Lagrangian fibrations (Q2107598)

A fibration of a hyperkähler manifold \(X\) of complex dimension \(2n\) is a proper surjective morphism \(f : X \rightarrow B\) with connected fibers. Assume that the base \(B\) is a normal variety of dimension \(0 < \dim (B) <2n\). By results of Matsushita, any fibration of a hyperkähler manifold is Lagrangian with respect to the holomorphic symplectic form. In this survey the authors review certain aspects of the theory of Langrangian fibrations of hyperkähler manifolds, emphasizing results by Matsushita about the general structure of a Lagrangian fibration, and results of \textit{J. Shen} and \textit{Q. Yin} [Duke Math. J. 171, No. 1, 209--241 (2022; Zbl 1490.14019)] and \textit{A. Harder} et al. [Forum Math. Sigma 9, Paper No. e50, 6 p. (2021; Zbl 1467.14101)] about \(P=W\) for Lagrangian fibrations and degenerations of hyperkähler manifolds.