The Topology of Equivariant Hilbert Schemes

The Topology of Equivariant Hilbert Schemes

Dori Bejleri &  Gjergji Zaimi
July 14, 2019
Abstract.

For a finite group acting linearly on , the equivariant Hilbert scheme is a natural resolution of singularities of . In this paper we study the topology of for abelian and how it depends on the group . We prove that the topological invariants of are periodic or quasipolynomial in the order of the group as varies over certain families of abelian subgroups of . This is done by using the Bialynicki-Birula decomposition to compute topological invariants in terms of the combinatorics of a certain set of partitions.

1. Introduction

Let be a smooth algebraic surface carrying the action of a finite group . The equivariant Hilbert scheme (Section 1.1) is a generalization of the Hilbert scheme of points on that parametrizes certain -equivariant subschemes. It is a natural resolution of singularities for the symmetric product of the quotient space. In this paper we study how the topology of these Hilbert schemes change as the group varies.

When is an abelian group acting linearly on , we exhibit (Main Theorems A and B) periodicity and quasipolynomiality for the Betti numbers and Euler characteristics of as the order of the group varies within certain familes of finite abelian subgroups of . The main tool is the combinatorics of balanced partitions (Section 1.3) and the proof is mostly combinatorial. To our knowledge, there is a priori no geometric relationship between the equivariant Hilbert schemes for the different groups we consider and it is an interesting question to understand why one might expect these results.

1.1. Statement of main results

Let be a finite subgroup of . The stack quotient of by the action of is a smooth two dimensional orbifold with singular coarse moduli space . The Hilbert scheme of points is a -dimensional quasiprojective scheme parametrizing flat families of substacks of with constant Hilbert polynomial [quot, Theorem 1.5]. Equivalently, is the moduli space of -equivariant ideals such that as representations [lili, Proposition 2.9]. It is a union of irreducible components of the fixed locus [brion, Proposition 4.1]. In fact is smooth (Section 2.3).

There is a Hilbert-Chow morphism

sending an ideal to its support in the coarse moduli space. The restriction of this morphism to the component of containing the locus of distinct free -orbits is a resolution of singularities 111When is abelian, is connected (Corollary 2.3) and so is itself a resolution. When , is a Nakajima quiver variety [weiqiang, Theorem 2] and so is connected [quiverkacmoody, Theorem 6.2]. The case for general is unknown to the authors.. When , is the minimal resolution [kidoh, Theorem 5.1] [ishii, Theorem 3.1].

From now on, we restrict to abelian. In Section 4.1 we will reduce our analysis to when the group is cyclic. To this end, we consider cyclic of order acting on by where is a primitive root of unity and . We will denote this group by and the equivariant Hilbert scheme by . The first result of this article concerns the behavior of the compactly supported Betti numbers .

Main Theorem A.

Fix integers and with , or equivalently having the same sign. Then for all .

That is, the Betti numbers of are eventually periodic in with period . The proof of Main Theorem A uses the Bialynicki-Birula decomposition to stratify by locally closed affine cells. Thus the statement of Main Theorem A lifts to the Grothendieck ring of varieties .

Theorem 1.1.

Fix integers with . Then the class in is a polynomial in whose coefficients are periodic in with period for . In particular, any motivic invariants of are eventually periodic in .

When and , this explains an observation of Gusein-Zade, Luengo, and Melle-Hernandez [GZLMH, pg. 601].

Our second main result examines the behavior of the topological invariants when . Recall that a function is called quasipolynomial of period if there are a polynomials such that where .

Main Theorem B.

Fix integers and such that , i.e. with opposite sign. Then the topological Euler characteristic is a quasipolynomial in with period for all .

Remark 1.1.

With a finer combinatorial analysis we can prove a strengthening of Main Theorem B to show that quasipolynomiality holds for Betti numbers and classes in the Grothendieck ring. Furthermore, one can show that quasipolynomiality holds for and that the quasipolynomial is of degree . This will appear in forthcoming work.

1.2. Background and motivation

Equivariant Hilbert schemes were first introduced by Ito and Nakamura [in1] for finite subgroups . They play a central role in the Mckay correspondence (see for example [reid, bkr, bezrufinkel]). Indeed much of the geometry of is determined in this case by the representation theory of . On the other hand, very little is known about equivariant Hilbert schemes for general finite subgroups (apart from the case , see for example [kidoh, ishii]).

