equivariant map f:C→X such that f(1)=x and C× acts on C by multiplication.
First example is a projective variety with a trivial (or any) C×-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 C×-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] .
to the affine GIT quotient M0=μ−1(0)//G.
The C×-action on M given by dilation will commute with the linear action of G on it so that the moment map (1.1.1) will be equivariant with respect to this and the weight 2 action of C× on g∗. This will induce a C×-action on Mσ, such that on the affine GIT quotient M0 it will have a single fixed point corresponding to the origin in μ−1(0)⊂M. This and the fact that the affinization map (1.1.3) is proper implies, that Mσρ is semiprojective, provided that Mσρ is non-singular, which we always assume.
A semiprojective hyperkähler variety with a symplectic form of homogeneity one as in (1.1.4) is called hyper-compact.
When G is a torus, Mσρ are the toric hyperkähler varieties of [HS] ; these always can be arranged to become hyper-compact. When the representation ρ arises from a quiver with a dimension vector Mσρ is a quiver variety as constructed by Nakajima in [Na2] . When the quiver has no edge loops, one can always arrange that Mσρ 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 1 in the simple vertex and n on the looped one, the Nakajima quiver variety becomes isomorphic with (C2)[n] the Hilbert scheme of n points on C2. This semiprojective hyperkähler variety however is not hyper-compact as we will see later.
turns out to be the famous Hitchin map [Hi2] , which by results of Hitchin [Hi1] when n=2 and Nitsure [Ni] for general n is a proper map. It is also C×-equivariant wich covers a C×-action on the affine A with a single fixed point. This implies that Mdn is indeed semiprojective.
Much in this section is due to Simpson [Si2] , Nakajima [Na2] and Atiyah–Bott [AB] .
Introduce Ui⊂X as the set of points x∈X for which limt→0tx∈Fi. Similarly, as above, we can define Di as the points x∈X for which limt→∞λx∈Fi. These are locally closed subsets and Bialynicki-Birula [Bia, Theorem 4.1] proves that both Ui and Di are subschemes of X which are isomorphic to certain affine bundles (so-called C×-fibrations) over Fi.
To see this we note that using the linearisation on the very ample line bundle L we can equivariantly embed X into some projective space PN with a linear C× action. (1.2.2) follows from the corresponding statement for the linear action of C× on PN, where it is clear.
which decomposes as ¯¯¯¯¯X=X∐Z corresponding to points in (X×C)ss with non-zero (respectively zero) second component. This thus yields an orbifold compactification of X, the algebraic analogue of Lerman’s symplectic cutting [Ler] , as studied in [Ha1] .
The core C of a semiprojective variety X is proper.
We denote by H∗(X;Z) the integer and by H∗(X;Q) the rational singular cohomology of a CW complex X. H∗(X;Z) is a graded anti commutative ring; while H∗(X;Q) is a graded anticommutative Q-algebra.
When X is a complex algebraic variety there is further structure on its rational cohomology. Motivated by the Frobenius action on the l-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 X.
for a projective variety X. We say that the weight filtration on H∗(X;Q) is pure when both (1.3.1) and (1.3.2) holds for every k. 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.
In the general case however there is no such relationships.
is the Thom isomorphism. Finally iλ:Fλ→Dλ, iJ:DJ→UJ and iJ+:DJ+→UJ+ denote the corresponding imbeddings.
The embedding i:C≅DI→X≅UI induces an isomorphism i∗:H∗(X;Z)≅H∗(C;Z).
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 X be a non-singular complex algebraic variety and f:X→C a smooth morphism, i.e. a surjective submersion. In addition, let X be semiprojective with a C× action making f equivariant covering a linear action of C× on C with positive weight. Then the fibers Xc:=f−1(c) have isomorphic cohomology supporting pure mixed Hodge structures.
In fact Simpson’s [Si2] main example for a semiprojective variety was M\rm% \scriptsize Hod, the moduli space of stable rank n degree 1 λ-connections on the curve which comes with f:M% \rm\scriptsize Hod→C satisfying the conditions of Corollary 1.3.3. Here f−1(0)≅M\rm\scriptsize Dol=Mgn is our moduli space of Higgs bundles while fλ=M\rm\scriptsize DR is the moduli space of certain holomorphic connections. The Corollary 1.3.3 then shows that H∗(M% \rm\scriptsize DR;Q)≅H∗(M\rm% \scriptsize Dol;Q) 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 E(Xλ;q) of an affine symplectic quotient, and to deduce that it gives the Poincaré polynomial we will put Xλ in a family f:X→C satisfying the conditions of Corollary 1.3.3.
The following result was discussed in [HS, Theorem 3.5] in the context of semiprojective toric varieties, and the proof was sketched in [Ha7] .
The core C is a deformation retract of the smooth semiprojective variety X.
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 Utubλ and Dtubλ retract to Fλ. Finally, the first and the third because these spaces all retract to Ftubλ∖Fλ.
