Euler characteristics of Hilbert schemes of points on simple surface singularities

Abstract

We study the geometry and topology of Hilbert schemes of points on the orbifold surface , respectively the singular quotient surface , where is a finite subgroup of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D.

Appendices

A Background on representation theory

This section plays no logical role in our paper, but it provides important background. For further discussion about the role of representation theory, see the announcement [21].

1.1 A.1 Affine Lie algebras and extended basic representations

Let $\Delta$ be an irreducible finite-dimensional root system, corresponding to a complex finite dimensional simple Lie algebra $\mathfrak {g}$ of rank n. Attached to $\Delta$ is also an (untwisted) affine Lie algebra ${\widetilde{\mathfrak {g}}}$, but a slight variant will be more interesting for us, see e.g. [11, Section 6]. Denote by the Lie algebra that is the direct sum of the affine Lie algebra ${\widetilde{\mathfrak {g}}}$ and an infinite Heisenberg algebra $\mathfrak {heis}$, with their centers identified.

Let $V_0$ be the basic representation of ${\widetilde{\mathfrak {g}}}$, the level-1 representation with highest weight $\omega _0$. Let be the standard Fock space representation of $\mathfrak {heis}$, having central charge 1. Then is a representation of that we may call the extended basic representation. By the Frenkel–Kac theorem [12], in fact

where $Q_\mathrm{\Delta }$ is the root lattice corresponding to the root system $\Delta$. Here, for , is the sum of weight subspaces of weight , $m \geqslant 0$. Thus, we can write the character of this representation as

(16)

where , and $\beta =(\beta _1, \ldots , \beta _n)$ is terms of the simple roots.

Example A.1

For $\Delta$ of type $A_n$, we have , , . In this case there is in fact a natural vector space isomorphism with Fock space itself, see e.g. [41, Section 3E].

1.2 A.2 Affine crystals

The basic representations $V_0, V$ of ${\widetilde{\mathfrak {g}}}$, respectively can be constructed on vector spaces spanned by explicit “crystal” bases. Crystal bases have many combinatorial models; in types A or D, the sets denoted with in the main part of our paper provide one possible combinatorial model [25] for the crystal basis for the basic representation. More precisely, given $\Delta$ of type A or D, there is a combinatorial condition which singles out a subset . The basic representation $V_0$ of $\widetilde{\mathfrak {g}}$ has a basis [24, 34] in bijection with elements of . The extended basic representation V of has a basis in bijection with elements of . The canonical embedding $V_0\subset V$, defined by the vacuum vector inside Fock space , is induced by the inclusion .

1.3 A.3 Affine Lie algebras and Hilbert schemes

As before, let be a finite subgroup and let $\mathrm{\Delta }\subset {\widetilde{\Delta }}$ be the corresponding finite and affine Dynkin diagrams. It is a well-known fact that the equivariant Hilbert schemes for all finite dimensional representations $\rho$ of G are Nakajima quiver varieties [37] associated to ${\widetilde{\mathrm{\Delta }}}$, with dimension vector determined by $\rho$, and a specific stability condition (see [13, 35] for more details for type A).

Nakajima’s general results on the relation between the cohomology of quiver varities and Kac–Moody algebras, specialized to this case, imply [37] that the direct sum of all cohomology groups is graded isomorphic to the extended basic representation V of the corresponding extended affine Lie algebra defined in A.1 above. By [36, Section 7], these quiver varieties have no odd cohomology. Thus the character formula (16) implies Theorem 1.3 in all types A, D, and E.

B Joins

Recall [1] that the join of two projective varieties is the locus of points on all lines joining a point of X to a point of Y in the ambient projective space. One well-known example of this construction is the following. Let and be two disjoint projective linear subspaces of .

Lemma B.1

The join equals . Moreover, the locus is covered by lines uniquely: for every , there exists a unique line with $p_i\in L_i$, containing p.

Let now be a hyperplane not containing the $L_i$, which we think of as the hyperplane “at infinity”. Let . Let $\overline{L}_i=L_i\cap H$, and let be the affine linear subspaces in V corresponding to $L_i$. Finally let and .

Lemma B.2

Projection away from $L_1$ defines a morphism $\phi :X\rightarrow L_2$, which is an affine fibration with fibres isomorphic to . $\phi$ restricts to a morphism , which is a trivial affine fibration over .

In geometric terms, the map $\phi$ is defined on as follows: take , find the unique line ${p_1p_2}$ passing through it, with $p_1\in \overline{L}_1$ and $p_2\in L_2$; then $\phi (p)=p_2$.

Let now U be a projective subspace of H which avoids $\overline{L}_2$. Let be a codimension one linear subspace, and its affine complement. In the main text, we need the following statement.

Lemma B.3

.

Proof

With the same argument as in Lemma B.2, $J(L_2^o,W) \cap V$ is a fibration over $L_2^o$ with fiber , and is a fibration over $L_2^o$ with fiber . Since , the projection from to has fibers . $\square$

We finally recall the base-change property of joins.

Lemma B.4

([1, B1.2]) Let S be an arbitrary scheme. Then for schemes and an S-scheme T, we have the following equality in :