On algebraically coisotropic submanifolds of holomorphic symplectic manifolds

2022/05/16 by Ekaterina Amerik, Frédéric Campana · 1 citation
Mathematics · #math.AG #math.CV

paper · pdf · doi:10.46298/epiga.2023.10493

Abstract

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× Y, N× Y) where N, Y are holomorphic symplectic and Z⊂ 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.

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• An explicit class of Lagrangian surfaces

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