Multiplicity of codimension 1 fibers in Lagrangian fibrations

Abstract

Let $\pi :X\to B$ be a Lagrangian fibration of a compact hyper-Kähler manifold. We prove that every codimension $1$ fiber of $\pi$ has multiplicity $1$.

Theorem 2.1 is false. The error appears when applying Lemma 2.5: the cyclic covering construction fixes an $m$-th root line bundle $L$. The pullback of the $m$-th root line bundle on $B$ does not coincide with the $m$-th root line bundle on $X$. The diagram (2.6) therefore does not exist, and $Y$ can be disconnected