Counterexample to the Generalized Bogomolov–Gieseker Inequality for Threefolds

Schmidt, Benjamin

doi:10.1093/imrn/rnw122

Cited by 24 publications

(34 citation statements)

References 13 publications

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“…For higher Picard rank it is known to be false in general. Counterexamples were given in [Sch17,Kos17,MS17]. In general, a relation between Assumpion C and Castelnuovo theory for projective curves ([Har82a, CCDG93]) was already observed in [BMT14,Tra14].…”

Section: Introductionmentioning

confidence: 92%

Derived categories and the genus of space curves

Schmidt¹

2020

Alg. Geom.

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We generalize a classical result about the genus of curves in projective space by Gruson and Peskine to principally polarized abelian threefolds of Picard rank one. The proof is based on wall-crossing techniques for ideal sheaves of curves in the derived category. In the process, we obtain bounds for Chern characters of other stable objects such as rank two sheaves. The argument gives a proof for projective space as well. In this case these techniques also indicate an approach for a conjecture by Hartshorne and Hirschowitz and we prove first steps towards it.

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Section: Introductionmentioning

confidence: 92%

Derived categories and the genus of space curves

Schmidt¹

2020

Alg. Geom.

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“…Counterexamples due to Schmidt [40] and Martinez [27] indicate that Conjectures 2.4 and 4.1 need to be modified in the case of a threefold obtained as the blowup at a point of another threefold; on the other hand, they have been verified for all Fano threefolds of Picard rank one [23].…”

Section: Update (March 2016)mentioning

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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“…These inequalities play a fundamental role in establishing the existence of stability conditions. We note, however, that a counter example to the conjectural inequality in [3] was found by Schmidt [27]. It would be very interesting to extend these inequalities to higher rank bundles admitting solutions of dHYM.…”

Section: Algebraic Aspects Of the Deformed Hermitian-yang-mills Equationmentioning

confidence: 73%

The Deformed Hermitian–Yang–Mills Equation in Geometry and Physics

Collins¹,

Xie²,

Yau³

2018

Geometry and Physics: Volume I

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To Nigel Hitchin, with admiration, on the occasion of his 70th birthday.Abstract. We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kähler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, discuss some recent progress towards its solution. In dimension 3 we prove a new Chern number inequality and discuss the relationship with algebraic stability conditions.

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Counterexample to the Generalized Bogomolov–Gieseker Inequality for Threefolds

Abstract: Abstract. We give a counterexample to the generalized Bogomolov-Gieseker inequality for threefolds conjectured by Bayer, Macrì and Toda using the blow up of a point over three dimensional projective space.

Cited by 24 publications

References 13 publications

Derived categories and the genus of space curves

Derived categories and the genus of space curves

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

The Deformed Hermitian–Yang–Mills Equation in Geometry and Physics

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