A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

Schmidt, Benjamin

doi:10.1112/blms/bdu048

Cited by 50 publications

(50 citation statements)

References 29 publications

(71 reference statements)

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“…Moreover, since Conjecture 4.1 is equivalent to Conjecture 2.4, and since the latter has been verified for P 3 in [11,26], and for the quadric threefold in [39], it also applies in these two cases. The inequality is new even in the case of P 3 : for sheaves of rank three, it is slightly weaker than classically known results, see [16,Theorem 4.3] and [31, Theorem 1.2], but no such results are known for higher rank.…”

Section: Remark 45mentioning

confidence: 99%

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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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Section: Remark 45mentioning

confidence: 99%

“…This was done in [11,26] for the case of P 3 , and in [39] for the case of the quadric in P 4 ; these are the only other cases in which Conjecture 2.4 is known.…”

Section: Related Workmentioning

confidence: 99%

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

2016

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“…This turned out to be true in various cases. The first proof was in the case of P 3 in [Mac14a] and a very similar proof worked for the smooth quadric hypersurface in P 4 in [Sch14]. These results were generalized with a fundamentally different proof to all Fano threefolds of Picard rank one in [Li15].…”

Section: Stability Conditions On Threefoldsmentioning

confidence: 93%

“…The first case was P 3 in [BMT14,Mac14a]. 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].…”

Section: Introductionmentioning

confidence: 92%

Lectures on Bridgeland Stability

Macrì

Schmidt

2017

Lecture Notes of the Unione Matematica Italiana

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Abstract. In these lecture notes we give an introduction to Bridgeland stability conditions on smooth complex projective varieties with a particular focus on the case of surfaces. This includes basic definitions of stability conditions on derived categories, basics on moduli spaces of stable objects and variation of stability. These notes originated from lecture series by the first author at the summer school Recent advances in algebraic and arithmetic geometry, Siena, Italy, August 24-28, 2015 and at the school Moduli of Curves, CIMAT, Guanajuato, Mexico, February 22 -March 4, 2016.

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“…Such an inequality provides a way to construct Bridgeland stability conditions on threefolds, and it was proved to be hold in the some cases: Proof. Please see [24], [29], [6] and [19].…”

Section: 3mentioning

confidence: 99%

Tilt-stability, vanishing theorems and Bogomolov-Gieseker type inequalities

Sun

2019

Advances in Mathematics

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We investigate the tilt-stability of stable sheaves on projective varieties with respect to certain tilt-stability conditions depends on two parameters constructed by Bridgeland [12] (see also [1,7,6]). For a stable sheaf, we give effective bounds of these parameters such that the stable sheaf is tiltstable. These allow us to prove new vanishing theorems for stable sheaves and an effective Serre vanishing theorem for torsion free sheaves. Using these results, we also prove Bogomolov-Gieseker type inequalities for the third Chern character of a stable sheaf on P 3 .

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A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

Abstract: Abstract. We prove a generalized Bogomolov-Gieseker inequality as conjectured by Bayer, Macrì and Toda for the smooth quadric threefold. This implies the existence of a family of Bridgeland stability conditions.

Cited by 50 publications

References 29 publications

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

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

Lectures on Bridgeland Stability

Tilt-stability, vanishing theorems and Bogomolov-Gieseker type inequalities

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