Skip to main content
SpringerLink
Log in

Sur la convexité holomorphe des revêtements linéaires réductifs d’une variété projective algébrique complexe

  • Published:
Inventiones mathematicae Aims and scope

Abstract

Let X be a smooth connected compact projective variety over ℂ. We study the Shafarevich conjecture concerning holomorphic convexity along the lines of Kollár’s approach, when the fundamental group of X admits large finite dimensionnal representations. We prove that, given n∈ℕ, the topological covering space \(\widetilde{X}_{n}\) of a projective algebraic compact complex manifold X corresponding to the intersection of the kernels of all linear reductive representations of π1(X) to GL n (ℂ) is holomorphically convex. In the surface case, this is a corollary of a theorem due to Katzarkov and Ramachandran.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Ergeb. Math. Grenzgeb., 3. Folge, Band 4. Springer 1989

  2. Bedford, E., Taylor, B.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)

    MathSciNet  MATH  Google Scholar 

  3. Bochnak, J., Coste, M., Roy, M.F.: Géometrie algébrique réelle. Ergeb. Math. Grenzgeb., 3. Folge, Band 12. Springer 1987

  4. Bogomolov, F., Katzarkov, L.: Complex projective surfaces and infinite groups. Geom. Funct. Anal. 8, 243–272 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buchdahl, N.: On compact Kähler surfaces. Ann. Inst. Fourier 49, 287–302 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Bruhat, F., Tits, J.: Groupes réductifs sur un corps local, I. Données radicielles valuées. Publ. Math., Inst. Hautes Étud. Sci. 41, 5–251 (1972)

    Google Scholar 

  7. Brown, K.: Buildings. Springer 1989

  8. Brylinski, J.L.: Loop Spaces, Characteristic Classes and Geometric Quantization. Prog. Math. 107. Birkhäuser 1992

  9. Campana, F.: Coréductions algébriques d’un espace analytique faiblement kählérien compact. Invent. Math. 63, 187–223 (1981)

    MathSciNet  MATH  Google Scholar 

  10. Campana, F.: Remarques sur le revêtement universel des variétés kähleriennes. Bull. Soc. Math. Fr. 122, 255–284 (1994)

    MathSciNet  MATH  Google Scholar 

  11. Campana, F.: Fundamental group and positivity of cotangent bundles of compact Kähler manifolds. J. Algebr. Geom. 4, 487–502 (1995)

    MathSciNet  MATH  Google Scholar 

  12. Campana, F., Peternell, T.: Algebraicity of the ample cone of projective varieties. J. Reine Angew. Math. 407, 160–166 (1990)

    MathSciNet  MATH  Google Scholar 

  13. Carlson, J., Toledo, D.: Harmonic mappings of Kähler manifolds to locally symmetric spaces. Publ. Math., Inst. Hautes Étud. Sci. 69, 173–201 (1989)

    Google Scholar 

  14. Cartan, H.: Oeuvres Collected Works. Remmert, R., Serre, J.P. (éd.). Springer 1979

  15. Castelnuovo, G.: Sulle superficie aventi il genere aritmetico negativo. Rend. Circ. Mat. Palermo, II. Ser. 20, 55–60 (1905)

    Google Scholar 

  16. Catanese, F.: Moduli and classification of irregular Kähler manifolds (and algebraic varieties) with Albanese general type fibrations. Invent. Math. 104, 263–288 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Corlette, K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28, 361–382 (1988)

    MathSciNet  MATH  Google Scholar 

  18. Corlette, K.: Non abelian Hodge theory. Proc. Symp. Pure Math. 54, 125–140 (1993)

    MATH  Google Scholar 

  19. De Franchis, M.: Sulle superficie algebriche le quali contengono un fascio irrazionale di curve. Rend. Circ. Mat. Palermo, II. Ser. 20, 49–54 (1905)

    Google Scholar 

  20. Demailly, J.P., Paun, M.: Numerical characterization of the Kähler cone of a compact Kähler manifold. math.AG/0105176. To appear in Ann. Math.

