Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds II

Abstract

We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian three-fold with Picard rank one by proving that tilt stable objects satisfy the strong Bogomolov-Gieseker type inequality. This is done by showing any Fourier-Mukai transform gives an equivalence of abelian categories which are double tilts of coherent sheaves.

24 pages, Spurious part of Props 5.5. and 5.6 removed and text adjusted. Contact details and references updated. Additional explanations added to note 3.2 and several minor corrections made following the referee's suggestions. To appear in Inter. J. Math

References in corpus (4)

• A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space
• A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold
• Some examples of tilt-stable objects on threefolds
• Semi-homogeneous sheaves, Fourier-Mukai transforms and moduli of stable sheaves on abelian surfaces

Cited by in corpus (7)

• A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold
• Stability conditions on Fano threefolds of Picard number one
• Moduli of Bridgeland semistable objects on 3-folds and Donaldson-Thomas invariants
• The Space of Stability Conditions on Abelian Threefolds, and on some Calabi-Yau Threefolds
• Derived category of coherent sheaves and counting invariants
• Categorical entropy for Fourier-Mukai transforms on generic abelian surfaces
• Preservation of semistability under Fourier-Mukai transforms