Local projectivity of Lagrangian fibrations on Hyperk\"ahler manifolds

TL;DR

The paper proves that Lagrangian fibrations from compact hyperk"ahler manifolds onto projective varieties are locally projective, answering a question by Kamenova and strengthening previous results about fiber projectivity.

Contribution

It establishes the local projectivity of Lagrangian fibrations on hyperk"ahler manifolds without relying on known fiber and base results.

Findings

Lagrangian fibrations are locally projective.

Smooth fibers of such fibrations are projective.

The proof avoids relying on existing fiber and base results.

Abstract

We show that if $f:X\to B$ is a Lagrangian fibration from a compact connected K\"ahler hyperk\"ahler manifold $X$ onto a projective normal variety $B$, then $f$ is locally projective. This answers a question raised by L. Kamenova and strengthens a former result (\cite{Ca05}, Proposition 2.1), according to which the smooth fibres of $f$ are projective. The proof below does not use the known additional results concerning fibres and base in this specific context).