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Donaldson-Thomas invariants of abelian threefolds and Bridgeland stability conditions

Abstract: We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical...

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Main Authors: Oberdieck, Georg (Author) , Piyaratne, Dulip (Author) , Toda, Yukinobu (Author)
Format: Article (Journal)
Language:English
Published: 2022
In: Journal of algebraic geometry
Year: 2022, Volume: 31, Issue: 1, Pages: 13-73
ISSN:1534-7486
DOI:10.1090/jag/788
Online Access:Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1090/jag/788
Verlag, lizenzpflichtig, Volltext: https://www.ams.org/jag/2022-31-01/S1056-3911-2021-00788-9/
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Author Notes:Georg Oberdieck, Dulip Piyaratne, and Yukinobu Toda

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520 |a Abstract: We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yields evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author. For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson–Thomas invariants for arbitrary Calabi–Yau threefolds with abelian actions is given. 
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