Title: Localized Donaldson-Thomas theory of surfaces

(To be published in American Journal of Mathematics)

Abstract: Let S be a projective simply connected complex surface and L be a line bundle on S. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of L. The moduli space admits a C ∗ – action induced by scaling the fibers of L. We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on S. We define the localized Donaldson-Thomas invariants of L by virtual localization in the case that L twisted by the anticanonical bundle of S admits a nonzero global section. When pg(S) > 0, in combination with Mochizuki’s formulas, we are able to express the localized DT invariants in terms of the invariants of the nested Hilbert schemes defined by the authors in [GSY17a], the Seiberg-Witten invariants of S, and the integrals over the products of Hilbert schemes of points on S. When L is the canonical bundle of S, the Vafa-Witten invariants defined recently by Tanaka-Thomas, can be extracted from these localized DT invariants. VW invariants are expected to have modular properties as predicted by S-duality.

Title: Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten Theories

(To be published in Notices of the International Chinese Congress of Mathematicians)

Abstract: This article provides the summary of [GSY17a] and [GSY17b] where the authors studied the enumerative geometry of “nested Hilbert schemes” of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and SeibergWitten theories.