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 étale cover, the pair $(X,M)$ is a product $(Z\times Y, N\times Y)$ where $N, Y$ are holomorphic symplectic and $Z\subset N$ is Lagrangian? We prove that this is indeed the case when $M$ is an abelian variety, and give some partial answer when the canonical bundle $K_X$ is semi-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 that Lagrangian submanifolds do not exist on a sufficiently general Abelian variety, in contrast to the case when $M$ is irreducible hyperkähler.

• 14J42 - Holomorphic symplectic varieties, hyper-Kähler varieties
• 14K12 - Subvarieties of abelian varieties
• 53D12 - Lagrangian submanifolds; Maslov index

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