On fundamental groups of algebraic varieties and value distribution theory

Yamanoi, Katsutoshi

Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 551-563.

Si une variété $X$ projective lisse admet une application holomorphe non-dégénérée $ℂ\to X$ du plan complexe $ℂ$, alors pour chaque représentation linéaire de dimension finie du groupe fondamental de $X$ l’image de cette représentation est presque abélienne. Cela soutient une conjecture proposée par F. Campana, parue dans ce même journal en 2004.

If a smooth projective variety $X$ admits a non-degenerate holomorphic map $ℂ\to X$ from the complex plane $ℂ$, 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.

Classification : 32H30, 14F35
Keywords: Value distribution theory, holomorphic map, fundamental group, algebraic variety
Mot clés : théorie de distributions des valeurs, application holomorphe, groupe fondamental, variété algébrique

[1] A’Campo, Norbert; Burger, Marc Réseaux arithmétiques et commensurateur d’après G. A. Margulis, Invent. Math., Volume 116 (1994) no. 1-3, pp. 1-25 | DOI | MR | Zbl

[2] Buzzard, Gregery T.; Lu, Steven S. Y. Algebraic surfaces holomorphically dominable by ${\mathbf{C}}^2$, Invent. Math., Volume 139 (2000) no. 3, pp. 617-659 | DOI | MR | Zbl

[3] Campana, F. Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler, J. Algebraic Geom., Volume 10 (2001) no. 4, pp. 599-622 | MR | Zbl

[4] Campana, Frédéric Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 3, pp. 499-630 | DOI | Numdam | MR | Zbl

[5] Eyssidieux, Philippe Sur la convexité holomorphe des revêtements linéaires réductifs d’une variété projective algébrique complexe, Invent. Math., Volume 156 (2004) no. 3, pp. 503-564 | DOI | MR | Zbl

[6] Griffiths, Phillip; Schmid, Wilfried Locally homogeneous complex manifolds, Acta Math., Volume 123 (1969), pp. 253-302 | DOI | MR | Zbl

[7] Griffiths, Phillip; Schmid, Wilfried 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, Bombay, 1975, pp. 31-127 | MR | Zbl

[8] Gromov, Mikhail; Schoen, Richard 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, pp. 165-246 | DOI | Numdam | MR | Zbl

[9] Jost, J.; Zuo, K. Harmonic maps into Bruhat-Tits buildings and factorizations of $p$-adically unbounded representations of ${\pi }_1$ of algebraic varieties. I, J. Algebraic Geom., Volume 9 (2000) no. 1, pp. 1-42 | MR | Zbl

[10] Katzarkov, L. On the Shafarevich maps, Algebraic geometry—Santa Cruz 1995 (Proc. Sympos. Pure Math.), Volume 62, Amer. Math. Soc., Providence, RI, 1997, pp. 173-216 | MR | Zbl

[11] Kollár, János Shafarevich maps and plurigenera of algebraic varieties, Invent. Math., Volume 113 (1993) no. 1, pp. 177-215 | DOI | MR | Zbl

[12] Noguchi, Junjiro Meromorphic mappings of a covering space over $C^m$ into a projective variety and defect relations, Hiroshima Math. J., Volume 6 (1976) no. 2, pp. 265-280 | MR | Zbl

[13] Noguchi, Junjiro On the value distribution of meromorphic mappings of covering spaces over ${\mathbf{C}}^m$ into algebraic varieties, J. Math. Soc. Japan, Volume 37 (1985) no. 2, pp. 295-313 | DOI | MR | Zbl

[14] Noguchi, Junjiro; Ochiai, Takushiro Geometric function theory in several complex variables, Translations of Mathematical Monographs, 80, American Mathematical Society, Providence, RI, 1990 (Translated from the Japanese by Noguchi) | MR | Zbl

[15] Noguchi, Junjiro; Winkelmann, Jörg; Yamanoi, Katsutoshi Degeneracy of holomorphic curves into algebraic varieties, J. Math. Pures Appl. (9), Volume 88 (2007) no. 3, pp. 293-306 | MR | Zbl

[16] Simpson, Carlos T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, pp. 5-95 | DOI | Numdam | MR | Zbl

[17] Yamanoi, Katsutoshi Holomorphic curves in abelian varieties and intersections with higher codimensional subvarieties II (preprint)

[18] Yamanoi, Katsutoshi Holomorphic curves in abelian varieties and intersections with higher codimensional subvarieties, Forum Math., Volume 16 (2004) no. 5, pp. 749-788 | DOI | MR | Zbl

[19] Zimmer, Robert J. Ergodic theory and semisimple groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984 | MR | Zbl

[20] Zuo, Kang Kodaira dimension and Chern hyperbolicity of the Shafarevich maps for representations of ${\pi }_1$ of compact Kähler manifolds, J. Reine Angew. Math., Volume 472 (1996), pp. 139-156 | DOI | MR | Zbl

[21] Zuo, Kang Representations of fundamental groups of algebraic varieties, Lecture Notes in Mathematics, 1708, Springer-Verlag, Berlin, 1999 | MR | Zbl