Homological Mirror Symmetry for Curves of Higher Genus

Katzarkov has proposed a generalization of Kontsevich’s mirror symmetry conjecture, covering some varieties of general type. Seidel [Se1] has proved a version of this conjecture in the simplest case of the genus two curve. In this paper we prove the conjecture (in the same version) for curves of genus g ≥ 3, relating the Fukaya category of a genus g curve to the category of Landau-Ginzburg branes on a certain singular surface. We also prove a kind of reconstruction theorem for hypersurface singularities. Namely, formal type of hypersurface singularity (i.e. a formal power series up to a formal change of variables) can be reconstructed, with some technical assumptions, from its D (Z/2)G category of Landau-Ginzburg branes. The precise statement is Theorem 1.2.