  21. Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)

    MATH  Google Scholar 

  22. Eyssidieux, P.: La caractéristique d’Euler du complexe de Gauss-Manin. J. Reine Angew. Math. 490,155–212 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Eyssidieux, P.: Un théoréme de Nakai-Moishezon pour certaines classes de type (1,1). Prépublication n. 133 du Laboratoire Emile Picard. math.AG/9811065. Unpublished

  24. Fornaess, J.E., Narasimhan, R.: The Levi problem on complex spaces with singularities. Math. Ann. 248, 47–72 (1980)

    MathSciNet  MATH  Google Scholar 

  25. Griffiths, P., Schmid, W.: Locally homogenous complex manifolds. Acta Math. 123, 254–301 (1969)

    Google Scholar 

  26. Gromov, M.: Kähler hyperbolicity and L 2 Hodge theory. J. Differ. Geom. 33, 263–291 (1991)

    MathSciNet  MATH  Google Scholar 

  27. Gromov, M., Lafontaine, J., Pansu, P.: Structures métriques pour les variétés riemanniennes. Cedic Fernand Nathan 1981

  28. Gromov, M., Schoen, R.: Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publ. Math., Inst. Hautes Étud. Sci. 76, 165–246 (1992)

    Google Scholar 

  29. Hartshorne, R.: Algebraic Geometry. Grad. Texts Math. 52. Springer 1977

  30. Hartman, P.: On homotopic harmonic maps. Can. J. Math. 19, 673–687 (1967)

    MATH  Google Scholar 

  31. Hironaka, H.: Resolution of singularities over a field of characteristic zero. Ann. Math. 79, 109–326 (1964)

    MATH  Google Scholar 

  32. Hitchin, N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc., III. Ser. 55, 59–126 (1987)

    Google Scholar 

  33. Jost, J.: Non linear methods in Riemannian and Kählerian Geometry. DMV Seminar 10. Birkhäuser 1991

  34. J. Jost Equilibrium maps between metric spaces. Calc. Var. 2, 173–204 (1994)

  35. Jost, J., Zuo, K.: Harmonic maps and SL(r,ℂ)-representations of π1 of quasi projective manifolds. J. Algebr. Geom. 5, 77–106 (1996)

    MathSciNet  MATH  Google Scholar 

  36. Jost, J., Zuo, K.: Harmonic maps into Bruhat-Tits Buildings and Factorizations of p-Adically Unbounded and Non Rigid Representations of π1 of Algebraic Varieties, I. J. Algebr. Geom. 9, 1–42 (2000)

    MathSciNet  MATH  Google Scholar 

  37. Katzarkov, L.: Nilpotent Groups and universal coverings of smooth projective manifolds. J. Differ. Geom. 45, 336–348 (1997)

    MathSciNet  MATH  Google Scholar 

  38. Katzarkov, L.: On the Shafarevich maps. Proc. Symp. Pure Math. 62, Santa Cruz 1995, Part 2, 173–216

  39. Katzarkov, L., Ramachandran, M.: On the universal coverings of algebraic surfaces. Ann. Sci. Éc. Norm. Supér., IV. Sér. 31, 525–535 (1998)

    Google Scholar 

  40. Kleiman, S.: Towards a numerical theory of ampleness. Ann. Math. 84, 293–344 (1966)

    MATH  Google Scholar 

  41. Kollár, J.: Shafarevich maps and plurigenera of algebraic varieties. Invent. Math. 113, 177–215 (1993)

    MathSciNet  Google Scholar 

  42. Korevaar, N., Schoen, R.: Sobolev spaces and harmonic maps for metric spaces targets. Commun. Anal. Geom. 1, 561–659 (1993)

    MathSciNet  MATH  Google Scholar 

  43. Lamari, A.: Courants kählériens et surfaces compactes. Ann. Inst. Fourier 49, 263–285 (1999)

    MathSciNet  MATH  Google Scholar 

  44. Lasell, B., Ramachandran, M.: Observations on harmonic maps and singular varieties. Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, 135–148 (1996)

    Google Scholar 

  45. Lojasiewicz, S.: Sur le problème de la division. Stud. Math. 18, 87–136 (1959)

    MATH  Google Scholar 

  46. Lojasiewicz, S., Zurro, M.A.: Una introduccion a la Geometria semi- y sub-analitica. Secretariado de Publicaciones, Universidad de Valladolid 1993

  47. Mok, N.: Factorization of semisimple discrete representations of Kähler groups. Invent. Math. 110, 557–614 (1992)

    MathSciNet  MATH  Google Scholar 

  48. Mok, N.: Harmonic forms with values in locally constant bundles. Proceedings of the Conference in Honor of J.P. Kahane, J. Fourier Anal. (special issue), 433–453 (1995)

