# Lagrangian fibrations of hyperk\"ahler fourfolds

## TL;DR

This paper proves that the base surface of a Lagrangian fibration on a projective irreducible symplectic fourfold is always isomorphic to the projective plane, revealing a specific geometric structure.

## Contribution

It establishes that the base of such a fibration must be the projective plane, providing a new classification result for these hyperk"ahler fourfolds.

## Key findings

- The base surface is isomorphic to ${
m P}^2$.
- Lagrangian fibrations on these fourfolds have a fixed base geometry.
- The result constrains the possible structures of hyperk"ahler fourfolds.

## Abstract

The base surface $B$ of a Lagrangian fibration $X\twoheadrightarrow B$ of a projective, irreducible symplectic fourfold $X$ is shown to be isomorphic to ${\mathbb P}^2$.