Mirror symmetry for lattice-polarized abelian surfaces

TL;DR

This paper explores mirror symmetry for lattice-polarized abelian surfaces, constructing moduli spaces, identifying mirror pairs, and classifying self-mirror cases through lattice analysis.

Contribution

It introduces lattice-polarized abelian surfaces, constructs their moduli spaces, and establishes a mirror symmetry framework analogous to K3 surfaces.

Findings

Identification of mirror pairs via moduli space isomorphism

Introduction of an involution pairing abelian surfaces with duals

Classification of self-mirror abelian surfaces based on Néron-Severi lattices

Abstract

Inspired by the Dolgachev-Nikulin-Pinkham mirror symmetry for lattice-polarized K3 surfaces, we study its analogue for abelian surfaces. In this paper, we introduce lattice-polarized abelian surfaces and construct their coarse moduli spaces. We then construct stringy K\"ahler moduli spaces for abelian surfaces and show that these two spaces are naturally identified for mirror pairs. We also introduce a natural involution on stringy K\"ahler moduli spaces which, under mirror symmetry, pairs abelian surfaces and their duals. Finally, we determine conditions for the existence of mirror partners and classify self-mirror abelian surfaces via their N\'eron-Severi lattices.