Mirror Symmetry in dimension one and Fourier-Mukai equivalences

Abstract

In this paper we will describe an approach to mirror symmetry for appropriate 1-dimensional DM stacks of arithmetic genus $g\le 1$, called tcnc curves, which was developed by the author with Treumann and Zaslow in arXiv:1103.2462 . This involves introducing a conjectural sheaf-theoretic model for the Fukaya category of punctured Riemann surfaces. As an application, we will investigate derived equivalences of tcnc curves, and generalize classic results of Mukai on dual abelian varieties (Mukai 1981).

arXiv comment17 pages, 7 figures. The first part is a survey and overlaps with arXiv:1103.2462, and arXiv:1111.5284. New results are contained in Section 4. To appear in Proceedings of the conference "Mirror Symmetry & Tropical Geometry," July 2-8 2011, Cetraro Italy

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