Semi-orthogonality in Fukaya-Seidel mirrors to blowups of abelian varieties

We prove a categorical homological mirror symmetry result for a blow-up of an abelian surface times $\mathbb{C}$, on the complex side. Specifically, the first author's paper arXiv:1908.04227 provides evidence for homological mirror symmetry for genus 2 curves $Σ_2$ on the complex side, where $Σ_2$ is a hypersurface in an abelian surface $V \cong T^4$. To obtain a mirror to $Σ_2$, the generalized SYZ approach arXiv:hep-th/0002222, arXiv:1205.0053 was used. That is, $(\text{Bl}_{Σ_2 \times \{0\}} V \times \mathbb{C}_y, y)=:(X,y)$ admits a Lagrangian torus fibration which has an SYZ mirror Landau-Ginzburg model $(Y,v_0)$, known as the generalized SYZ mirror of $Σ_2$. The mirror to $X$ - without the superpotential - is obtained by removing a generic fiber from $(Y,v_0)$. In this paper, we prove a categorical homological mirror symmetry result for $X$. To do so, we equip the symplectic mirror with a Fukaya category which involves both partial and full wrapping in the base of a symplectic Landau-Ginzburg model. Semi-orthogonality appears in the categorical invariants on both sides of HMS.