Categorical mirror symmetry on cohomology for a complex genus 2 curve

Abstract

Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs X and Y such that the complex geometry on X mirrors the symplectic geometry on Y. It allows one to deduce symplectic information about Y from known complex properties of X. Strominger-Yau-Zaslow [61] described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich [43] conjectured that a complex invariant on X (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of Y (the Fukaya category, see [9], [29], [49], [1]). This is known as homological mirror symmetry. In this project, we first use the construction of “generalized SYZ mirrors” for hypersurfaces in toric varieties following Abouzaid-Auroux-Katzarkov [6], in order to obtain X and Y as manifolds. The complex manifold is the genus 2 curve Σ2 (so of general type c1<0) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model (Y,v0) equipped with a holomorphic function v0:Y→C which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of DbCoh(Σ2) into a cohomological Fukaya-Seidel category of Y as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations in [55] and in Abouzaid-Seidel [3].