The Space of Stability Conditions on Abelian Threefolds, and on some Calabi-Yau Threefolds

Abstract

We describe a connected component of the space of stability conditions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. Our proof includes the following essential steps: 1. We simultaneously strengthen a conjecture by the first two authors and Toda, and prove that it follows from a more natural and seemingly weaker statement. This conjecture is a Bogomolov-Gieseker type inequality involving the third Chern character of "tilt-stable" two-term complexes on smooth projective threefolds; we extend it from complexes of tilt-slope zero to arbitrary tilt-slope. 2. We show that this stronger conjecture implies the so-called support property of Bridgeland stability conditions, and the existence of an explicit open subset of the space of stability conditions. 3. We prove our conjecture for abelian threefolds, thereby giving reproving and generalizing a result by Maciocia and Piyaratne. Important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall-crossing, closely related to the support property.

45 pages, 1 figure. v2: addressed referee comments. To appear in Inventiones Math

References in corpus (9)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Bridgeland-Stable Moduli Spaces for K-Trivial Surfaces
• On equivariant triangulated categories
• Stability conditions on Fano threefolds of Picard number one
• Derived category of coherent sheaves and counting invariants
• Some examples of tilt-stable objects on threefolds
• The genus of projective curves on complete intersection surfaces
• On the Kummer construction
• Constant families of t-structures on derived categories of coherent sheaves

Cited by in corpus (8)

• Riemann-Hilbert problems from Donaldson-Thomas theory
• Counterexample to the Generalized Bogomolov-Gieseker Inequality for Threefolds
• Moduli of Bridgeland semistable objects on 3-folds and Donaldson-Thomas invariants
• The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface
• Hilbert scheme of twisted cubics as simple wall-crossing
• Brill-Noether theory for curves on generic abelian surfaces
• Bogomolov-Gieseker Type Inequality on Calabi-Yau and Fano 3-folds
• Preservation of semistability under Fourier-Mukai transforms