The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds. (English) Zbl 1360.14057
This very well-written paper studies Bridgeland stability conditions on Calabi-Yau threefolds along the line of Bogomolov-Gieseker type inequality conjecture proposed by 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 A. Maciocia and 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 §§2–4. The construction of stability conditions for abelian threefolds is given in §§7–9, footing on the argument of isogeny given in §6. §10 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.
Some general discussion on Bogomolov-Gieseker type inequality and the main conjecture are given in §§2–4. The construction of stability conditions for abelian threefolds is given in §§7–9, footing on the argument of isogeny given in §6. §10 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.
Reviewer: Shintaro Yanagida (Nagoya)
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 14J30 | \(3\)-folds |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
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