Characteristic foliations -- a survey

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 hyperkähler 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 hyperkähler 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 $D$.

21 pages, minor corrections. To appear in Bulletin of the London Mathematical Society