  49. Mok, N.: Steinness of universal covers of certain compact Kähler manifolds. J. Reine Angew. Math. 489, 1–14 (1997)

    MathSciNet  MATH  Google Scholar 

  50. Mok, N.: The generalized Castelnuovo-de Franchis for unitary representations. Berrick, Loo, Wang (eds.) Proceedings: Geometry from the Pacific Rim, pp. 262–284. de Gruyter 1997

  51. Mumford, D.: Abelian Varieties. Tata Stud. Math. Vol. 5, 2nd edn. Oxford University Press 1974

  52. Napier, T.: Convexity properties of smooth projective varieties. Math. Ann. 286, 433–480 (1990)

    MathSciNet  MATH  Google Scholar 

  53. Narasimhan, R.: The Levi problem for complex spaces, II. Math. Ann. 146, 195–216 (1962)

    MATH  Google Scholar 

  54. Peters, C.: Curvature for period domains. Proc. Symp. Pure Math. 53 (1991)

  55. Procesi, C.: Finite dimensionnal representations of algebras. Israel J. Math. 19, 169–182 (1974)

    MATH  Google Scholar 

  56. Procesi, C.: The invariant theory of n×n matrices. Adv. Math. 19, 306–381 (1976)

    MATH  Google Scholar 

  57. Ran, Z.: Subvarieties of abelian varieties. Invent. Math. 62, 459–479 (1981)

    MathSciNet  MATH  Google Scholar 

  58. Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann. 175, 257–268 (1968)

    MATH  Google Scholar 

  59. Sampson, J.: Applications of harmonic maps to Kähler geometry. Contemp. Math. 49, 125–134 (1986)

    MATH  Google Scholar 

  60. Serre, J.P.: Représentations linéaires des groupes finis, 3eme éd. Hermann 1978

  61. Simpson, C.: Constructing Variations of Hodge Structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)

    MathSciNet  MATH  Google Scholar 

  62. Simpson, C.: Higgs bundles and local systems. Publ. Math., Inst. Hautes Étud. Sci. 75, 5–95 (1992)

    Google Scholar 

  63. Simpson, C.: Some families of local systems over smooth projective varieties. Ann. Math. 138, 337–425 (1993)

    MathSciNet  MATH  Google Scholar 

  64. Simpson, C.: Subspaces of moduli spaces of rank one local systems. Ann. Sci. Éc. Norm. Supér., IV. Sér. 26, 361–401 (1993)

    Google Scholar 

  65. Simpson, C.: A Lefschetz theorem for π0 of the integral leaves of a holomorphic one-form. Comput. Math. 87, 99–113 (1993)

    MATH  Google Scholar 

  66. Simpson, C.: Moduli of representations of the fundamental group of a smooth projective variety, I. Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994); II, Publ. Math., Inst. Hautes Étud. Sci. 80, 5–79 (1994)

    Google Scholar 

  67. Simpson, C.: The Hodge filtration on nonabelian cohomology. Proc. Symp. Pure Math. 62, Part 2, 217–281 (1997)

  68. Siu, Y.T.: The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. Math. 112, 73–111 (1980)

    MathSciNet  MATH  Google Scholar 

  69. Skoda, H.: Prolongement des courants positifs fermés de masses finies. Invent. Math. 66, 361–376 (1982)

    MathSciNet  MATH  Google Scholar 

  70. Zuo, K.: Factorization of rigid Zariski dense representations of π1 of projective manifolds. Invent. Math. 118, 37–46 (1994)

    MathSciNet  MATH  Google Scholar 

  71. Zuo, K.: Kodaira dimension and Chern hyperbolicity of the Shafarevich maps for representations of π1 of compact Kähler manifolds. J. Reine Angew. Math. 472, 139–156 (1996)

    MathSciNet  MATH  Google Scholar 

  72. Zuo, K.: Representations of fundamental groups of algebraic varieties. Lect. Notes Math. 1708. Springer 1999

  73. Ziemer, W.: Weakly differentiable functions. Grad. Texts Math. 120. Springer 1989

Download references

Author information

Authors and Affiliations

  1. CNRS-UMR 5580, Laboratoire Emile Picard, Université Paul Sabatier, UFR MIG, 118, Route de Narbonne, 31062, Toulouse, France

    Philippe Eyssidieux

Authors
  1. Philippe Eyssidieux
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philippe Eyssidieux.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eyssidieux, P. Sur la convexité holomorphe des revêtements linéaires réductifs d’une variété projective algébrique complexe. Invent. math. 156, 503–564 (2004). https://doi.org/10.1007/s00222-003-0345-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-003-0345-0

Search

Navigation