Homological mirror symmetry for invertible polynomials in two variables

TL;DR

This paper proves homological mirror symmetry for two-variable invertible polynomials by explicitly constructing Milnor fibres and establishing derived equivalences between complex nodal stacky curves.

Contribution

It provides a detailed proof of homological mirror symmetry in a specific setting involving invertible polynomials and maximal symmetry groups.

Findings

Established homological mirror symmetry for two-variable invertible polynomials.

Constructed explicit gluing of Milnor fibres.

Proved derived equivalences between complex nodal stacky curves.

Abstract

In this paper, we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the $B$-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and as an application, we prove derived equivalences between certain nodal stacky curves, some of whose irreducible components have non-trivial generic stabiliser.