Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds

By Antony Maciocia, D. Piyaratne. 2013

79 citations recorded by the source corpus.

We show that the construction of Bayer, Bertram, Macri and Toda gives rise to a Bridgeland stability condition on a principally polarized abelian threefold with Picard rank one by establishing their conjectural generalized Bogomolov-Gieseker inequality for certain tilt stable objects. We do this by proving that a suitable Fourier-Mukai transform preserves the heart of a particular conjectural stability condition. We also show that the only reflexive sheaves with zero first and second Chern classes are the flat line bundles.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/607d181df221d506142cfacfd8778f5ff4c95cfc

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Citations in the corpus

Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014, background): “Assuming a conjecture on ‘Bogomolov–Gieseker type inequalities’, Bayer, Macr̀ı and Toda [7] construct Bridgeland stability conditions on Db coh(M) for M a Calabi–Yau 3-fold; the conjecture is proved by Macioca and Piyaratne [48, 49] when M is an abelian 3-fold.”
Stability conditions on Fano threefolds of Picard number 1 (2015, background): “The abelian 3-fold of Picard number one case has been done in [7] and [8] by Maciocia and Piyaratne.”
Lectures on Bridgeland Stability (2016, result): “The conjecture is also known to be true for all abelian threefolds with two independent proofs by [MP16] and [BMS14].”
The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds (2014, background): “[MP13b] Antony Maciocia and Dulip Piyaratne.”