Characteristic foliation on vertical hypersurfaces on holomorphic symplectic manifolds with Lagrangian Fibration

Abstract: Let $Y$ be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold $X$ of dimension $2n$. The characteristic foliation $F$ is the kernel of the symplectic form restricted to $Y$. Assume that $X$ is equipped with a Lagrangian fibration $\pi:X\to B$ and $Y=\pi^{-1}D$, where $D$ is a hypersurface in $B$. It is easy to see that the leaves of $F$ are contained in the fibers of $\pi$. We prove that a very general leaf is Zariski dense in a fiber of $\pi$.