Lagrangian fibrations on symplectic fourfolds

For a Lagrangian fibration from a projective irreducible symplectic manifold to a normal variety one is interested in a description of the base variety. In all known examples of such fibrations the base is isomorphic to $\mathbb{P}^n $. According to Matsushita, the base of such a fibration is a $\mathbb{Q}$- factorial log terminal Fano variety of Picard number $1$. Following Wenhao Ou, we will discuss the proof of a theorem asserting that in the case of a Lagrangian fibration with the $4$-dimensional total space the base is either isomorphic to $\mathbb{P}^2$ or a del Pezzo surface with an $E_8$ singularity and two nodal rational curves in its anticanonical linear system.