Lagrangian fibrations onto varieties with isolated quotient singularities

In this note, we show that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi>M</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi>X</m:mi> </m:mrow> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi>M</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi>X</m:mi> </m:mrow> </m:mrow> </m:math> f M→ X is a Lagrangian fibration from a hyper-Kähler fourfold 𝑀 onto a normal surface 𝑋, then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo>≅</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> X² , which recovers a recent result of Huybrechts–Xu and Ou.