Jump to ContentJump to Main Navigation
Show Summary Details
More options …

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

See all formats and pricing
More options …
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


Let k be an algebraically closed field of characteristic 0, and let f:XY 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 fan:XanYan 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 n-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.


  • [1]

    V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr. 33, American Mathematical Society, Providence 1990. Google Scholar

  • [2]

    C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. Google Scholar

  • [3]

    F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), no. 5, 539–545. CrossrefGoogle Scholar

  • [4]

    O. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer, New York 2001. Google Scholar

  • [5]

    T. de Fernex, J. Kollár and C. Xu, The dual complex of singularities, preprint (2012), http://arxiv.org/abs/1212.1675.

  • [6]

    T. Graber, J. Harris and J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67. CrossrefGoogle Scholar

  • [7]

    M. J. Greenberg, Rational points in Henselian discrete valuation rings, Publ. Math. Inst. Hautes Études Sci. 31 (1966), 59–64. CrossrefGoogle Scholar

  • [8]

    A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci. 28 (1966), 1–255. Google Scholar

  • [9]

    A. Hogadi and C. Xu, Degenerations of rationally connected varieties, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3931–3949. CrossrefGoogle Scholar

  • [10]

    J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin 1996. Google Scholar

  • [11]

    J. Kollár, A conjecture of Ax and degenerations of Fano varieties, Israel J. Math. 162 (2007), 235–251. Web of ScienceCrossrefGoogle Scholar

  • [12]

    J. Kollár, Y. Miyaoka and S. Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. Google Scholar

  • [13]

    J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge 1998. Google Scholar

  • [14]

    J. Kollár, J. Nicaise and C. Xu, Semi-stable extensions over 1-dimensional bases, preprint (2015), http://arxiv.org/abs/1510.02446.

  • [15]

    M. Kontsevich and Y. Soibelman, Affine structures and non-Archimedean analytic spaces, The unity of mathematics, Progr. Math. 244, Birkhäuser, Boston (2006), 321–385. Google Scholar

  • [16]

    M. Mustaţă and J. Nicaise, Weight functions on non-Archimedean analytic spaces and the Kontsevich–Soibelman skeleton, preprint (2013), http://arxiv.org/abs/1212.6328v3.

  • [17]

    J. Nicaise, Singular cohomology of the analytic Milnor fiber, and mixed Hodge structure on the nearby cohomology, J. Algebraic Geom. 20 (2011), no. 2, 199–237. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    J. Nicaise, Berkovich skeleta and birational geometry, preprint (2014), http://arxiv.org/abs/1409.5229.

  • [19]

    J. Nicaise and C. Xu, The essential skeleton of a degeneration of algebraic varieties, preprint (2013), http://arxiv.org/abs/1307.4041.

  • [20]

    P. Vojta, Nagata’s embedding theorem, preprint (2007), http://arxiv.org/abs/0706.1907.

  • [21]

    T. Y. Yu, Gromov compactness in non-archimedean analytic geometry, preprint (2014), http://arxiv.org/abs/1401.6452.

About the article

Received: 2014-07-02

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, DOI: https://doi.org/10.1515/crelle-2016-0019.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in