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 arXiv:hep-th/9606040 described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich arXiv:alg-geom/9411018 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 references in article abstract). 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 arXiv:1205.0053v4, in order to obtain $X$ and $Y$ as manifolds. The complex manifold is the genus 2 curve $\Sigma_2$ (so of general type $c_1<0$) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model $(Y,v_0)$ equipped with a holomorphic function $v_0:Y \to \mathbb{C}$ which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of $D^bCoh(\Sigma_2)$ 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 and in Abouzaid-Seidel.