On the characteristic foliation on a smooth hypersurface in a holomorphic symplectic fourfold

TL;DR

This paper investigates the characteristic foliation on smooth hypersurfaces within holomorphic symplectic fourfolds, establishing conditions under which the fourfold admits a Lagrangian fibration and describing the hypersurface as a pullback of a curve.

Contribution

It proves that if the Zariski closure of a general leaf of the characteristic foliation is a surface, then the fourfold admits a Lagrangian fibration with the hypersurface as a pullback of a curve.

Findings

The fourfold admits a Lagrangian fibration under the given conditions.

The hypersurface is the inverse image of a curve on the base of the fibration.

The characteristic foliation's leaves have Zariski closures that influence the global geometry.

Abstract

Let $X$ be an irreducible holomorphic symplectic fourfold and $D$ a smooth hypersurface in $X$. It follows from a result by Amerik and Campana that the characteristic foliation (that is the foliation given by the kernel of the restriction of the symplectic form to $D$) is not algebraic unless $D$ is uniruled. Suppose now that the Zariski closure of its general leaf is a surface. We prove that $X$ has a lagrangian fibration and $D$ is the inverse image of a curve on its base.