The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds
type: Item
instance of: scholarly article
title: The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds
zbMATH Open document ID: 1360.14057
author: Arend Bayer
author: Emanuele Macrì
author: Paolo Stellari
published in: Inventiones Mathematicae
publication date: 2016-12-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1410.1585
review text: This very well-written paper studies Bridgeland stability conditions on Calabi-Yau threefolds along the line of Bogomolov-Gieseker type inequality conjecture proposed by \textit{A. Bayer} et al. [J. Algebr. Geom. 23, No. 1, 117--163 (2014; Zbl 1306.14005)]. Precisely speaking, this paper proposes a stronger conjecture, and shows that the stronger one implies the support property of Bridgeland stability conditions, and by a deformation argument that an open family of stability conditions do exist. This stronger conjecture is proved for abelian threefolds, generalizing the result by \textit{A. Maciocia} and \textit{D. Piyaratne} [Algebr. Geom. 2, No. 3, 270--297 (2015; Zbl 1322.14040); Int. J. Math. 27, No. 1, Article ID 1650007, 27 p. (2016; Zbl 1360.14064)] in the case of Picard number one. Some general discussion on Bogomolov-Gieseker type inequality and the main conjecture are given in \S\S2--4. The construction of stability conditions for abelian threefolds is given in \S\S7--9, footing on the argument of isogeny given in \S6. \S10 gives a construction of stability conditions for quotients of abelian threefolds. Appendices on support property and deformation argument are also well worth of read. As mentioned in the ``related work'' subsection in the introduction, the method of constructing stability condition given in this paper is different from the one by Maciocia and Piyaratne who used Fourier-Mukai transforms. Since semi-homogeneous vector bundles give essentially all the Fourier-Mukai transforms on abelian varieties, and semi-homogeneous bundles are constructed by isogeny, there should be some conceptual explanation of similarity between these two methods.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6664764
zbMATH Keywords: Bridgeland stability condition
zbMATH Keywords: abelian threefold
Wikidata QID: Q59469457
reviewed by: Shintarou Yanagida
MaRDI profile type: Publication
OpenAlex ID: W1666560682
arXiv ID: 1410.1585
cites work: Bridgeland-stable moduli spaces for \(K\)-trivial surfaces
cites work: Bridgeland stability of line bundles on surfaces
cites work: On the Kummer construction
cites work: Polynomial Bridgeland stability conditions and the large volume limit
cites work: Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds
cites work: Bridgeland stability conditions on threefolds. II: An application to Fujita's conjecture
cites work: A note on Bogomolov-Gieseker type inequality for Calabi-Yau 3-folds
cites work: The McKay correspondence as an equivalence of derived categories
cites work: Q4830002
cites work: The space of stability conditions on the local projective plane
cites work: Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities
cites work: Q4232461
cites work: Stability conditions on triangulated categories
cites work: Stability conditions on \(K3\) surfaces
cites work: On Kummer 3-folds
cites work: Restriction theorems for stable rank 3 vector bundles on \(P^ n\).
cites work: Inducing stability conditions
cites work: Very ampleness and higher syzygies for Calabi-Yau threefolds
cites work: Q3579591
cites work: Tilting in abelian categories and quasitilted algebras
cites work: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
cites work: On the S-fundamental group scheme
cites work: Some Examples of Tilt-Stable Objects on Threefolds
cites work: Computing the walls associated to Bridgeland stability conditions on projective surfaces
cites work: A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space
cites work: Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds
cites work: Chern classes of rank 3 stable reflexive sheaves
cites work: Semi-homogeneous vector bundles on an abelian variety
cites work: On polarized canonical Calabi-Yau threefolds
cites work: Calabi-Yau threefolds of quotient type
cites work: Q5421225
cites work: Phases of Lagrangian-invariant objects in the derived category of an abelian variety
cites work: Regularity on abelian varieties I
cites work: Regularity on abelian varieties II: Basic results on linear series and defining equations
cites work: A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold
cites work: Counterexample to the Generalized Bogomolov–Gieseker Inequality for Threefolds
cites work: Higgs bundles and local systems
cites work: Curve counting theories via stable objects I. DT/PT correspondence
cites work: Bogomolov-Gieseker-type inequality and counting invariants
cites work: Stability conditions and extremal contractions
cites work: Derived category of coherent sheaves and counting invariants
cites work: Gepner type stability conditions on graded matrix factorizations
cites work: Q4296929
DOI: 10.1007/S00222-016-0665-5
description: scientific article