Characteristic foliations -- a survey

TL;DR

This survey comprehensively reviews recent results on the geometry of characteristic foliations on smooth divisors in compact hyperk"ahler manifolds, highlighting the role of the Beauville-Bogomolov form and the abundance conjecture.

Contribution

It consolidates and explains recent advances in understanding characteristic foliations on hyperk"ahler divisors, including proofs and the influence of the Beauville-Bogomolov form.

Findings

Complete description of leaf behavior in dimension four

Dependence of results on the Beauville-Bogomolov square

Higher-dimensional results linked to the abundance conjecture

Abstract

This is a survey article, with essentially complete proofs, of a series of recent results concerning the geometry of the characteristic foliation on smooth divisors in compact hyperk\"ahler manifolds, starting with work by Hwang-Viehweg, but also covering articles by Amerik-Campana and Abugaliev. The restriction of the holomorphic symplectic form on a hyperk\"ahler manifold $X$ to a smooth hypersurface $D\subset X$ leads to a regular foliation ${\mathcal F}\subset{\mathcal T}_D$ of rank one, the characteristic foliation. The picture is complete in dimension four and shows that the behavior of the leaves of ${\mathcal F}$ on $D$ is determined by the Beauville-Bogomolov square $q(D)$ of $D$. In higher dimensions, some of the results depend on the abundance conjecture for $D$.