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Katsutoshi Yamanoi
On fundamental groups of algebraic varieties and value distribution theory
(Groupes fondamentaux des variétés algébriques et théorie de distributions des valeurs)
Annales de l'institut Fourier, 60 no. 2 (2010), p. 551-563, doi: 10.5802/aif.2532
Article: subscription required (your ip address: 54.242.233.11) | Reviews MR 2667786 | Zbl 1193.32010
Class. Math.: 32H30, 14F35
Keywords: Value distribution theory, holomorphic map, fundamental group, algebraic variety
[1] Norbert A’Campo & Marc Burger, “Réseaux arithmétiques et commensurateur d’après G. A. Margulis”, Invent. Math. 116 (1994) no. 1-3, p. 1-25
Article | MR 1253187 | Zbl 0833.22014
[2] Gregery T. Buzzard & Steven S. Y. Lu, “Algebraic surfaces holomorphically dominable by $\mathbf{C}^2$”, Invent. Math. 139 (2000) no. 3, p. 617-659
Article | MR 1738063 | Zbl 0967.14025
[3] F. Campana, “Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler”, J. Algebraic Geom. 10 (2001) no. 4, p. 599-622 MR 1838973 | Zbl 1072.14512
[4] Frédéric Campana, “Orbifolds, special varieties and classification theory”, Ann. Inst. Fourier (Grenoble) 54 (2004) no. 3, p. 499-630
Cedram | MR 2097416 | Zbl 1062.14014
[5] Philippe Eyssidieux, “Sur la convexité holomorphe des revêtements linéaires réductifs d’une variété projective algébrique complexe”, Invent. Math. 156 (2004) no. 3, p. 503-564
Article | MR 2061328 | Zbl 1064.32007
[6] Phillip Griffiths & Wilfried Schmid, “Locally homogeneous complex manifolds”, Acta Math. 123 (1969), p. 253-302
Article | MR 259958 | Zbl 0209.25701
[7] Phillip Griffiths & Wilfried Schmid, Recent developments in Hodge theory: a discussion of techniques and results, Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, 1975, p. 31–127 MR 419850 | Zbl 0355.14003
[8] Mikhail Gromov & Richard Schoen, “Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one”, Inst. Hautes Études Sci. Publ. Math. (1992) no. 76, p. 165-246
Numdam | MR 1215595 | Zbl 0896.58024
[9] J. Jost & K. Zuo, “Harmonic maps into Bruhat-Tits buildings and factorizations of $p$-adically unbounded representations of $\pi _1$ of algebraic varieties. I”, J. Algebraic Geom. 9 (2000) no. 1, p. 1-42 MR 1713518 | Zbl 0984.14011
[10] L. Katzarkov, On the Shafarevich maps, Algebraic geometry—Santa Cruz 1995, Amer. Math. Soc., 1997, p. 173–216 MR 1492537 | Zbl 0906.14007
[11] János Kollár, “Shafarevich maps and plurigenera of algebraic varieties”, Invent. Math. 113 (1993) no. 1, p. 177-215
Article | MR 1223229 | Zbl 0819.14006
[12] Junjiro Noguchi, “Meromorphic mappings of a covering space over $C^{m}$ into a projective variety and defect relations”, Hiroshima Math. J. 6 (1976) no. 2, p. 265-280
Article | MR 422694 | Zbl 0338.32016
[13] Junjiro Noguchi, “On the value distribution of meromorphic mappings of covering spaces over ${\bf C}^m$ into algebraic varieties”, J. Math. Soc. Japan 37 (1985) no. 2, p. 295-313
Article | MR 780664 | Zbl 0566.32019
[14] Junjiro Noguchi & Takushiro Ochiai, Geometric function theory in several complex variables, Translations of Mathematical Monographs 80, American Mathematical Society, 1990, Translated from the Japanese by Noguchi MR 1084378 | Zbl 0713.32001
[15] Junjiro Noguchi, Jörg Winkelmann & Katsutoshi Yamanoi, “Degeneracy of holomorphic curves into algebraic varieties”, J. Math. Pures Appl. (9) 88 (2007) no. 3, p. 293-306 MR 2355461 | Zbl 1135.32018
[16] Carlos T. Simpson, “Higgs bundles and local systems”, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, p. 5-95
Numdam | MR 1179076 | Zbl 0814.32003
[17] Katsutoshi Yamanoi, “Holomorphic curves in abelian varieties and intersections with higher codimensional subvarieties II”, preprint
[18] Katsutoshi Yamanoi, “Holomorphic curves in abelian varieties and intersections with higher codimensional subvarieties”, Forum Math. 16 (2004) no. 5, p. 749-788
Article | MR 2096686 | Zbl 1073.32007
[19] Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics 81, Birkhäuser Verlag, 1984 MR 776417 | Zbl 0571.58015
[20] Kang Zuo, “Kodaira dimension and Chern hyperbolicity of the Shafarevich maps for representations of $\pi _1$ of compact Kähler manifolds”, J. Reine Angew. Math. 472 (1996), p. 139-156
Article | MR 1384908 | Zbl 0838.14017
[21] Kang Zuo, Representations of fundamental groups of algebraic varieties, Lecture Notes in Mathematics 1708, Springer-Verlag, 1999 MR 1738433 | Zbl 0987.14014
Katsutoshi Yamanoi
On fundamental groups of algebraic varieties and value distribution theory
(Groupes fondamentaux des variétés algébriques et théorie de distributions des valeurs)
Annales de l'institut Fourier, 60 no. 2 (2010), p. 551-563, doi: 10.5802/aif.2532
Article: subscription required (your ip address: 54.242.233.11) | Reviews MR 2667786 | Zbl 1193.32010
Class. Math.: 32H30, 14F35
Keywords: Value distribution theory, holomorphic map, fundamental group, algebraic variety
Résumé - Abstract
If a smooth projective variety $X$ admits a non-degenerate holomorphic map $\mathbb{C}\rightarrow X$ from the complex plane $\mathbb{C}$, then for any finite dimensional linear representation of the fundamental group of $X$ the image of this representation is almost abelian. This supports a conjecture proposed by F. Campana, published in this journal in 2004.
