Lagrangian Fibrations onto Varieties with Isolated Quotient Singularities

TL;DR

The paper proves that a germ of a projective Lagrangian fibration from a holomorphic symplectic manifold to a normal analytic base with isolated quotient singularities must have a smooth base, using a base-change by a finite group and the Baum–Fulton–Quart singular Lefschetz–Riemann–Roch formula to force a contradiction unless the group action is trivial. This yields a dimension-four corollary: if a fourfold hyper-Kähler manifold maps to a normal surface, the base is $\mathbb{P}^2$, aligning with and extending prior results by Huybrechts–Xu and Ou. The approach closely mirrors the strategy in earlier work but relies on local-to-global fixed-point arguments via singular RR formulas, avoiding some previous global classifications. The paper also discusses extensions to dimension six under terminal singularities and provides examples showing sharpness of the assumptions, including cases with non-isolated singularities and with singular $M$.

Abstract

In this note, we show that if $f\colon M\rightarrow X$ 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 $f\colon M\rightarrow X$ 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.

Theorems & Definitions (26)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof