Foliations on hypersurfaces in holomorphic symplectic manifolds

TL;DR

This paper studies the structure of foliations induced by holomorphic symplectic forms on hypersurfaces within holomorphic symplectic manifolds, revealing conditions under which the space of leaves inherits a symplectic structure.

Contribution

It characterizes when the foliation on a hypersurface has compact leaves, leading to a new understanding of the symplectic structure on the space of leaves.

Findings

The foliation on the hypersurface can have compact leaves under certain conditions.

The space of leaves inherits a holomorphic symplectic form.

This provides new insights into the geometry of hypersurfaces in symplectic manifolds.

Abstract

Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction $\sigma|_Y$ of the holomorphic symplectic form induces a rank one foliation on Y. We investigate situations where this foliation has compact leaves; in such cases we obtain a space of leaves Y/F which has dimension 2n-2 and admits a holomorphic symplectic form.