Bibliography
Article | MR 1253187 | Zbl 0833.22014
[2] Gregery T. Buzzard & Steven S. Y. Lu, “Algebraic surfaces holomorphically dominable by $\mathbf{C}^2$”, Invent. Math. 139 (2000) no. 3, p. 617-659
Article | MR 1738063 | Zbl 0967.14025
[3] F. Campana, “Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler”, J. Algebraic Geom. 10 (2001) no. 4, p. 599-622 MR 1838973 | Zbl 1072.14512
[4] Frédéric Campana, “Orbifolds, special varieties and classification theory”, Ann. Inst. Fourier (Grenoble) 54 (2004) no. 3, p. 499-630
Cedram | MR 2097416 | Zbl 1062.14014
[5] Philippe Eyssidieux, “Sur la convexité holomorphe des revêtements linéaires réductifs d’une variété projective algébrique complexe”, Invent. Math. 156 (2004) no. 3, p. 503-564
Article | MR 2061328 | Zbl 1064.32007
[6] Phillip Griffiths & Wilfried Schmid, “Locally homogeneous complex manifolds”, Acta Math. 123 (1969), p. 253-302
Article | MR 259958 | Zbl 0209.25701
[7] Phillip Griffiths & Wilfried Schmid, Recent developments in Hodge theory: a discussion of techniques and results, Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, 1975, p. 31–127 MR 419850 | Zbl 0355.14003
[8] Mikhail Gromov & Richard Schoen, “Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one”, Inst. Hautes Études Sci. Publ. Math. (1992) no. 76, p. 165-246
Numdam | MR 1215595 | Zbl 0896.58024
[9] J. Jost & K. Zuo, “Harmonic maps into Bruhat-Tits buildings and factorizations of $p$-adically unbounded representations of $\pi _1$ of algebraic varieties. I”, J. Algebraic Geom. 9 (2000) no. 1, p. 1-42 MR 1713518 | Zbl 0984.14011
[10] L. Katzarkov, On the Shafarevich maps, Algebraic geometry—Santa Cruz 1995, Amer. Math. Soc., 1997, p. 173–216 MR 1492537 | Zbl 0906.14007
[11] János Kollár, “Shafarevich maps and plurigenera of algebraic varieties”, Invent. Math. 113 (1993) no. 1, p. 177-215
Article | MR 1223229 | Zbl 0819.14006
[12] Junjiro Noguchi, “Meromorphic mappings of a covering space over $C^{m}$ into a projective variety and defect relations”, Hiroshima Math. J. 6 (1976) no. 2, p. 265-280
Article | MR 422694 | Zbl 0338.32016
[13] Junjiro Noguchi, “On the value distribution of meromorphic mappings of covering spaces over ${\bf C}^m$ into algebraic varieties”, J. Math. Soc. Japan 37 (1985) no. 2, p. 295-313
Article | MR 780664 | Zbl 0566.32019
[14] Junjiro Noguchi & Takushiro Ochiai, Geometric function theory in several complex variables, Translations of Mathematical Monographs 80, American Mathematical Society, 1990, Translated from the Japanese by Noguchi MR 1084378 | Zbl 0713.32001
[15] Junjiro Noguchi, Jörg Winkelmann & Katsutoshi Yamanoi, “Degeneracy of holomorphic curves into algebraic varieties”, J. Math. Pures Appl. (9) 88 (2007) no. 3, p. 293-306 MR 2355461 | Zbl 1135.32018
[16] Carlos T. Simpson, “Higgs bundles and local systems”, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, p. 5-95
Numdam | MR 1179076 | Zbl 0814.32003
[17] Katsutoshi Yamanoi, “Holomorphic curves in abelian varieties and intersections with higher codimensional subvarieties II”, preprint
[18] Katsutoshi Yamanoi, “Holomorphic curves in abelian varieties and intersections with higher codimensional subvarieties”, Forum Math. 16 (2004) no. 5, p. 749-788
Article | MR 2096686 | Zbl 1073.32007
[19] Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics 81, Birkhäuser Verlag, 1984 MR 776417 | Zbl 0571.58015
[20] Kang Zuo, “Kodaira dimension and Chern hyperbolicity of the Shafarevich maps for representations of $\pi _1$ of compact Kähler manifolds”, J. Reine Angew. Math. 472 (1996), p. 139-156
Article | MR 1384908 | Zbl 0838.14017
[21] Kang Zuo, Representations of fundamental groups of algebraic varieties, Lecture Notes in Mathematics 1708, Springer-Verlag, 1999 MR 1738433 | Zbl 0987.14014
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310