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Simply connected varieties in characteristic $p>0$

Part of: Arithmetic problems. Diophantine geometry Local theory (Co)homology theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  09 October 2015

Hélène Esnault ,
Vasudevan Srinivas  and
Jean-Benoît Bost
Hélène Esnault
Affiliation:
Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany email esnault@math.fu-berlin.de
Vasudevan Srinivas
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai-400005, India email srinivas@math.tifr.res.in
Jean-Benoît Bost
Affiliation:
Département de Mathématiques, Université Paris-Sud, Bât. 425, 91405, Orsay, France email jean-benoit.bost@math.u-psud.fr
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Abstract

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We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over an algebraic closure of a finite field if the variety admits a normal projective compactification with boundary locus of codimension greater than or equal to $2$.

Keywords

stratified bundlesfundamental groupsformal and algebraic geometry

MSC classification

Primary: 11G99: None of the above, but in this section 14B20: Formal neighborhoods 14G17: Positive characteristic ground fields 14G99: None of the above, but in this section 14F35: Homotopy theory; fundamental groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 2 , February 2016 , pp. 255 - 287
Copyright
© The Authors 2015 

References

Andreotti, A., Théorèmes de dépendance algébrique sur les espaces complexes pseudo-concaves, Bull. Soc. Math. France 91 (1963), 1–38.Google Scholar
Bădescu, L., Algebraic surfaces, Universitext (Springer, 2001).CrossRefGoogle Scholar
Bădescu, L., Projective geometry and formal geometry (Birkhäuser, Basel, 2004).CrossRefGoogle Scholar
Bost, J.-B., Algebraic leaves of algebraic foliations over number fields, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 161–221.CrossRefGoogle Scholar
Bost, J.-B. and Chambert-Loir, A., Analytic curves in algebraic varieties over number fields, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, Vol. I, Progress in Mathematics, vol. 269 (Birkhäuser, Boston, MA, 2009), 69–124.CrossRefGoogle Scholar
Chen, H., Algebraicity of formal varieties and positivity of vector bundles, Math. Ann. 354 (2012), 171–192.CrossRefGoogle Scholar
Debarre, O., Théorèmes de connexité pour les produits d’espaces projectifs et les grassmanniennes, Amer. J. Math. 118 (1996), 1347–1367.Google Scholar
Deligne, P. and Milne, J., Tannakian categories, Lecture Notes in Mathematics, vol. 900 (Springer, 1982).CrossRefGoogle Scholar
Deligne, P., Letter to Hélène Esnault, dated 7 January 2014.Google Scholar
dos Santos, J.-P.-S., Fundamental group schemes for stratified sheaves, J. Algebra 317 (2007), 691–713.CrossRefGoogle Scholar
Grothendieck, A. and Dieudonné, J. A., Eléments de géométrie algébrique. I, Grundlehren der Mathematischen Wissenschaften, vol. 166 (Springer, Berlin, 1971).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Publ. Math. Inst. Hautes Études Sci. 11 (1961), 167.Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. Inst. Hautes Études Sci. 32 (1967), 361.Google Scholar
Grothendieck, A., Élements de géométrie algébrique: IV Études locales des schémas et des morphismes de schémas, Publ. Math. Inst. Hautes Études Sci. 24 (1965), 28 (1966).Google Scholar
Esnault, H., On flat bundles in characteristic 0 and p > 0, in European congress of mathematics, Krakow, 2–7 July (European Mathematical Society, 2012), 301–313.Google Scholar
Esnault, H. and Mehta, V., Simply connected projective manifolds in characteristic p > 0 have no nontrivial stratified bundles, Invent. Math. 181 (2010), 449–465 (Erratum http://www.mi.fu-berlin.de/users/esnault/helene_publ.html).CrossRefGoogle Scholar
Fulton, W. and Hansen, J., A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings, Ann. of Math. (2) 110 (1979), 159–166.CrossRefGoogle Scholar
Fulton, W. and Lazarsfeld, R., Connectivity and its applications in algebraic geometry, in Algebraic geometry (Chicago, Ill., 1980), Lecture Notes in Mathematics, vol. 862 (Springer, Berlin–New York, NY, 1981), 26–92.CrossRefGoogle Scholar
