Communications in Analysis and Geometry

Volume 21 (2013)

Number 3

Donaldson–Thomas invariants of certain Calabi–Yau 3-folds

Pages: 541 – 578

DOI: http://dx.doi.org/10.4310/CAG.2013.v21.n3.a4

Authors

Wei-Ping Li (Department of Mathematics, HKUST, Clear Water Bay, Kowloon, Hong Kong)

Zhenbo Qin (Department of Mathematics, University of Missouri, Columbia, Mo., U.S.A.)

Abstract

We compute the Donaldson–Thomas invariants for two types of Calabi–Yau 3-folds. These invariants are associated to the moduli spaces of rank-2 Gieseker semistable sheaves. None of the sheaves are locally free, and their double duals are locally free stable sheaves investigated earlier in [12, 25, 33]. We show that these Gieseker moduli spaces are isomorphic to some Quot-schemes. We prove a formula for Behrend’s $ν$-functions when torus actions present with positive dimensional fixed point sets, and use it to obtain the generating series of the relevant Donaldson–Thomas invariants in terms of the McMahon function.

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