  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.
