On the discriminant locus of a Lagrangian fibration

Let $X\to\P^n$ be an irreducible holomorphic symplectic manifold of dimension $2n$ fibred over $\P^n$. Matsushita proved that the generic fibre is a holomorphic Lagrangian abelian variety. In this article we study the discriminant locus $\Delta\subset\P^n$ parametrizing singular fibres. Our main result is a formula for the degree of $\Delta$, leading to bounds on the degree when $X$ is a four-fold.