Lagrangian Fibrations onto Varieties with Isolated Quotient Singularities

In this note, we show that if $f\text{\hspace{0.17em}⁣}:M\to X$ is a germ of a projective Lagrangian fibration from a holomorphic symplectic manifold $M$ onto a normal analytic variety $X$ with isolated quotient singularities, then $X$ is smooth. In particular, if $f\text{\hspace{0.17em}⁣}:M\to X$ is a Lagrangian fibration from a hyper-Kähler fourfold $M$ onto a normal surface $X$, then $X\cong {\mathbb{P}}^2$, which recovers a recent result of Huybrechts–Xu and Ou.