The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

By Arend Bayer, Emanuele Macrì, P. Stellari. 2014

169 citations recorded by the source corpus.

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 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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8093299e56adc40da982c7af41a19ddae777e21b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

Citations in the corpus

Birational models of moduli spaces of coherent sheaves on the projective plane (2016, background): “10 in [8], the discriminants of w0 and w w0 are less than that of w .”
Bridgeland Stability Conditions on Fano Threefolds (2016, methodology): “ are able to prove the original conjecture with = 0 by using the toric Frobenius morphism. The proof in these two cases will then follow a similar \limit" argument of Euler characteristics as in [BMS16]. Theorem 1.2. Let Xbe a P2-bundle over P1. Let Hbe an ample divisor such that, for all eecti”
On stability conditions for the quintic threefold (2018, methodology): “Stability conditions are only known to exist on few families of smooth projective threefolds: Fano threefolds [6,18,24,29,31], Abelian threefolds [5,26,27] and Kummer type threefolds [5].”
Lectures on Bridgeland Stability (2016, background): “d with a fundamentally diﬀerent proof to all Fano threefolds of Picard rank one in [Li19]. The conjecture is also known to be true for all abelian threefolds with two independent proofs by [MP16] and [BMS16]. Unfortunately, it turned out to be problematic in the case of the blow up of P3 in a point ”