This is the first paper in a project to understand the geometry of equivariant Hilbert schemes for abelian subgroups of using the combinatorics of balanced partitions (Section 1.3). The main theorems of this paper show new phenomena that appear only when we let the group vary outside of . These results are similar in spirit to the work of Göttsche [gottsche], Nakajima [nakajima1] and others which show that one should study Hilbert schemes all at once, though in our case for all groups rather than for all .

Balanced partitions carry much more geometric information than just topological invariants. For example they determine an open affine cover of whose coordinate rings can be written purely combinatorially from the partitions (Section 2). The hope is that the combinatorial bijections used in the proofs of Theorems 1.2 and 1.3 have an interpretation on the level of the equivariant Hilbert schemes themselves that will lead to a geometric explanation for the periodicity and quasipolynomiality phenomena.

1.2.1. Toric resolutions and continued fractions

The particular case of , and is instructive. Then is the affine toric variety corresponding to the cone generated by and and is the toric minimal resolution. It then follows from a result of Hirzebruch [cls, Theorem 10.2.3] that the Poincare polynomial of is of the form where is the length of Hirzebruch-Jung continued fraction expansion

and . This is evidently periodic in with period .

Figure 1. The supplementary cones which correspond to the affine toric varieties and .

A similar computation when , and yields the singular toric variety with supplementary cone. Then the Poincare polynomial takes the same form where is the length of the continued fraction expansion of . Quasipolynomiality can then be deduced from a geometric duality between the continued fractions of supplementary cones [contfrac, Proposition 2.7]. In fact, the length of the continued fraction expansion of is a linear quasipolynomial in .

For , we provide an analogue of the continued fraction expansion given by the set of balanced partitions defined below. We will see that the balanced partitions control the topology of the Hilbert scheme resolution of the same way the continued fraction controls the topology of the minimal resolution of . Furthermore, Theorems 1.2 and 1.3 below, from which we deduce the main theorems, can be seen as a higher dimensional analogue of the geometric duality for continued fractions.

1.2.2. Future work and speculations

Ultimately, the goal is to understand the total cohomology

and compute its graded character which is the generating function of the Betti numbers . When so that , is diffeomorphic to [nagao, Lemma 4.1.3] and the Göttsche formula [gottsche, Theorem 0.1] computes this generating function as an infinite product. After specializing to the Euler characteristic, we can deduce the formula

from the cores-and-quotients bijection (see Proposition LABEL:prop:trivialcore).

The work of Nakajima [nakajima1, quiverkacmoody] explains these infinite product formulas using representation theory of infinite dimensional Lie algebras. In particular, is a highest weight irreducible representation of a certain Heisenberg Lie algebra and this action intertwines two natural bases of coming from cores-and-quotients [nagao]. We expect a similar picture to be true for the more general equivariant Hilbert schemes .

Question 1.1.

Does carry a natural action of an infinite dimensional Lie algebra that can be described combinatorially in terms of balanced partitions?

Computer computations with balanced partitions suggest the answer to Question 1.1 is yes and furthermore that is generated in degrees for . This particular bound is interesting because it is the bound appearing in Main Theorem A. This suggests that if Question 1.1 has an affirmative answer, then there is some relationship between the Lie algebras and and their representations on the corresponding cohomologies at least when .

Moreover, these computations suggests that the Betti number generating function for is in general not an infinite product when is not in , but rather is a quasimodular form that can be written as a finite sum of infinite products. This is part of a general picture that generating functions for sheaf counting invariants on surfaces have modular properties (see [modularforms] for a survey on this phenomena). Indeed the Euler characteristics and Poincaré polynomials of are naive Donaldson-Thomas type invariants 222See for example [bridgeland] and [BBS] for Hilbert scheme invariants from the point of view of Donaldson-Thomas theory. and the modularity property, if true, would be an analogue of S-duality [vafawitten] for the the quotient orbifolds . It would then be an interesting question to consider how the structure of these generating functions interacts with the stabilization properties from Main Theorems A and B.

1.3. Balanced partitions

Main Theorems A and B are proved by expressing the invariants above in terms of counting certain colored partitions or Young diagrams. We call these balanced partitions.

