Fixed divisors on hyperk\"ahler manifolds

TL;DR

This paper investigates the properties of fixed divisors on hyperk"ahler manifolds, demonstrating that the fixed part of certain linear systems is reduced and exploring the implications for fibrations in four-dimensional cases.

Contribution

It establishes the reducedness of fixed parts of linear systems on hyperk"ahler manifolds and links fixed divisors to Lagrangian fibrations in four dimensions.

Findings

Fixed part of |A| is reduced for nef and big divisors A.

|2A| is shown to be mobile.

In dimension four, fixed divisors induce Lagrangian fibrations.

Abstract

Let $X$ be a hyperk\"ahler manifold, and let $A$ be a nef and big divisor on $X$. We show that the fixed part of the linear system $|A|$ is reduced and as a consequence $|2A|$ is mobile. If $X$ has dimension four we also show that if the fixed part of $|A|$ is not empty, the mobile part induces a (rational) Lagrangian fibration.