Abstract:
This is a joint work with F. Campana. Recall that a
submanifold $X$ in a holomorphic symplectic manifold $M$ is said to
be coisotropic if the corank of the restriction of the holomorphic
symplectic form $s$ is maximal possible, that is equal to the
codimension of $X$. In particular a hypersurface is always
coisotropic. The kernel of the restriction of $s$ defines a
foliation on $X$; if it is a fibration, $X$ is said to be
algebraically coisotropic. A few years ago we proved that a
non-uniruled algebraically coisotropic hypersurface $X\subset M$ is
a finite etale quotient of $C\times Y\subset S\times Y$, where
$C\subset S$ is a curve in a holomorphic symplectic surface, and $Y$
is arbitrary holomorphic symplectic. We prove some partial results
on the higher-codimensional analogue of this, with emphasis on the
abelian case.