A partition of a natural number is a sequence of nonnegative integers such that . We identify with its Young diagram, which is a subset of boxes arranged as left justified rows so that the row contains boxes. We view this as living inside the lattice and use notation as in the diagram below. We denote by (resp ) the number of blocks in the row (resp column) of .

Figure 2. The Young diagram corresponding to the partition of . It is balanced and the colors are the residue classes .

Anticipating that the boxes correspond to monomials having -weight , we color the partitions by the monoid homomorphism that assigns to each . In particular this assigns a color viewed as an element of to each box in . We say is an -balanced partition if there exists an such that contains exactly boxes colored by for each residue class modulo . In particular, any such must be a partition of .

Denote the set of all -balanced partitions of by . There is a function we call the Betti statistic (Definition 3.1). We will show the following proposition using the Bialynicki-Birula decomposition.

Proposition 1.1.

The Betti numbers of are given by

In particular, the Poincaré polynomial of satisfies

The main theorems will then follow from the following combinatorial results.

Theorem 1.2.

Fix integers and with . There is a natural bijection that preserves the Betti statistic for .

Theorem 1.3.

Fix integers and with . The cardinality is a quasipolynomial in of period for .

1.4. Acknowledgments

The authors would like to thank T. Graber for suggesting this project and helping with the early stages. We are grateful to W. Hann-Caruthers for helping with the computational aspects that led us to conjecture the main theorems, and to L. Li for providing us with a draft of the unfinished manuscript [lili] from which we learned many of the ideas in Sections 2 and 3. D.B. would like to thank his advisor D. Abramovich for his constant help and encouragement without which this paper would have never materialized. Finally, we would like to thank J. Ali, K. Ascher, S. Asgarli, D. Ranganathan and A. Takeda for many helpful comments on this draft. D.B. was partially supported by a Caltech Summer Undergaduate Research Fellowship and NSF grant DMS-1162367.

2. The geometry of

In this section we give a systematic description of the geometry of . We discuss the natural torus action on as well as smoothness and irreducibility.

2.1. Torus actions

The algebraic torus acts naturally on or equivalently on by . This induces an action on by pulling back ideals,

The fixed points of this action are the doubly homogeneous ideals, that is, the monomial ideals. These are in one-to-one correspondence with partitions of by the assignment

Define . It is clear that forms a basis for so that .

Every monomial ideal is fixed by . However, if and only if is isomorphic as a representation to . The space decomposes as a direct sum of irreducible representations for each with weight . Since decomposes as a direct sum of one copy of each irreducible representation, must have copies of each. Thus each weight must appear times in the decomposition of so we have proved the following:

Lemma 2.1.

The -fixed points in are in one to one correspondence with , the set of -balanced partitions of .

2.2. Local theory of Hilbert schemes

In this section we recall facts about the local geometry of following Haiman’s description given in [haiman].

One can define a torus invariant open affine neighborhood of given by

The coordinate functions on are given by for and where

(1)

Multiplying (1) by we obtain

Therefore the coefficients satisfy the relations

(2)

Similarly, we obtain the relation

(3)

by multiplying by .

We will often denote the function as an arrow on the on the grid pointing from box to box . These functions are torus eigenfunctions with action given by

Consequently, acts by

The actions commute so that , and thus , inherits a action.

For each box , define two distinguished coordinate functions

(4)

where is the size of the row and the size of the column of . We can picture and as southwest and northeast pointing arrows hugging the diagram. Note that each diagram has such distinguished arrows associated to it, two for each box.

Figure 3. The distinguished arrows and for the box in dark gray.

Now we can use these arrows to understand cotangent space to which we will denote . The set of vanishing at are precisely the ones for . These form generators for . The relation (2) expresses as since and for . Thus

(5)

as local parameters in . Similarly, (3) implies that

(6)

in .

If we denote as an arrow, then (5) and (6) imply that if we slide an arrow horizontally or vertically while keeping and then the arrow represents the same local parameter in . Furthermore, if an arrow can be moved so that the head leaves the grid, then it is identically zero in because only positive degree monomials appear in . In this way every northwest pointing arrow vanishes in and any southwest or northeast pointing arrow can be moved until it either vanishes or is of the form or respectively. This proves the following:

Proposition 2.1.

([haiman, Proposition 2.4], [fogarty, Theorem 2.4]) The set over forms a system of local parameters generating the cotangent space of . In particular, is smooth.

