Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Donaldson–Thomas invariants of abelian threefolds and Bridgeland stability conditions


Authors: Georg Oberdieck, Dulip Piyaratne and Yukinobu Toda
Journal: J. Algebraic Geom.
DOI: https://doi.org/10.1090/jag/788
Published electronically: September 14, 2021
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Abstract | References | Additional Information

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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Additional Information

Georg Oberdieck
Affiliation: Mathematisches Institut, Universität Bonn, Germany
MR Author ID: 1175196
Email: georgo@math.uni-bonn.de

Dulip Piyaratne
Affiliation: Department of Mathematics, Xiamen University of Malaysia, Malaysia
MR Author ID: 1116811
ORCID: 0000-0002-6945-8664
Email: piyaratne@xmu.edu.my

Yukinobu Toda
Affiliation: Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo (WPI), Japan
MR Author ID: 792353
Email: yukinobu.toda@ipmu.jp

Received by editor(s): August 22, 2018
Received by editor(s) in revised form: May 25, 2021
Published electronically: September 14, 2021
Additional Notes: The first author was supported by the National Science Foundation Grant DMS-1440140 while in residence at MSRI, Berkeley. The second author was supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan. The third author was supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and Grant-in Aid for Scientific Research grant (No. 26287002) from MEXT, Japan
Article copyright: © Copyright 2021 University Press, Inc.