Title: A Global Liouville–Arnold Theorem for Algebraic Completely Integrable Systems

Abstract: I will report on joint work with Y. Kim and C. Schnell. We prove that a Lagrangian fibration with a smooth base admits a smooth group scheme over the entire base, acting on the total space in such a way that the action is simply transitive on the smooth fibers.

The group scheme is constructed by integrating vertical vector fields, in the spirit of the Liouville–Arnold theorem, but significant difficulties arise from possible singularities of both the total space and the fibration.

Our results apply to complex algebraic varieties as well as Kähler spaces. If time permits, I will also discuss topological consequences for Lagrangian fibrations.