Kodaira-type classification of singular fibers of some minimal abelian fibrations

TL;DR

This paper generalizes Kodaira's classification of singular fibers from elliptic fibrations to minimal abelian fibrations of higher dimensions, providing a comprehensive framework for understanding their fiber types.

Contribution

It introduces a unified classification scheme for singular fibers in minimal abelian fibrations, extending classical results to higher-dimensional cases.

Findings

Classifies all possible singular fibers with abelian variety parts

Divides fibers into semistable, unstable, and multiple categories

Further subdivides multiple fibers into three specific types

Abstract

Let $X \to S$ be a minimal abelian fibration of relative dimension $n$ over a curve. We classify all possible singular fibers $X_s$ having $(n-1)$-dimensional ``abelian variety parts''. This generalizes Kodaira's work on elliptic fibrations, and Matsushita and Hwang--Oguiso's work on Lagrangian fibrations into a single framework. The classification is divided into three parts: semistable, unstable, and multiple. Multiple fibers are again divided into three types: semistable-like, mixed, and unstable-like.