2.3. The cotangent space to

We give a description of the weight space decomposition of the cotangent space to any monomial ideal . This will be used later to compute the Bialynicki-Birula cells.

By Proposition 2, is smooth. It follows that the -fixed locus is also smooth [fogarty2, Proposition 4]. In particular, the component is smooth. Moreover, since acts by scaling on , then restricts to be nonzero on the fixed locus if and only if acts trivially on . Thus the functions for generate the coordinate ring of . These correspond to the arrows that start and end on a box with the same color. We call these arrows invariant.

Figure 4. Invariant arrows on corresponding to the box .
Proposition 2.2.

Let . The cotangent space to has basis given by the set of and that are invariant.

Proof.

By the discussion above, these are the only local parameters of that restrict to be nonzero in a neighborhood of in . On the other hand, acts trivially on the invariant arrows so they remain linearly independent in the cotangent space of the fixed locus. ∎

Corollary 2.1.

Let be an -balanced partition of . Then exactly of the arrows of the form or are invariant.

Proof.

The number of such arrows is the dimension of the cotangent space to which is since is smooth and dimensional. ∎

Let denote the cotangent space to the torus fixed point . Denote by for the irreducible representation of on which acts by .

Corollary 2.2.

The weight space decomposition of as a representation of is given by

Proof.

acts on by . Then we get the result by Proposition 2.2 as well as the definition (4) of and . ∎

Remark 2.1.

In the literature, the weight space decomposition of the tangent space is often described in terms of the arm and leg of a box . This description is equivalent because

2.4. Connectedness of

We explain why is connected. The idea is that for any ideal , picking a monomial order and taking initial degeneration to a monomial ideal gives a rational curve in so that every ideal lies in the same connected component as a monomial ideal. Then one must show that all the monomial ideals are connected by chains of rational curves. This is done more generally in [maclagan-smith] for multigraded Hilbert schemes. In this section we will deduce connectedness from the results of [maclagan-smith].

Let be the polynomial ring graded by some abelian group . For any function , the multigraded Hilbert scheme is the subvariety of parametrizing homogeneous ideals such that

That is, is the moduli space of homogeneous ideals with Hilbert function .

The equivariant Hilbert scheme is a special case as follows. Let be a finite abelian group and let be the dual group of characters of . Then the action of on induces an -grading on by

It is easy to see that an ideal is homogeneous if and only if it is -invariant. Furthermore, each is the character of some irreducible representation of so the condition as representations of is equivalent to for each . Therefore

where is -graded by the action of and for all .

Connectedness now follows from the following theorem of Maclagan and Smith:

Theorem 2.1.

([maclagan-smith, Theorem 3.15]) is rationally chain connected for any function satisfying

Corollary 2.3.

is irreducible and the Hilbert-Chow morphism

is a resolution of singularities.

3. The Bialynicki-Birula stratification

In this section we will show how to reduce the problem of computing Betti numbers of to counting -balanced partitions of with the Betti statistic (see Definition 3.1). The idea is to use the action of an algebraic torus on and the theory of Bialynicki-Birula [bialynicki] to stratify into affine cells. Then a local analysis of the torus action at fixed points yields the appropriate statistic giving the Betti numbers.

These techniques are standard in the theory of Hilbert schemes of points (see for example [es1, es2, lili, buryak-feigin]).

3.1. The Bialynicki-Birula Decomposition Theorem

Let be an algebraic torus and a smooth quasiprojective variety on which acts. Suppose the fixed point locus is finite. Then for a generic one-dimensional subtorus , we have . We further assume that

exists for all .

In this case, define

Then the are locally closed and .

The action of on induces an action of on the tangent space . Define to be the subspace of vectors on which acts with positive weight and let be its dimension.

Theorem 3.1.

(Bialynicki-Birula Decomposition Theorem [bialynicki, Theorem 4.4]333Bialynicki-Birula originally proved this theorem for projective. The version we use here for quasiprojective is obtained by taking a torus equivariant compactification. See for example [BBS, Lemma B.2]) Let and as above. Then each locally closed stratum is isomorphic to an affine space so that

Furthermore, the compactly supported Betti number is given by

3.2. The stratification of

We will apply the above results to the action of on . As we saw (Lemma 2.1) the fixed points are indexed by balanced partitions . We pick for generic so that consists of only the monomial ideals.

