Homological mirror symmetry for orbifold log Calabi-Yau surfaces

TL;DR

This paper establishes homological mirror symmetry for orbifold log Calabi-Yau surfaces by constructing an abstract Lefschetz fibration and relating it to explicit Laurent polynomial mirrors, advancing understanding of mirror symmetry in orbifold settings.

Contribution

It introduces a novel abstract Lefschetz fibration construction for orbifold log Calabi-Yau surfaces and connects it to explicit Laurent polynomial mirrors, including a case with orbifold del Pezzo surfaces.

Findings

Constructed an abstract Lefschetz fibration for orbifold surfaces.

Linked the construction to the special McKay correspondence.

Provided an explicit Laurent polynomial mirror example.

Abstract

We prove homological mirror symmetry for orbifold log Calabi-Yau surfaces at the large complex structure limit by constructing an abstract Lefschetz fibration associated to each pair $(\mathcal{X},\mathcal{D})$ with $\mathcal{X}$ a projective rational surface with isolated cyclic quotient orbifold points and $\mathcal{D}$ a stacky anticanonical divisor. We describe a Lefschetz stabilization procedure which, on the mirror, corresponds to the special McKay correspondence of Ishii and Ueda arXiv:1104.2381v2 [math.AG]. Moreover, we relate our abstract construction to an explicit Laurent polynomial mirror in an example consisting of a family of orbifold del Pezzo surfaces.