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 involv- ing 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.

Lectures on Bridgeland Stability

“…The case of abelian threefolds was independently proved in [MP15,MP16] and [BMS14]. Moreover, as pointed out in [BMS14], this also implies the case ofétale quotients of abelian threefolds and gives the existence of Bridgeland stability condition on orbifold quotients of abelian threefolds (this includes examples of Calabi-Yau threefolds which are simply-connected). The latest progress is the proof of the conjecture for all Fano threefolds of Picard rank one in [Li15] and a proof of a modified version for all Fano threefolds independently in [BMSZ16] and [Piy16].…”

“…A similar argument was then successfully applied to the smooth quadric hypersurface in P 4 in [Sch14]. The case of abelian threefolds was independently proved in [MP15,MP16] and [BMS14]. Moreover, as pointed out in [BMS14], this also implies the case ofétale quotients of abelian threefolds and gives the existence of Bridgeland stability condition on orbifold quotients of abelian threefolds (this includes examples of Calabi-Yau threefolds which are simply-connected).…”

“…These results were generalized with a fundamentally different proof to all Fano threefolds of Picard rank one in [Li15]. The conjecture is also known to be true for all abelian threefolds with two independent proofs by [MP16] and [BMS14]. Unfortunately, it turned out to be problematic in the case of the blow up of P 3 in a point as shown in [Sch16].…”

“…Arcara and Bertram realized that the construction can be generalized to any surface by using the Bogomolov Inequality in [AB13]. The proof of the support property is in [BM11,BMT14,BMS14]. As a consequence, these stability conditions vary continuously in ω and B.…”

“…In particular, it requires the construction of stability conditions on Calabi-Yau threefolds. It is an open problem to even find a single example of a stability condition on a simply connected projective Calabi-Yau threefold where skyscraper sheaves are stable (examples where they are semistable are in [BMS14]). Most successful attempts on threefolds trace back to a conjecture in [BMT14].…”