Lagrangian Fibrations onto Varieties with Isolated Quotient Singularities

TL;DR

This paper proves that a Lagrangian fibration from a holomorphic symplectic manifold onto a variety with isolated quotient singularities implies the base is smooth, with specific results for hyper-Kähler fourfolds mapping onto surfaces.

Contribution

It establishes the smoothness of the base variety in Lagrangian fibrations with isolated quotient singularities, extending recent results for hyper-Kähler fourfolds.

Findings

Base variety is smooth under the given conditions.

For hyper-Kähler fourfolds, the base is isomorphic to the projective plane.

Recovers recent results by Huybrechts--Xu and Ou.

Abstract

In this note, we show that if $f\colon M\rightarrow X$ is a germ of a projective Lagrangian fibration from a holomorphic symplectic manifold $M$ onto a normal analytic variety $X$ with isolated quotient singularities, then $X$ is smooth. In particular, if $f\colon M\rightarrow X$ is a Lagrangian fibration from a hyper-K\"ahler fourfold $M$ onto a normal surface $X$, then $X\cong \mathbb{P}^2$, which recovers a recent result of Huybrechts--Xu and Ou.