Gieseker, D., Flat vector bundles and the fundamental group in non-zero characteristics, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), 1–31.Google Scholar
Gieseker, D., On two theorems of Griffiths about embeddings with ample normal bundle, Amer. J. Math. 99 (1977), 1137–1150.CrossRefGoogle Scholar
Grothendieck, A., Représentations linéaires et compactifications profinies des groupes discrets, Manuscripta Math. 2 (1970), 375–396.CrossRefGoogle Scholar
Hartshorne, R., Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403–450.CrossRefGoogle Scholar
Hartshorne, R., Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, vol. 156 (Springer, Berlin, 1970).CrossRefGoogle Scholar
Hartshorne, R., Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176.CrossRefGoogle Scholar
Hartshorne, R., Algebraic geometry., Graduate Texts in Mathematics, vol. 52, corrected third printing (Springer, New York, NY–Heidelberg–Berlin, 1983).Google Scholar
Hironaka, H., On some formal imbeddings, Illinois J. Math. 12 (1968), 587–602.CrossRefGoogle Scholar
Hironaka, H. and Matsumura, H., Formal functions and formal embeddings, J. Math. Soc. Japan 20 (1968), 52–82.CrossRefGoogle Scholar
Hrushovski, E., The elementary theory of Frobenius automorphisms, Preprint (2004),arXiv:math/0406514v1 [math.LO].Google Scholar
Huybrechts, D. and Lehn, M., The geometry of moduli spaces of sheaves, Aspects of Mathematics, vol. E31 (Vieweg, Braunschweig, 1997).CrossRefGoogle Scholar
Jouanolou, J.-P., Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42 (Birkhäuser Boston, Boston, MA, 1983).Google Scholar
Kindler, L., Regular stratified bundles and tame ramification, Trans. Amer. Math. Soc. 367 (2015), 6461–6485.CrossRefGoogle Scholar
Langer, A., Chern classes of reflexive sheaves on normal surfaces, Math. Z. 235 (2000), 591–614.CrossRefGoogle Scholar
Langer, A., Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251–276.CrossRefGoogle Scholar
Lange, H. and Stuhler, U., Vektorbündel auf kurven und darstellungen der algebraischen fundamentalgruppe, Math. Z. 156 (1977), 73–83.CrossRefGoogle Scholar
Malčev, A., On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S. 8 (1940), 405–422.Google Scholar
Milne, J., Étale cohomology (Princeton University Press, 1980).Google Scholar
Nagata, M., Local rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13 (John Wiley & Sons, New York–London, 1962).Google Scholar
Osserman, B., The generalized Verschiebung map for curves of genus 2, Math. Ann. 336 (2006), 963–986.CrossRefGoogle Scholar
Phùng, H.-H., Gauss–Manin stratification and stratified fundamental group schemes, Ann. Inst. Fourier (Grenoble) 63 (2013), 2267–2285.CrossRefGoogle Scholar
Poonen, B., Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), 1099–1127.CrossRefGoogle Scholar
Ramanujam, C. P., Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. (N.S.) 36 (1972), 41–51.Google Scholar
Saavedra Rivano, N., Catégories tannakiennes, Lecture Notes in Mathematics, vol. 265 (Springer, 1972).CrossRefGoogle Scholar
Seidenberg, A., The hyperplane sections of normal varieties, Trans. Amer. Math. Soc. 69 (1950), 357–386.CrossRefGoogle Scholar
Grothendieck, A., Séminaire de Géométrie Algébrique du Bois-Marie – 1960–61 – Revêtements étales et groupe fondamental (SGA1), Lecture Notes in Mathematics, vol. 224, édition recomposée et annotée (Springer, 1971).Google Scholar
Grothendieck, A., Séminaire de Géométrie Algébrique du Bois-Marie – 1962 – Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA2), Advanced Studies in Pure Mathematics, vol. 2 (North-Holland, Amsterdam, 1968).Google Scholar
Artin, M., Grothendieck, A. and Verdier, J.-L., Séminaire de Géométrie Algébrique du Bois-Marie – 1963–64 – Théorème de finitude pour un morphisme propre; dimension cohomologique des schémas algébriques affines (SGA4), Lecture Notes in Mathematics, vol. 305 (Springer, 1973), 145–168.Google Scholar
Zariski, O. and Samuel, P., Commutative algebra, vol. II, Graduate Texts in Mathematics, vol. 29 (Springer, New York, NY, 1975); reprint of the 1960 edition.Google Scholar