Duke Mathematical Journal

Néron–Severi groups under specialization

Davesh Maulik and Bjorn Poonen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


André used Hodge-theoretic methods to show that in a smooth proper family XB of varieties over an algebraically closed field k of characteristic zero, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to André’s theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is p-adically nowhere dense. Our proof uses the “p-adic Lefschetz (1,1)-theorem” of Berthelot and Ogus, combined with an analysis of p-adic power series. We prove analogous statements for cycles of higher codimension, assuming a p-adic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties.

Article information

Duke Math. J., Volume 161, Number 11 (2012), 2167-2206.

First available in Project Euclid: 24 July 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C25: Algebraic cycles
Secondary: 14D07: Variation of Hodge structures [See also 32G20] 14F25: Classical real and complex (co)homology 14F30: $p$-adic cohomology, crystalline cohomology


Maulik, Davesh; Poonen, Bjorn. Néron–Severi groups under specialization. Duke Math. J. 161 (2012), no. 11, 2167--2206. doi:10.1215/00127094-1699490. https://projecteuclid.org/euclid.dmj/1343133926

Export citation


  • [1] D. Abramovich, K. Karu, K. Matsuki, and J. Włodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), 531–572.
  • [2] Y. André, Pour une théorie inconditionnelle des motifs, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 5–49.
  • [3] P. Berthelot, Cohomologie cristalline des schémas de caractéristique $p>0$, Lecture Notes in Math. 407, Springer, Berlin, 1974.
  • [4] P. Berthelot, A. Grothendieck, and L. Illusie, Théorie des intersections et théorème de Riemann-Roch, with the collaboration of D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud, and J. P. Serre, Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Math. 225, Springer, Berlin, 1971.
  • [5] P. Berthelot and L. Illusie, Classes de Chern en cohomologie cristalline, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1695–A1697; ibid. 270 (1970), A1750–A1752.
  • [6] P. Berthelot and A. Ogus, $F$-isocrystals and de Rham cohomology, I, Invent. Math. 72 (1983), 159–199.
  • [7] E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. $p$, III, Invent. Math. 35 (1976), 197–232.
  • [8] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin, 1990.
  • [9] N. Bourbaki, Elements of Mathematics: Commutative Algebra, chapters 1–7, translated from the French, reprint of the 1989 English translation, Elem. Math. (Berlin), Springer, Berlin, 1998.
  • [10] J. W. S. Cassels, Local Fields, London Math. Soc. Student Texts 3, Cambridge Univ. Press, Cambridge, 1986.
  • [11] O. Debarre, Higher-Dimensional Algebraic Geometry, Universitext, Springer, New York, 2001.
  • [12] P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 259–278.
  • [13] P. Deligne, Théorie de Hodge, II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.
  • [14] P. Deligne, J. S. Milne, A. Ogus, and K. Shih, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900, Springer, Berlin, 1982.
  • [15] M. Emerton, A $p$-adic variational Hodge conjecture and modular forms with complex multiplication, preprint, http://www.math.uchicago.edu/~emerton/pdffiles/cm.pdf (accessed 8 June 2012)
  • [16] G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, with an appendix by David Mumford, Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin, 1990.
  • [17] H. Gillet and W. Messing, Cycle classes and Riemann–Roch for crystalline cohomology, Duke Math. J. 55 (1987), 501–538.
  • [18] M. Green, P. A. Griffiths, and K. H. Paranjape, Cycles over fields of transcendence degree $1$, Michigan Math. J. 52 (2004), 181–187.
  • [19] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95–103.
  • [20] A. Grothendieck, “Crystals and the de Rham cohomology of schemes” in Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, 306–358.
  • [21] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, I: Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960).
