On the characteristic foliation on a smooth hypersurface in a holomorphic symplectic fourfold
Let X be an irreducible holomorphic symplectic fourfold and D a smooth hypersurface in X. It follows from a result by Amerik and Campana that the characteristic foliation (that is the foliation given by the kernel of the restriction of the symplectic form to D) is not algebraic unless D is uniruled. Suppose now that the Zariski closure of its general leaf is a surface. We prove that X has a lagrangian fibration and D is the inverse image of a curve on its base.