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Donaldson–Thomas invariants for complexes on abelian threefolds

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Abstract

Donaldson–Thomas invariants for moduli spaces M of perfect complexes on an abelian threefold X are usually zero. A better object is the quotient \({K=[M/X\times\widehat{X}]}\) of complexes modulo twist and translation. Roughly speaking, this amounts to fixing not only the determinant of the complexes in M, but also that of their Fourier–Mukai transform. We modify the standard perfect symmetric obstruction theory for perfect complexes to obtain a virtual fundamental class, giving rise to a DT-type invariant of the quotient K. It is insensitive to deformations of X, and respects derived equivalence. As illustrations we examine the case of Picard bundles and of Hilbert schemes of points.

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  1. Stord/Haugesund University College, Haugesund, Norway

    Martin G. Gulbrandsen

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  1. Martin G. Gulbrandsen
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Correspondence to Martin G. Gulbrandsen.

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Gulbrandsen, M.G. Donaldson–Thomas invariants for complexes on abelian threefolds. Math. Z. 273, 219–236 (2013). https://doi.org/10.1007/s00209-012-1002-3

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  • DOI: https://doi.org/10.1007/s00209-012-1002-3

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