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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2017: 1.49

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1435-5345
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Volume 2019, Issue 747

# Rational connectivity and analytic contractibility

Morgan Brown
/ Tyler Foster
Published Online: 2016-07-12 | DOI: https://doi.org/10.1515/crelle-2016-0019

## Abstract

Let $k$ be an algebraically closed field of characteristic 0, and let $f:X\to Y$ be a morphism of smooth projective varieties over the ring $k\left(\left(t\right)\right)$ of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the induced map ${f}^{\mathrm{an}}:{X}^{\mathrm{an}}\to {Y}^{\mathrm{an}}$ 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 ${ℙ}^{n}$-bundle over a smooth projective $k\left(\left(t\right)\right)$-variety is homotopy equivalent to the Berkovich analytification of the base, and that the Berkovich analytification of a rationally connected smooth projective variety over $k\left(\left(t\right)\right)$ is contractible.

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Revised: 2016-03-12

Published Online: 2016-07-12

Published in Print: 2019-02-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-0943832

Both authors are partially supported by NSF RTG grant DMS-0943832.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 747, Pages 45–62, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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