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Mirror symmetry for lattice-polarized abelian surfaces

Yu-Wei Fan, Kuan-Wen Lai

TL;DR

This work extends Dolgachev’s lattice-polarized mirror symmetry to abelian surfaces by defining lattice-polarized abelian surfaces and their coarse moduli spaces, and by formulating a stringy Kahler moduli space via Bridgeland stability conditions. It proves that for mirror pairs $X$ and $Y$, the complex moduli of one correspond to the stringy Kahler moduli of the other and vice versa, with a natural involution on the Kahler side that implements the dual abelian partner under mirror symmetry. Existence criteria for mirrors are established, notably requiring $\rho(X)\le 2$ or $\rho(X)=3$ with $\operatorname{NS}(X)\cong \mathbb{Z}(-2n)\oplus U$, and a detailed analysis is given for the Picard number $2$ case in terms of discriminant forms, including a criterion for self-mirror abelian surfaces. The paper further reduces self-mirror questions to local anti-automorphisms of discriminant forms and provides explicit examples, including products of elliptic curves and $U(n)$-type lattices, with a precise criterion for principal polarizations. Overall, the results illuminate how lattice data and derived-category stability conditions govern mirror duality, and they extend the lattice-mirror dictionary to abelian surfaces, enriching the landscape of Calabi–Yau mirror phenomena beyond K3 surfaces.

Abstract

Inspired by Dolgachev's 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.

Mirror symmetry for lattice-polarized abelian surfaces

TL;DR

This work extends Dolgachev’s lattice-polarized mirror symmetry to abelian surfaces by defining lattice-polarized abelian surfaces and their coarse moduli spaces, and by formulating a stringy Kahler moduli space via Bridgeland stability conditions. It proves that for mirror pairs and , the complex moduli of one correspond to the stringy Kahler moduli of the other and vice versa, with a natural involution on the Kahler side that implements the dual abelian partner under mirror symmetry. Existence criteria for mirrors are established, notably requiring or with , and a detailed analysis is given for the Picard number case in terms of discriminant forms, including a criterion for self-mirror abelian surfaces. The paper further reduces self-mirror questions to local anti-automorphisms of discriminant forms and provides explicit examples, including products of elliptic curves and -type lattices, with a precise criterion for principal polarizations. Overall, the results illuminate how lattice data and derived-category stability conditions govern mirror duality, and they extend the lattice-mirror dictionary to abelian surfaces, enriching the landscape of Calabi–Yau mirror phenomena beyond K3 surfaces.

Abstract

Inspired by Dolgachev's 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.
Paper Structure (13 sections, 24 theorems, 146 equations)

This paper contains 13 sections, 24 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Lattice-polarized abelian surfaces
  3. Marked complex 2-tori
  4. A lattice-theoretic lemma
  5. Lattice-polarization and complex moduli
  6. Mirror pairs of abelian surfaces
  7. Stringy Kähler moduli spaces
  8. Mirror pairs and symplectic duals
  9. Existence of mirror partners
  10. Self-mirror abelian surfaces
  11. Anti-automorphisms on discriminant forms
  12. Anti-automorphisms on 2-Sylow summands
  13. Examples of self-mirror abelian surfaces

Key Result

Theorem 1.2

For each abelian surface $X$, one can associate two spaces such that if $Y$ is a mirror partner of $X$, there are natural isomorphisms

Theorems & Definitions (62)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Lemma 2.1
  • proof
  • proof
  • ...and 52 more