Lemma 3.1.

For all , the limit

exists in .

Proof.

Consider the monomial partial order given by weight . That is, if and only if . Let be any polynomial with leading term under this monomial partial order. Then for ,

Multiplying through by gives . So in the limit as , we get . The dimension is fixed so the degree of the polynomials in a Gröbner basis of is bounded [grobner, Theorem 8.2]. Since we are taking all polynomials of bounded degree have a unique leading term under this monomial partial order so the limit ideal is the initial monomial ideal generated by these leading terms. Taking initial ideal is a flat limit so is a monomial ideal corresponding to some balanced partition.

Applying Theorem 3.1 gives a decomposition of indexed by balanced partitions :

where is the dimension of the positive weight subspace of the tangent space at .

Definition 3.1.

Define the Betti statistic function as follows:

That is, is the number of invariant arrows on that are pointing either strictly north or weakly southwest.

Remark 3.1.

Note from the definition (4) of , it is vertical if and only if .

Figure 5. This diagram has Betti statistic three.
Proposition 3.1.

For any , we have .

Proof.

Corollary 2.2 gives us the weight space decomposition of the cotangent space . The tangent space is the dual space and so has weight space decomposition

Considering the subtorus , we see the weight spaces for this subtorus are generated by the invariant with weight and invariant with weight . The arrows point southwest and so satisfy . Since , this means the weight .

On the other hand, a vector points northeast. If it points strictly northeast, then but and so the weight . If it points strictly north, then and so that . Therefore the positive weight vectors are exactly counted by the Betti statistic.

This proves Proposition 1.1 which we repeat here for convenience:

Proposition 1.1.

The Betti numbers of are given by

In particular, the Poincare polynomial of satisfies

and the topological Euler characteristic is given by .

This reduces Main Theorems A and B to the combinatorial statements in Theorems 1.2 and 1.3. The proofs of these will be given in Section 4.

3.3. Grothendieck ring of varieties

In this section we will discuss the Grothendieck ring of varieties. Due to the Bialynicki-Birula decomposition, any statements about Betti numbers (for example Main Theorem A) lift to the Grothendieck ring of varieties.

Recall the Grothendieck ring of varieties is the ring generated by isomorphism classes of varieties under the cut-and-paste relations:

The ring structure is given by with unit . We denote by . Then .

If where are a finite collection of locally closed subvarieties, then

Thus the Bialynicki-Birula decomposition induces a decomposition of the class in . We get the following:

Proposition 3.2.

The class of in is given by

The ring is universal with respect to ring valued invariants of varieties satisfying cut-and-paste and splitting as a product for . These include compactly supported Euler characteristic, virtual Poincare polynomials, and virtual mixed Hodge polynomials. These are often called motivic invariants.

Proposition 3.2 allows us to compute all motivic invariants of in terms of the Betti statistic on the set of balanced partitions. Then we apply Theorems 1.2 and 1.3 proven below to obtain Theorem 1.1.

4. Proofs of the theorems

In Section 3, we showed how the main theorems follow from Theorems 1.2 and 1.3. In this section we will give combinatorial proofs of these results after making an initial reduction.

4.1. The Chevalley-Shephard-Todd Theorem

Here we reduce to the case where both and are coprime to using the Chevalley-Shephard-Todd theorem.

Let be a finite group acting linearly and faithfully on . We say that an element is a pseudoreflection if it fixes a hyperplane in . We recall the following classical theorem:

Theorem 4.1.

(Chevalley-Shephard-Todd [CST, §5 Thm 4]) The following are equivalent:

  1. is generated by pseudoreflections,

  2. is smooth,

  3. ,

  4. the natural map is flat.

Let denote the irreducible component containing the locus of distinct free -orbits in .

Corollary 4.1.

The restriction of the Hilbert-Chow morphism to is an isomorphism if and only if any of the equivalent conditions of the Chevalley-Shephard-Todd theorem hold.

Proof.

Suppose the conditions of the theorem hold so that is flat. Then this is a flat family of -orbits in and so induces a map

which is a section to . This is an isomorphism on a dense open subset of with inverse given by and so is an isomorphism everywhere.

For the converse suppose is an isomorphism. We have a commutative diagram

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