On algebraically coisotropic submanifolds

This is a joint work with F. Campana. Recall that a submanifold $X$ in a holomorphic symplectic manifold $M$ is said to be coisotropic if the corank of the restriction of the holomorphic symplectic form $s$ is maximal possible, that is equal to the codimension of $X$. In particular a hypersurface is always coisotropic. The kernel of the restriction of $s$ defines a foliation on $X$; if it is a fibration, $X$ is said to be algebraically coisotropic. A few years ago we proved that a non-uniruled algebraically coisotropic hypersurface $X\subset M$ is a finite etale quotient of $C\times Y\subset S\times Y$, where $C\subset S$ is a curve in a holomorphic symplectic surface, and $Y$ is arbitrary holomorphic symplectic. We prove some partial results on the higher-codimensional analogue of this, with emphasis on the abelian case.