Stability conditions and extremal contractions

Toda, Yukinobu

doi:10.1007/s00208-013-0915-4

Cited by 32 publications

(41 citation statements)

References 19 publications

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“…We will therefore review these statements in more detail in Sect. 3 and Appendix 2; one can also deduce them with the same arguments as in the surface case, treated in detail in [44 …”

Section: Properties Of Tilt-stabilitymentioning

confidence: 93%

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

2016

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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.

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“…We will therefore review these statements in more detail in Sect. 3 and Appendix 2; one can also deduce them with the same arguments as in the surface case, treated in detail in [44 …”

Section: Properties Of Tilt-stabilitymentioning

confidence: 93%

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

2016

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“…• A mathematical formulation of Denef-Moore's formula derived in the study of Ooguri-Strominger-Vafa's conjecture, relating black hole entropy and topological string, [Tod11a]; • The possibility to realize extremal contractions for threefolds as moduli spaces of semistable objects in the derived category, [Tod12b].…”

Section: Introductionmentioning

confidence: 99%

A generalized Bogomolov–Gieseker inequality for the three-dimensional projective space

Macrì

2014

Algebra Number Theory

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“…However this is not the case for an arbitrary projective surface, and we need to find a general argument proving such an inequality. In the previous paper [21], we established such a BG type inequality for semistable objects on an arbitrary projective surface, and proved the support property for some stability conditions in U (X). We use this result to show the support property for stability conditions contained in other subsets U(Y ).…”

Section: Main Resultmentioning

confidence: 95%

“…Motivation. This paper is a continuation of the previous paper [21], in which the following question on the relationship between minimal model program (MMP) and Bridgeland stability conditions [7] was addressed (cf. [21, Question 1.1]): Question 1.1.…”

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confidence: 99%