Mirror symmetry for lattice-polarized abelian surfaces

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 Kähler 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 Kähler 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 Néron-Severi lattices.

💡 Research Summary

The paper extends the Dolgachev‑Nikulin‑Pinkham mirror symmetry, originally formulated for lattice‑polarized K3 surfaces, to the setting of abelian surfaces. The authors first introduce the notion of a lattice‑polarized abelian surface. Unlike K3 surfaces, where the marking is placed on the second cohomology group, for abelian surfaces the marking is placed on the first cohomology group H¹, reflecting the fact that the period map for abelian surfaces lives in the Grassmannian Gr(2,4). By fixing a free ℤ‑module L≅ℤ⁴ and its exterior square Λ=∧²L≅U⊕³, an admissible marking is defined as an isomorphism µ:H¹(X,ℤ)→L whose induced map ∧²µ:H²(X,ℤ)→Λ is an isometry. The period of a marked torus is then a point