Lagrangian fibrations on hyperkahler manifolds have no multiple fibers in codimension one

Abstract

Let $f:M\to \mathbb{C}{P}^n$ be a Lagrangian fibration on a compact hyperkahler manifold. We prove that $f$ has no multiple fibers in codimension 1. We prove that this condition is equivalent to the primitivity of the fundamental class of ${\pi }^{-1}(H)$, where $H\subset \mathbb{C}{P}^n$ denotes a hyperplane divisor.

26 pages, v. 1.1, a couple of minor corrections, thanks to E. Shinder