Appears in collection : Conférence à la mémoire de Jean-Pierre Demailly

A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. This is quite surprising since there are several examples where the base locus of the anticanonical system has codimension two. In a joint work with Saverio Secci we show that for four-dimensional Fano manifolds the behaviour is completely opposite: if the base locus is a normal surface, hence has codimension two, all the anticanonical divisors are singular. In this talk I will explain how this statement is related to extension problems on K-trivial varieties with a fibre space structure.