Nested Hilbert schemes on surfaces: Virtual fundamental class

Abstract

We construct natural virtual fundamental classes for nested Hilbert schemes on a nonsingular projective surface S. This allows us to define new invariants of S that recover some of the known important cases such as Poincaré invariants of Dürr-Kabanov-Okonek and the stable pair invariants of Kool-Thomas. In certain cases, we can express these invariants in terms of integrals over the products of Hilbert scheme of points on S, and relate them to the vertex operator formulas found by Carlsson-Okounkov. The virtual fundamental classes of the nested Hilbert schemes play a crucial role in the Donaldson-Thomas theory of local-surface-threefolds that we study in [GSY17b].

Hilbert schemes, Donaldson–Thomas theory, Vafa–Witten and Seiberg–Witten theory

“…Their rich geometric structures have proved to have many applications in mathematics and physics (see [N99] for a survey). The current article is a survey of the articles [GSY17a] and [GSY17b], where the authors studied the enumerative geometry of "nested Hilbert schemes" of points and curves on algebraic surfaces.…”

“…In [GSY17a] and [GSY17b], in order to define invariants for the nested Hilbert schemes (see [GSY17a, Definitions 2.13, 2.14]), the authors constructed a virtual fundamental class [S [n] β ] vir and then considered cases where one integrates appropriate cohomology classes against it. More precisely, they constructed a natural perfect obstruction theory over S [n] β .…”

“…This is done by studying the deformation/obstruction theory of the maps of coherent sheaves given by the natural inclusions (1) following Illusie. It turned out that this construction in particular provides a uniform way of studying all known obstruction theories of the Hilbert schemes of points and curves, as well as the stable pair moduli spaces on S. The first main result of [GSY17a] is as follows (see also [GSY17a, Propositions 2.5, and Corollary 2.6]):…”

“…The focus of SW theory in physics is the study of moduli space of vacua in N = 2, D = 4 super Yang-Mills theory, and in particular certain dualities of the theory such as electric-magnetic duality (Montonen-Olive duality). In [GLSY17] Gukov-Liu-Sheshmani-Yau conjectured a relation between SW invariants (associate to SU (2) gauge invariant theory) of a projective surface S and DT invariants of a noncompact threefold X obtained by total space of a line bundle L on S. In a later work [GSY17a,GSY17b] , which will be elaborated below, Gholampour-Sheshmani-Yau proved the conjecture in [GLSY17,Equation 56].…”

“…sheaves are supported on S or a fat neighborhood of S in X). The authors showed in [GSY17a] that the DT invariants of X satisfy an equation in terms of SW invariants of S and certain correction terms governed by invariants of nested Hilbert schemes:…”