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 $D$.
Categories: math.AG