2014/10/06 by Arend Bayer, Bayer, Arend, Emanuele Macrì +3 · 12 citations
Mathematics · #14F05 (Primary) #14J30 #18E30 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1410.1585
openalex publication_date 2014/10/06 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We describe a connected component of the space of stability conditions on\nabelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant\nresolution of) a finite quotient of an abelian threefold. Our proof includes\nthe following essential steps:\n 1. We simultaneously strengthen a conjecture by the first two authors and\nToda, and prove that it follows from a more natural and seemingly weaker\nstatement. This conjecture is a Bogomolov-Gieseker type inequality involving\nthe third Chern character of "tilt-stable" two-term complexes on smooth\nprojective threefolds; we extend it from complexes of tilt-slope zero to\narbitrary tilt-slope.\n 2. We show that this stronger conjecture implies the so-called support\nproperty of Bridgeland stability conditions, and the existence of an explicit\nopen subset of the space of stability conditions.\n 3. We prove our conjecture for abelian threefolds, thereby giving reproving\nand generalizing a result by Maciocia and Piyaratne.\n Important in our approach is a more systematic understanding on the behaviour\nof quadratic inequalities for semistable objects under wall-crossing, closely\nrelated to the support property.\n