On algebraically coisotropic submanifolds of holomorphic symplectic manifolds

TL;DR

This paper studies algebraically coisotropic submanifolds in holomorphic symplectic manifolds, proving certain structural results especially for abelian varieties and cases where the canonical bundle is semi-ample.

Contribution

It establishes that non-uniruled coisotropic submanifolds in abelian varieties are essentially products involving Lagrangian submanifolds, extending understanding of their structure.

Findings

When $M$ is an abelian variety, the pair $(X,M)$ decomposes into a product up to finite étale cover.

If $K_X$ is nef and big, then $X$ is Lagrangian in $M$.

Lagrangian submanifolds do not exist on sufficiently general Abelian varieties.

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 \'etale 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 hyperk\"ahler.