Multiplicity of codimension 1 fibers in Lagrangian fibrations

TL;DR

This paper proves that in a Lagrangian fibration of a compact hyper-Kähler manifold, all codimension 1 fibers are multiplicity one, establishing a uniformity property of such fibers.

Contribution

It establishes that every codimension 1 fiber in a Lagrangian fibration of a hyper-Kähler manifold has multiplicity one, a new result in the geometry of these fibrations.

Findings

All codimension 1 fibers have multiplicity 1.

The result applies to Lagrangian fibrations of compact hyper-Kähler manifolds.

It contributes to understanding the structure of fibers in hyper-Kähler fibrations.

Abstract

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-K\"ahler manifold. We prove that every codimension $1$ fiber of $\pi$ has multiplicity $1$.