Obstructions for codimension one multiple fibers of Lagrangian and Calabi--Yau fibrations

Abstract

We prove that compact hyper-K\"ahler manifolds with a Lagrangian fibration over a projective space have no multiple fibers in codimension one. This has several consequences for the structure of Lagrangian fibrations, including progress on Sawon's conjecture on general singular fibers, Kamenova--Lu anti-hyperbolicity, and the extension of the N\'eron model action to a big open subset of the base. We prove the same result for Calabi--Yau fibrations on simply-connected K-trivial varieties, including elliptic fibrations, with a single exceptional case: an odd-dimensional K-trivial variety with a single fiber of multiplicity 2 over the projective line. This exceptional case is realized by examples of Borisov--Nuer and their generalizations.