Characteristic foliation on a hypersurface of general type in a projective symplectic manifold

TL;DR

This paper investigates when the characteristic foliation on a hypersurface in a projective symplectic manifold is algebraic, proving it is not algebraic if the hypersurface is of general type, using stability theorems and positivity results.

Contribution

It establishes an étale Reeb stability theorem for foliations and applies it to show non-algebraicity of characteristic foliations on general type hypersurfaces.

Findings

Characteristic foliation on hypersurface of general type is not algebraic.

Established an étale Reeb stability theorem in foliation theory.

Connected positivity of direct image sheaves to foliation properties.

Abstract

Given a projective symplectic manifold $M$ and a non-singular hypersurface $X \subset M$, the symplectic form of $M$ induces a foliation of rank 1 on $X$, called the characteristic foliation. We study the question when the characteristic foliation is algebraic, namely, all the leaves are algebraic curves. Our main result is that the characteristic foliation of $X$ is not algebraic if $X$ is of general type. For the proof, we first establish an \'etale version of Reeb stability theorem in foliation theory and then combine it with the positivity of the direct image sheaves associated to families of curves.