On algebraically coisotropic submanifolds of holomorphic symplectic manifolds

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 . 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.

Endomorphism Algebras and Automorphism Groups of certain Complex Tori

“…It also follows that both H −1,0 (T ) and H 0,−1 (T ) have the same dimension 2g/2 = g. 1 A rational Hodge structure H is called irreducible or simple if its only rational Hodge substructures are H itself and {0} [6, Sect. 2.2].…”

“…The following property of X was introduced and studied by F. Campana [5,Definition 3.3]. (Recently, it was used in the study of coisotropic and lagrangian submanifolds of symplectic manifolds [1].) Definition 1.1.…”