# Paper accepted in American Journal of Mathematics

Associate Professor Artan Sheshmani together with his collaborators publish in the American Journal of Mathematics

2019.09.11 | Jane Jamshidi

Front page of American Journal of Mathematics

The title of the paper is: "Localized Donaldson-Thomas Theory of surfaces" and the authors are Amin Gholampour, Artan Sheshmani and Shing-Tung Yau.

The abstract of the paper reads:  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.

The paper is currenty available as preprint on arXiv.org: https://arxiv.org/pdf/1701.08902.pdf

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