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

TL;DR

This paper provides evidence for homological mirror symmetry between a blow-up of an abelian surface times the complex plane and its symplectic Landau-Ginzburg mirror, highlighting semi-orthogonality in categorical invariants.

Contribution

It establishes a categorical HMS result for the blow-up of an abelian surface times the complex plane, involving semi-orthogonality and wrapped Fukaya categories.

Findings

Proves categorical HMS for the blow-up and its mirror.

Introduces a Fukaya category with partial and full wrapping.

Identifies semi-orthogonality in categorical invariants.

Abstract

We prove evidence of Kontsevich's homological mirror symmetry conjecture (HMS) for a blow-up of an abelian surface times the complex plane, on the complex side, and its symplectic Landau-Ginzburg mirror. Specifically, the first author proved evidence of HMS for a 1-parameter family of genus 2 curves on the complex side, as a hypersurface in an abelian surface. The generalized SYZ mirror to the hypersurface is then the SYZ mirror to the Landau-Ginzburg model given by the blow-up of the abelian surface times the complex plane, along the hypersurface times zero, with superpotential given by projection to the complex plane. The mirror to the blow-up - without the superpotential - is obtained by removing a generic smooth fiber from the generalized SYZ mirror superpotential. We prove a categorical HMS result for the latter pair, between categories expected to split-generate. To do so, we equip the punctured superpotential with a Fukaya category which involves both partial and full wrapping in the base of the symplectic Landau-Ginzburg model due to the removal of a generic fiber. Semi-orthogonality appears in the categorical invariants on both sides of HMS.