TY - JOUR

T1 - Localized donaldson-thomas theory of surfaces

AU - Gholampour, Amin

AU - Sheshmani, Artan

AU - Yau, Shing Tung

N1 - Funding Information:
Manuscript received February 25, 2017; revised August 30, 2018. Research of the first author supported in part by NSF grant DMS-1406788; research of the second author supported in part by NSF grants DMS-1607871, DMS-1306313, and Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No 14.641.31.0001; research of the third author supported in part by NSF grants DMS-0804454 and PHY-1306313, and Simons 38558. American Journal of Mathematics 142 (2020), 405–442. ©c 2020 by Johns Hopkins University Press.
Publisher Copyright:
© 2020 by Johns Hopkins University Press.

PY - 2020/4

Y1 - 2020/4

N2 - 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 (reduced) Donaldson-Thomas invariants of L by virtual localization in the case that L twisted by the anti-canonical bundle of S admits a nonzero global section. When pg (S) > 0, in combination with Mochizuki’s formulas, we are able to express these invariants in terms of the invariants from the nested Hilbert schemes defined by the authors, 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.

AB - 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 (reduced) Donaldson-Thomas invariants of L by virtual localization in the case that L twisted by the anti-canonical bundle of S admits a nonzero global section. When pg (S) > 0, in combination with Mochizuki’s formulas, we are able to express these invariants in terms of the invariants from the nested Hilbert schemes defined by the authors, 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.

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U2 - 10.1353/ajm.2020.0011

DO - 10.1353/ajm.2020.0011

M3 - Article

AN - SCOPUS:85082960090

VL - 142

SP - 405

EP - 442

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 2

ER -