# Singular fibres of very general Lagrangian fibrations

## TL;DR

This paper investigates the nature of singular fibres in very general Lagrangian fibrations, proposing a conjecture that they are semistable degenerations of abelian varieties, and provides partial results and examples supporting this.

## Contribution

It formulates a conjecture on the structure of singular fibres in Lagrangian fibrations and offers partial proofs and examples that support this conjecture.

## Key findings

- Partial proof supporting the conjecture on singular fibres.
- Description of an example illustrating the conjectured behaviour.
- Evidence suggesting singular fibres are semistable degenerations of abelian varieties.

## Abstract

Let $\pi:X\rightarrow\mathbb{P}^n$ be a (holomorphic) Lagrangian fibration that is very general in the moduli space of Lagrangian fibrations. We conjecture that the singular fibres in codimension one must be semistable degenerations of abelian varieties. We prove a partial result towards this conjecture, and describe an example that provides further evidence.