Homological mirror symmetry for curves of higher genus

Abstract

This paper is devoted to homological mirror symmetry conjecture for curves of higher genus. It was proposed by Katzarkov as a generalization of original Kontsevichʼs conjecture.A version of this conjecture in the case of the genus two curve was proved by Seidel [25]. Based on the paper of Seidel, we prove the conjecture (in the same version) for curves of genus g⩾3. Namely, we relate the Fukaya category of a genus g curve to the category of singularities of zero fiber in the mirror dual Landau–Ginzburg model.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.