Base manifolds for fibrations of projective irreducible symplectic manifolds

TL;DR

This paper proves that for a certain class of symplectic manifolds with a fibration structure, the base manifold must be a projective space, using geometric structures like affine structures and minimal rational tangents.

Contribution

It establishes that the base of a Lagrangian fibration of a projective irreducible symplectic manifold is necessarily a projective space, combining affine and rational tangent structures.

Findings

The base manifold is biholomorphic to a projective space.

The proof uses affine structures from Lagrangian fibrations.

Variety of minimal rational tangents characterizes the base.

Abstract

Given a projective irreducible symplectic manifold $M$ of dimension $2n$, a projective manifold $X$ and a surjective holomorphic map $f:M \to X$ with connected fibers of positive dimension, we prove that $X$ is biholomorphic to the projective space of dimension $n$. The proof is obtained by exploiting two geometric structures at general points of $X$: the affine structure arising from the action variables of the Lagrangian fibration $f$ and the structure defined by the variety of minimal rational tangents on the Fano manifold $X$.