Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-19T19:25:22.160Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigid-analytic varieties with projective reduction violating Hodge symmetry

Part of: Abelian varieties and schemes Arithmetic problems. Diophantine geometry Cycles and subschemes

Published online by Cambridge University Press:  19 March 2021

Alexander Petrov
Alexander Petrov*
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA02138, USAapetrov@math.harvard.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct examples of smooth proper rigid-analytic varieties admitting formal models with projective special fibers and violating Hodge symmetry for cohomology in degrees ${\geq }3$. This answers negatively the question raised by Hansen and Li.

Keywords

rigid-analytic varietiesHodge symmetryp-adic Hodge theory

MSC classification

Primary: 14G20: Local ground fields
Secondary: 14G22: Rigid analytic geometry 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14K22: Complex multiplication
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 3 , March 2021 , pp. 625 - 640
Check for updates
Copyright
© The Author(s) 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achinger, P., Hodge symmetry for rigid varieties via log Hard Lefschetz, Preprint (2020), arXiv:2005.02246.Google Scholar
Antieau, B., Bhatt, B. and Mathew, A., Counter examples to Hochschild–Kostant–Rosenberg in characteristic $p$, Forum Math. Sigma, to appear. Preprint (2019), arXiv:1909.11437.Google Scholar
Bhatt, B., Morrow, M. and Scholze, P., Integral $p$-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219–397.10.1007/s10240-019-00102-zCrossRefGoogle Scholar
Bosch, S. and Lütkebohmert, W., Formal and rigid geometry. I. Rigid spaces, Math. Ann. 295 (1993), 291–317.10.1007/BF01444889CrossRefGoogle Scholar
Chai, C.-L., Conrad, B. and Oort, F., Complex multiplication and lifting problems, Mathematical Surveys and Monographs, vol. 195 (American Mathematical Society, Providence, RI, 2014).Google Scholar
Fontaine, J.-M., Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, in Journées de Géométrie Algébrique de Rennes, Rennes, 1978, volume III, Astérisque, vol. 65 (Société Mathématique de France, Paris, 1979), 3–80.Google Scholar
Fontaine, J.-M. and Messing, W., p-adic periods and p-adic étale cohomology, in Current trends in arithmetical algebraic geometry, Arcata, CA, 1985, Contemporary Mathematics, vol. 67 (American Mathematical Society, Providence, RI, 1987), 179–207.10.1090/conm/067/902593CrossRefGoogle Scholar
Fontaine, J.-M. and Rapoport, M., Existence de filtrations admissibles sur des isocristaux, Bull. Soc. Math. France 133 (2005), 73–86.10.24033/bsmf.2479CrossRefGoogle Scholar
Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics, second edition, vol. 2 (Springer, Berlin, 1998).10.1007/978-1-4612-1700-8CrossRefGoogle Scholar
Hansen, D. and Li, S., Line bundles on rigid varieties and Hodge symmetry, Math. Z. 296 (2020), 1777–1786.10.1007/s00209-020-02535-3CrossRefGoogle Scholar
Katz, N.M., Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Publ. Math. Inst. Hautes Études Sci. 39 (1970), 175–232.10.1007/BF02684688CrossRefGoogle Scholar
Katz, N.M. and Messing, W., Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77.10.1007/BF01405203CrossRefGoogle Scholar
Messing, W., The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, vol. 264 (Springer, Berlin, 1972).10.1007/BFb0058301CrossRefGoogle Scholar
Raynaud, M., “p-torsion” du schéma de Picard, in Journées de Géométrie Algébrique de Rennes, Rennes, 1978, volume II, Astérisque, vol. 64 (Société Mathématique de France, Paris, 1979), 87–148.Google Scholar
Rogov, V., Non-algebraic deformations of flat Kähler manifolds, Math. Res. Lett., to appear. Preprint (2019), arXiv:1911.00798.Google Scholar
Scholze, P., $p$-adic Hodge theory for rigid-analytic varieties, Forum Math. Pi 1 (2013), 77.10.1017/fmp.2013.1CrossRefGoogle Scholar
Scholze, P., p-adic geometry, in Proceedings of the International Congress of Mathematicians, Rio de Janeiro, 2018, volume I, Plenary lectures (World Scientific, Hackensack, NJ, 2018).Google Scholar
Grothendieck, A., Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Lecture Notes in Mathematics, vol. 224 (Springer Berlin 1971).Google Scholar
Deligne, P. and Katz, N., Groupes de monodromie en géométrie algébrique II, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lecture Notes in Mathematics, vol. 340 (Springer Berlin 1973).Google Scholar
Suh, J., Symmetry and parity in Frobenius action on cohomology, Compos. Math. 148 (2012), 295–303.10.1112/S0010437X11007056CrossRefGoogle Scholar
Tate, J., Classes d'isogénie des variétés abéliennes sur un corps fini (d'après T. Honda), in Séminaire Bourbaki 1968/69: Exposés 347–363, Lecture Notes in Mathematics, vol. 175 (Springer, Berlin, 2071), Exposé 352, 95–110.Google Scholar
Totaro, B., Hodge theory of classifying stacks, Duke Math. J. 167 (2018), 1573–1621.10.1215/00127094-2018-0003CrossRefGoogle Scholar