  Given a projective irreducible symplectic manifold $M$ of dimension $2n$, a
projective manifold $X$ and a surjective holomorphic map $f:M \to X$ with
connected fibers of positive dimension, we prove that $X$ is biholomorphic to
the projective space of dimension $n$. The proof is obtained by exploiting two
geometric structures at general points of $X$: the affine structure arising
from the action variables of the Lagrangian fibration $f$ and the structure
defined by the variety of minimal rational tangents on the Fano manifold $X$.
