Local structure of principally polarized stable Lagrangian fibrations

TL;DR

This paper characterizes the local structure of principally polarized stable Lagrangian fibrations, providing explicit period maps and constructing examples to illustrate the behavior of characteristic cycles.

Contribution

It offers a complete description of the local structure and explicit period maps for principally polarized stable Lagrangian fibrations, including construction methods from period maps.

Findings

Explicit form of the period map for such fibrations

Construction of fibrations from given period maps

Examples demonstrating characteristic cycle behaviors

Abstract

A holomorphic Lagrangian fibration is stable if the characteristic cycles of the singular fibers are of type $I_m, 1 \leq m <\infty,$ or $A_{\infty}$. We will give a complete description of the local structure of a stable Lagrangian fibration when it is principally polarized. In particular, we give an explicit form of the period map of such a fibration and conversely, for a period map of the described type, we construct a principally polarized stable Lagrangian fibration with the given period map. This enables us to give a number of examples exhibiting interesting behavior of the characteristic cycles.