TITLE: Density of non-algebraic leaves in the characteristic foliation
QUESTION [5 upvotes]: Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has one-dimensional kernel. This gives a non-singular foliation $F$ on $D$.
Is it true that if there exists at least one non-algebraic leaf then the union of all non-algebraic leaves can not be contained in a closed subvariey of $D$?

REPLY [3 votes]: For a foliation on a projective manifold $Y$, the set of points belonging to invariant subschemes with a given Hilbert polynomial is a Zariski closed subset of $Y$. Therefore the set of algebraic leaves is a countable union of Zariski closed subsets. In particular, the set of points belonging to nonalgebraic leaves, if not empty, is dense both in Zariski and in Euclidean topology.
If the ambient is compact Kahler then the analogue statement probably holds true. Without the Kahler assumption I am not sure what can be said.