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

Macrì, Emanuele

doi:10.2140/ant.2014.8.173

Cited by 71 publications

(90 citation statements)

References 27 publications

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“…Therefore, we will replace B by β and ω by α in the notation of slope functions and categories. Due to Proposition 2.7 and Lemma 3.2 in [Mac12] it suffices to prove the statement for α < (ii) The inclusion C ⊂ A α,β , A α,β [1] holds.…”

Section: Resultsmentioning

confidence: 99%

A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

Schmidt

2014

Bulletin of the London Mathematical Society

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Section: Resultsmentioning

confidence: 99%

A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

Schmidt

2014

Bulletin of the London Mathematical Society

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“…The reduction is based on the methods of [26]: as we approach this limit, either E remains stable, in which case the above inequality is enough to ensure that E satisfies our conjecture everywhere. Otherwise, E will be strictly semistable at some point; we then show that all its Jordan-Hölder factors have strictly smaller "H -discriminant" (which is a variant of the discriminant appearing in the classical Bogomolov-Gieseker inequality).…”

Section: Reduction To a Limit Casementioning

confidence: 99%

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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“…An important inequality introduced in [BMT14] and proved in [Mac14] for ν α,β -semistable objects is the following: Theorem 2.3. (Generalized Bogomolov-Gieseker inequality) For any ν α,β -semistable object E ∈ Coh β (P 3 ) satisfying ν α,β (E) = 0, we have the following inequality…”

Section: Definition 22 Let Cohmentioning

confidence: 99%