Abstract In this note, we show that if f: M → X f M 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 M X is a Lagrangian fibration from a hyper-Kähler fourfold 𝑀 onto a normal surface 𝑋, then X ≅ P 2 X^2, which recovers a recent result of Huybrechts–Xu and Ou.
Müller et al. (Wed,) studied this question.