Characteristic foliation on the discriminant hypersurface of a holomorphic Lagrangian fibration

We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujiki's class C. Our approach is based on the study of the characteristic vector field of the discriminant hypersurface, which naturally arises from the defining equation of the hypersurface via the symplectic form. As an application, we show that the characteristic foliation of the discriminant hypersurface has algebraic leaves which are either rational curves or smooth elliptic curves.