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 $F\subset T_D$ of rank 1, the characteristic foliation. The picture is complete in dimension 4 and shows that the behaviour of the leaves of $F$ on $D$ is determined by the Beauville–Bogomolov square $q\left(D\right)$ of $D$. In higher dimensions, some of the results depend on the abundance conjecture for $D$.