The Bialinycki-Birula decomposition X=∐i∈IUi of a semiprojective variety is perfect. In particular P(X;t)=∑λ∈IP(Fλ;t)t2kλ.
Theorem 1.4.1 (Weak Hard Lefschetz).
for the Betti numbers of the smooth semiprojective variety. As a consequence both sequences of odd and even Betti numbers grow until k and satisfy bk−i(X)≤bk+i(X).
Possibly the analogous result to (1.4.1) holds when C is not equidimensional and k is the smallest dimension of the irreducible components of C. 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 Dλ is Lagrangian. In other words, dimDλ=dimX2 and hence k=dimX2 as codimUmin=0. 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 Mgn, 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 (C2)[n] and more generally the ADHM spaces Mn,m. Nevertheless, in these cases we know by [Br] and respectively [EL] and [Ba] that the cores are irreducible and in particular equidimensional of dimension n−1 and mn−1 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 Y, the Hard Lefschetz theorem, together with the fact that H2(Y) generates H∗(Y), 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.
The above Weak Hard Lefschetz theorem was used in [HS] and [Ha2] to give new restrictions on the Poincaré polynomials of toric hyperkähler varieties and, in turn, on the face vectors of rational hyperplane arrangements. However a complete classification in this case has not even been conjectured.
For the moduli space of Higgs bundles Mgn Theorem 1.4.1 is a consequence of the Relative Hard Lefschetz theorem [dCHM] using the argument of [HV, 4.2.8] .
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 k=dimC?
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 k=dimC.
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.
In the previous section we collected results on the cohomology of a general semiprojective variety X. In this section we show that when X arises as symplectic quotient of a vector space, we can use “arithmetic harmonic analysis” to count points on X 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.
This result gives the Betti numbers of a strongly polynomial count variety X, 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 PX(t). This technique from [Ha5, Ha6] we explain in the next section.
Thus in order to count the Fq points of μ−1(ξ) we only need to determine the function aϱ as defined in (2.2.1) and compute its Fourier transform as in (2.2.2). In turn we assume that ξ∈(g∗)G and we use this to count the Fq points of the affine GIT quotients X:=μ−1(ξ)//G, in cases when G acts freely on μ−1(ξ), when the number of Fq points on μ−1(ξ)//G is just #μ−1(ξ)/|G|. In our cases considered below this quantity will turn out to be a polynomial in q, yielding by (2.1.1) a formula for the virtual weight polynomials of affine GIT quotient μ−1(ξ)//G.
Finally we can connect the affine GIT quotient to the GIT quotients with generic linarization as in §1.1.1 by considering X:=μ−1(¯¯¯¯¯¯¯¯¯¯C×ξ)//σG a non-singular semiprojective variety with a projection f:X→C≅¯¯¯¯¯¯¯¯¯¯C×ξ⊂g∗ with generic fiber Xλξ=f−1(λξ)=μ−1(λξ)//σG=μ−1(λξ)//G the affine GIT quotient when λ≠0 and X0=μ−1(0)//σG the GIT quotient with linearization σ. Now Corollary 1.3.3 can be applied to show that X0 and Xξ 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.
Let H⊂Qn be a rational hyperplane arrangement. In this case the toric hyperkähler variety MH arises as linear symplectic quotient, with generic linearization, induced by a torus action ρH:T→GL(V) constructed from H as in [HS, §6] . The variety MH is an orbifold and is non-singular when H is unimodular. In the unimodular case it was first constructed in [BD] by differential geometric means.
where the Betti numbers hi(H) are the h-numbers of the hyperplane arrangement H; 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] .
where k(A) denotes the number of connected components of the subgraph QA⊆Q with edge set A and the same set V=V(Q) of vertices as Q. Note that the exponent k(A)+#A−#V equals b1(QA), the first Betti number of QA.
where the sum is over all connected subgraphs Q′⊆Q with vertex set V. (This polynomial is related to the reliability polynomial of Q by a simple change of variables, hence the choice of name.) A remarkable theorem of Tutte guarantees that TQ, and hence also RQ, has non-negative (integer) coefficients.
with A,B∈gl(V), I∈Hom(W,V) and J∈Hom(V,W).
where l(λ) is the number of parts in the partition λ of n; this was the original computation of Ellingsrud-Stromme in [ES] .
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 P(s) be the set of partitions of s∈Z≥0. For two partitions λ=(λ1,…,λl)∈P(s) and μ=(μ1,…,μm)∈P(s) we define n(λ,μ)=∑i,jmin(λi,μj). Writing λ=(1m1(λ),2m2(λ),…)∈P(s) we let l(λ)=∑mi(λ)=l be the number of parts in λ. For any λ∈P(s) we have n(λ,(1s))=sl(λ).

References: §11
 §5
 §1
 §1
 §1
 §2
 §1
 §6
 V. 
 §2