  • [22] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961).
  • [23] Grothendieck, A., Éléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, I, Inst. Hautes Études Sci. Publ. Math. 11 (1961).
  • [24] Grothendieck, A., Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, III, Inst. Hautes Études Sci. Publ. Math. 28 (1966).
  • [25] Grothendieck, A., Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967).
  • [26] R. Hartshorne, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 5–99.
  • [27] J. Igusa, An Introduction to the Theory of Local Zeta Functions, AMS/IP Stud. Adv. Math. 14, Amer. Math. Soc., Providence, 2000.
  • [28] S. L. Kleiman, “Algebraic cycles and the Weil conjectures” in Dix esposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, 359–386.
  • [29] S. L. Kleiman, “The Picard scheme” in Fundamental Algebraic Geometry, Math. Surveys Monogr. 123, Amer. Math. Soc., Providence, 2005, 235–321.
  • [30] J. Kürschák, Über Limesbildung und allgemeine Körpertheorie, J. Reine Angew. Math. 142 (1913), 211–253.
  • [31] D. Lampert, Algebraic $p$-adic expansions, J. Number Theory 23 (1986), 279–284.
  • [32] W. E. Lang, On Enriques surfaces in characteristic $p$, I, Math. Ann. 265 (1983), 45–65.
  • [33] J. Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), 151–207.
  • [34] D. W. Masser, Specializations of endomorphism rings of abelian varieties, Bull. Soc. Math. France 124 (1996), 457–476.
  • [35] J. S. Milne, Étale Cohomology, Princeton Math. Ser. 33, Princeton Univ. Press, Princeton, 1980.
  • [36] D. Mumford, Abelian Varieties, with appendices by C. P. Ramanujam and Y. Manin, corrected reprint of the 2nd ed. (1974), Tata Inst. Fund. Res. Stud. Math. 5, published for the Tata Institute of Fundamental Research, Bombay, by Hindustan Book Agency, New Delhi, 2008.
  • [37] D. Mumford, The Red Book of Varieties and Schemes, 2nd expanded ed., includes the Michigan lectures (1974) on curves and their Jacobians, with contributions by Enrico Arbarello, Lecture Notes in Math. 1358, Springer, Berlin, 1999.
  • [38] J. P. Murre, On contravariant functors from the category of pre-schemes over a field into the category of abelian groups (with an application to the Picard functor), Inst. Hautes Études Sci. Publ. Math. 23 (1964), 5–43.
  • [39] A. Néron, Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps, Bull. Soc. Math. France 80 (1952), 101–166.
  • [40] R. Noot, Abelian varieties: Galois representation and properties of ordinary reduction, special issue in honour of Frans Oort, Compositio Math. 97 (1995), 161–171.
  • [41] A. Ogus, $F$-isocrystals and de Rham cohomology, II. Convergent isocrystals, Duke Math. J. 51 (1984), 765–850.
  • [42] F. Oort, Sur le schéma de Picard, Bull. Soc. Math. France 90 (1962), 1–14.
  • [43] M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Math. 119, Springer, Berlin, 1970.
  • [44] P. Ribenboim, The theory of classical valuations, Springer Monogr. Math., Springer, New York, 1999.
  • [45] J.-P. Serre, Lectures on the Mordell-Weil Theorem, 3rd ed., translated from the French and edited by M. Brown from notes by M. Waldschmidt, with a foreword by Brown and Serre, Aspects Math., Vieweg, Braunschweig, 1997.
  • [46] J.-P. Serre, Œuvres. Collected papers, IV: 1985–1998, Springer, Berlin, 2000.
  • [47] T. Shioda, On the Picard number of a complex projective variety, Ann. Sci. École Norm. Sup. (4) 14 (1981), 303–321.
  • [48] T. Terasoma, Complete intersections with middle Picard number $1$ defined over $\textbf{Q}$, Math. Z. 189 (1985), 289–296.
  • [49] R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007), 1–15.
  • [50] C. Voisin, Hodge theory and complex algebraic geometry, II, reprint of the 2003 ed., translated from the French by L. Schneps, Cambridge Stud. Adv. Math. 77, Cambridge Univ. Press, Cambridge, 2007.
  • [51] Voisin, Claire, Hodge loci, preprint, 2010.
  • [52] G. Yamashita, The $p$-adic Lefschetz $(1;1)$ theorem in the semistable case, and the Picard number jumping locus, Math. Res. Letters 18 (2011), 109–126.