Lagrangian fibrations onto varieties with isolated quotient singularities

Abstract In this note, we show that if f : M → X f\colon M\rightarrow 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 : M → X f\colon M\rightarrow X is a Lagrangian fibration from a hyper-Kähler fourfold 𝑀 onto a normal surface 𝑋, then X ≅ P 2 X\cong\mathbb{P}^{2} , which recovers a recent result of Huybrechts–Xu and Ou.