Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds

TL;DR

This paper constructs moduli spaces for lattice polarized holomorphic symplectic manifolds, explores their compactifications, and examines the relationship between boundary components and rational Lagrangian fibrations on mirror manifolds.

Contribution

It introduces a construction of quasi-projective moduli spaces for lattice polarized irreducible holomorphic symplectic manifolds and analyzes their boundary structure and mirror symmetry relations.

Findings

Constructed quasi-projective moduli spaces for lattice polarized manifolds.

Analyzed Baily--Borel compactification of these moduli spaces.

Established a link between boundary components and rational Lagrangian fibrations.

Abstract

We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.