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We investigate algebraically coisotropic submanifolds $X$ in a holomorphicsymplectic projective manifold $M$. Motivated by our results in thehypersurface 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$(Z\times Y, N\times Y)$ where $N, Y$ are holomorphic symplectic and $Z\subsetN$ is Lagrangian? We prove that this is indeed the case when $M$ is an abelianvariety, and give some partial answer when the canonical bundle $K_X$ issemi-ample. In particular, when $K_X$ is nef and big, $X$ is Lagrangian in $M$(in fact this also holds without nefness assumption). We also remark thatLagrangian submanifolds do not exist on a sufficiently general Abelian variety,in contrast to the case when $M$ is irreducible hyperk\"ahler.

On algebraically coisotropic submanifolds of holomorphic symplectic manifolds