[Paper Review] Prime order automorphisms of abelian surfaces: a lattice-theoretic point of view

Giovanni Mongardi, Kévin Tari+1 more|arXiv (Cornell University)|Jun 18, 2015

Algebraic Geometry and Number Theory5 references5 citations

TL;DR

This paper provides a lattice-theoretic classification of prime order automorphisms on complex abelian surfaces by analyzing their action on the second integral cohomology lattice $H^2(A,\mathbb{Z}) \cong U^{\oplus 3}$. Using the Torelli theorem and Hodge theory, it classifies non-symplectic and symplectic automorphisms of prime order $p = 2,3,5$, identifying invariant lattices and constructing moduli spaces via period domains and arithmetic groups. The key contribution is a complete list of possible invariant lattices $T(G_\sigma)$ for each prime order, establishing a correspondence between automorphism types and lattice invariants.

ABSTRACT

In this article, we focus on a new perspective of automorphisms of complex 2-tori, reviewing previous works from a lattice-theoretic point of view. In particular, we give a classification of families of symplectic and non-symplectic automorphisms of prime order.

Motivation & Objective

• To classify prime order automorphisms of complex abelian surfaces using the action on $H^2(A,\mathbb{Z})$.
• To determine the invariant lattices $T(G_\sigma)$ for non-symplectic and symplectic automorphisms of prime order.
• To construct moduli spaces for families of abelian surfaces with such automorphisms using period domains and arithmetic groups.
• To provide a reference framework for studying natural automorphisms of generalized Kummer manifolds in higher dimensions.

Proposed method

• Apply the Torelli theorem of Shioda to relate Hodge structures on $H^1(A,\mathbb{Z})$ to the cohomology lattice $H^2(A,\mathbb{Z}) \cong U^{\oplus 3}$.
• Use the action of automorphisms on $H^2(A,\mathbb{Z})$ to define the invariant lattice $T(G_\sigma)$ as the fixed sublattice under the group $G_\sigma = \langle \sigma, -\mathrm{id} \rangle$.
• For non-symplectic automorphisms, construct the period domain $D^H$ and the stabilizer group $\Gamma^H$ to parametrize isomorphism classes of $[H]$-polarized pairs.
• For symplectic automorphisms, define the restricted period domain $\Omega_{T(G)}$ within $\mathbb{P}(T(G) \otimes \mathbb{C})$ and use twistor families to lift period lines to actual families of tori.
• Leverage the surjectivity of the period map and monodromy invariance to show that the moduli space $\mathcal{M}_G$ parametrizes all marked abelian surfaces with a symplectic $G$-action.
• Use $p$-elementary lattice theory and the classification of $p$-elementary lattices to identify all possible $T(G_\sigma)$ for $p=2,3,5$.

Experimental results

Research questions

• RQ1 Which $p$-elementary lattices arise as the invariant lattice $T(G_\sigma)$ for non-symplectic automorphisms of prime order $p$ on abelian surfaces?
• RQ2 How do the invariants $\mathrm{rank}(T(G_\sigma))$, $a$, and $\dim$ of the invariant lattice classify families of abelian surfaces with such automorphisms?
• RQ3 What is the structure of the moduli space for abelian surfaces admitting a symplectic automorphism of prime order?
• RQ4 How does the period domain $D^H$ and the group $\Gamma^H$ parametrize isomorphism classes of $[H]$-polarized pairs $(A,G)$?
• RQ5 Can every generic twistor line in the period domain of abelian surfaces be realized as the period locus of a twistor family from a hyperkähler metric?

Key findings

• For non-symplectic automorphisms of prime order, the invariant lattice $T(G_\sigma)$ is isomorphic to one of the lattices listed in the table: $U$, $U(2)$, $\langle 2\rangle \oplus \langle -2\rangle$, $U \oplus \langle -2\rangle^{\oplus 2}$, $U(3)$, $U \oplus A_2(-1)$, or $H_5$, depending on $p$ and the invariants $r$, $\dim$, and $a$.
• The moduli space $\Gamma^H \backslash D^H$ parametrizes isomorphism classes of $[H]$-polarized pairs $(A,G)$ for non-symplectic automorphisms, with dimension $4 - r$ for $p=2$ and $\frac{6 - r}{p-1} - 1$ for $p=3,5$.
• For symplectic automorphisms, the moduli space $\mathcal{M}_G$ is a union of connected components, each rationally connected, and parametrizes all marked abelian surfaces with a symplectic $G$-action.
• Every generic twistor line in the period domain of abelian surfaces lifts to a twistor family of complex tori via a $G$-invariant hyperkähler metric, ensuring the moduli space is well-defined and geometrically realizable.
• The generic point of $\mathcal{M}_G$ corresponds to a marked abelian surface with $T(G) = T_A$ and $S(G) = NS(A)$, confirming the lattice-theoretic classification is geometrically meaningful.
• The classification of $T(G_\sigma)$ via $p$-elementary lattices fully determines the Fujiki family to which the automorphism belongs, providing a complete invariant for non-symplectic prime order automorphisms.