### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 45-62 |

Number of pages | 18 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 2019 |

Issue number | 747 |

DOIs | |

State | Published - Feb 1 2019 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal fur die Reine und Angewandte Mathematik*,

*2019*(747), 45-62. https://doi.org/10.1515/crelle-2016-0019