In this talk, we first use the construction of SYZ mirrors for hypersurfaces in abelian varieties following Abouzaid-Auroux-Katzarkov. We obtain a Landau-Ginzburg model $(Y,v_0)$ that is "generalized SYZ mirror" to the genus 2 curve $\Sigma_2$, thought of as a hypersurface in its Jacobian torus $Jac(\Sigma_2)$. We equip this Landau-Ginzburg model with a Fukaya category and then describe an embedding of the category $D^b_{\mathcal{L}}Coh(\Sigma_2)$ generated by powers of the defining line bundle for $\Sigma_2$ restricted to the hypersurface, into the cohomological Fukaya-Seidel category of the mirror LG model. 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 work of Abouzaid-Seidel.

This work is the speaker's PhD thesis under advisor Denis Auroux. It was partially supported by NSF grants DMS-1264662, DMS-1406274, and DMS-1702049, and by a Simons Foundation grant (# 385573, Simons Collaboration on Homological Mirror Symmetry).