On algebraically coisotropic submanifolds of holomorphic symplectic manifolds

We investigate algebraically coisotropic submanifolds X in a holomorphic symplectic projective manifold M. Motivated by our results in the hypersurface case, we raise the following question: when X is not uniruled, is it true that up to a finite etale cover, the pair (X, M) is a product (Z x Y,N x Y ), where N, Y are holomorphic symplectic and Z subset of N is Lagrangian? We prove that this is indeed the case when M is an Abelian variety and give a partial answer when the canonical bundle K-X is semiample. In particular, when K(X )is nef and big, X is Lagrangian in M (using a recent text of Taji, we could also obtain this for X of general type). We also remark that Lagrangian submanifolds do not exist on a sufficiently general Abelian variety, in contrast to the case when M is irreducible hyperkahler.