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