Euler Characteristics of Hilbert Schemes of Points On Surfaces with Simple Singularities

Abstract

Abstract. This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C 2 /G with G < SL(2, C) a finite subgroup, we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type A singularities. We announce a proof of our conjecture for singularities of type D. The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding (extended) affine Lie algebra; we discuss possible representation-theoretic consequences of this fact. Our results, respectively conjectures, imply the modularity of the generating function for surfaces with type A and type D, respectively arbitrary, simple singularities, confirming predictions of S-duality. Euler characteristics of Hilbert schemes of pointsLet X be a quasiprojective variety X over the field C of complex numbers. Let Hilb m (X) denote the Hilbert scheme of m points on X, the quasiprojective scheme parametrizing 0-dimensional subschemes of X of length m. Consider the generating series of topological Euler characteristicsFor a smooth variety X, the series Z X (q), as well as various refinements, have been extensively studied. For a nonsingular curve X = C, we have MacDonald's result [8]For a nonsingular surface X = S, we have (a specialization of) Göttsche's formula [5] (1).There are also results for higher-dimensional varieties [1]. For singular varieties X, very little is known about the series Z X (q). For a singular curve X = C with a finite set {P 1 , . . . , P k } of planar singularities however, we have the beautiful conjecture of Oblomkov and Shende [14], proved by Maulik [9], which specializes to the following:Here each Z (Pi,C) (q) is a highly nontrivial local term that can be expressed in terms of the HOMFLY polynomial of the embedded link of the singularity P i ∈ C. Simple surface singularitiesIn this announcement, we consider the generating series Z S (q) for X = S a singular surface with simple (Kleinian, rational double point) singularities. First, we discuss the local situation. As is well known, locally analytically S is a quotient singularity S = C 2 /G ∆ . Here G ∆ < SL(2, C) is a finite subgroup corresponding to an irreducible simply-laced Dynkin diagram ∆, the dual graph of the exceptional components in the minimal resolution of the singularity. There are three possible types: ∆ can be of type A n for n ≥ 1, type D n for n ≥ 4 and type E n for n = 6, 7, 8. The following is our main conjecture.

Euler characteristics of Hilbert schemes of points on simple surface singularities

“…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].…”

Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras

“…In this article we prove a conjecture by Gyenge, Némethi, and Szendrői in [13] and [14] that gives a formula of the generating function of Euler numbers of Hilbert schemes of points Hilb N (C 2 /Γ) on a simple singularity C 2 /Γ, where Γ is a finite subgroup of SL (2). When Γ is of type A, Euler numbers were computed by Dijkgraaf and Su lkowski [9], and Toda [38].…”

“…When Γ is of type A, Euler numbers were computed by Dijkgraaf and Su lkowski [9], and Toda [38]. The formula in [13] and [14] is given in a different form and makes sense for arbitrary Γ. The formula was proved for type D, as well as type A, in [14].…”

“…The formula in [13] and [14] is written as a specialization of the character for the "Fock space," which is the tensor product of the basic representation of the affine Lie algebra g aff corresponding to Γ, and the usual Fock space representation of the Heisenberg algebra. The Fock space was realized in a geometric way as the direct sum of homology groups of various connected components of Hilb N (C 2 ) Γ for all N (see [31]).…”

Atiyah class and sheaf counting on local Calabi Yau fourfolds

“…This is an important step in the proof since it provides a direct connection with the domain of the α T in diagram (23). The starting point is the exact sequence of O Z U ×Z U -modules…”