Let k be an algebraically closed field of characteristic 0, and let f : X → Y be a morphism of smooth projective varieties over the ring k((t)) of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the induced map f an W Xan → Y an between the Berkovich analytifications of X and Y is a homotopy equivalence. Two important consequences of this result are that the Berkovich analytification of any Pn-bundle over a smooth projective k((t))-variety is homotopy equivalent to the Berkovich analytification of the base, and that the Berkovich analytification of a rationally connected smooth projective variety over k((t)) is contractible.
ASJC Scopus subject areas
- Applied Mathematics