Stability conditions and Calabi–Yau fibrations

Toda, Yukinobu

doi:10.1090/s1056-3911-08-00511-0

Cited by 12 publications

(13 citation statements)

References 14 publications

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“…the total space of the canonical bundle of a surface. For instance, see [Bri06,BM11,IUU10,Tod08,Tod09b]. The case of non-projective complex tori has been studied in [Mei07].…”

Section: Background On Stability Conditionsmentioning

confidence: 99%

Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

et al. 2014

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ABSTRACT. We construct new t-structures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a BogomolovGieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples.Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.

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“…the total space of the canonical bundle of a surface. For instance, see [Bri06,BM11,IUU10,Tod08,Tod09b]. The case of non-projective complex tori has been studied in [Mei07].…”

Section: Background On Stability Conditionsmentioning

confidence: 99%

Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

et al. 2014

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“…The space Stab(X) have been studied in several examples. For instance see [7], [8], [9], [17], [24], [30], [31], [32]. Although the notion of stability conditions on triangulated categories has drawn much interest recently, we are not able to study the most important case, D = D b (X) for a projective Calabi-Yau 3-fold X at this time.…”

Section: Stability Conditions On Triangulated Categoriesmentioning

confidence: 99%

Limit stable objects on Calabi-Yau 3-folds

Toda

2009

Duke Math. J.

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105

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In this paper, we introduce new enumerative invariants of curves on Calabi-Yau 3-folds via certain stable objects in the derived category of coherent sheaves. We introduce the notion of limit stability on the category of perverse coherent sheaves, a subcategory in the derived category, and construct the moduli spaces of limit stable objects. We then define the counting invariants of limit stable objects using Behrend's constructible functions on that moduli spaces. It will turn out that our invariants are generalizations of counting invariants of stable pairs introduced by Pandharipande and Thomas. We will also investigate the wall-crossing phenomena of our invariants under change of stability conditions. I(X) = {I C ⊂ O X | C ⊂ X is a subscheme with dim C ≤ 1}.

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“…Some examples of stability conditions on projective spaces were studied in [Mac07,ABL07,Ohk10]. Other local Calabi-Yau threefold cases were studied in [Tod08,Tod09], and, as already mentioned, an open subset of Stab † (D 0 ) has been described in [Bri06].…”

Section: Introductionmentioning

confidence: 99%

The space of stability conditions on the local projective plane

Bayer¹,

Macrì²

2011

Duke Math. J.

122

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ABSTRACT. We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simply-connected. We determine the group of autoequivalences preserving this connected component.Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with level structure, with the morphism given by solutions of Picard-Fuchs equations. This result is motivated by the notion of Π-stability and by mirror symmetry.

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Stability conditions and Calabi–Yau fibrations

Cited by 12 publications

References 14 publications

Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

Limit stable objects on Calabi-Yau 3-folds

The space of stability conditions on the local projective plane

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