  Let k be an algebraically closed field of characteristic 0, and let f be a
morphism of smooth projective varieties from X to Y over the ring k((t)) of
formal Laurent series. We prove that if a general geometric fiber of f is
rationally connected, then the Berkovich analytifications of X and Y are
homotopy equivalent. Two important consequences of this result are that the
homotopy type of the Berkovich analytification of any smooth projective variety
X over k((t)) is a birational invariant of X, and that the Berkovich
analytification of a rationally connected smooth projective variety over k((t))
is contractible.
