On polarization types of Lagrangian fibrations

TL;DR

This paper investigates the polarization types of Lagrangian fibrations on holomorphic symplectic manifolds, proving their constancy in families and determining types for specific cases, with conjectures on deformation dependence.

Contribution

It introduces the concept of polarization types for Lagrangian fibrations and proves their invariance in families, providing explicit results for K3^[n]-type fibrations and proposing a deformation-based conjecture.

Findings

Polarization type is constant in families of Lagrangian fibrations.

Determined polarization types for K3^[n]-type fibrations.

Conjecture that polarization type depends only on deformation class.

Abstract

The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. Associate a polarization type to such Lagrangian fibrations coming from polarizations on a generic fiber. We prove that this polarization type is constant in families of Lagrangian fibrations. Further, we determine the polarization type of K3^[n]-type fibrations and conjecture that the polarization type should only depend on the deformation type of the total space.