On fundamental groups of algebraic varieties and value distribution theory
Annales de l'Institut Fourier, Volume 60 (2010) no. 2, p. 551-563

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

Si une variété X projective lisse admet une application holomorphe non-dégénérée X du plan complexe , 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.

DOI : https://doi.org/10.5802/aif.2532
Classification:  32H30,  14F35
Keywords: Value distribution theory, holomorphic map, fundamental group, algebraic variety
@article{AIF_2010__60_2_551_0,
     author = {Yamanoi, Katsutoshi},
     title = {On fundamental groups of algebraic varieties and value distribution theory},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {2},
     year = {2010},
     pages = {551-563},
     doi = {10.5802/aif.2532},
     mrnumber = {2667786},
     zbl = {1193.32010},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_2_551_0}
}
Yamanoi, Katsutoshi. On fundamental groups of algebraic varieties and value distribution theory. Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 551-563. doi : 10.5802/aif.2532. http://www.numdam.org/item/AIF_2010__60_2_551_0/

[1] A’Campo, Norbert; Burger, Marc Réseaux arithmétiques et commensurateur d’après G. A. Margulis, Invent. Math., Tome 116 (1994) no. 1-3, pp. 1-25 | Article | MR 1253187 | Zbl 0833.22014

[2] Buzzard, Gregery T.; Lu, Steven S. Y. Algebraic surfaces holomorphically dominable by C 2 , Invent. Math., Tome 139 (2000) no. 3, pp. 617-659 | Article | MR 1738063 | Zbl 0967.14025

[3] Campana, F. Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler, J. Algebraic Geom., Tome 10 (2001) no. 4, pp. 599-622 | MR 1838973 | Zbl 1072.14512

[4] Campana, Frédéric Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), Tome 54 (2004) no. 3, pp. 499-630 | Article | Numdam | MR 2097416 | Zbl 1062.14014

[5] Eyssidieux, Philippe Sur la convexité holomorphe des revêtements linéaires réductifs d’une variété projective algébrique complexe, Invent. Math., Tome 156 (2004) no. 3, pp. 503-564 | Article | MR 2061328 | Zbl 1064.32007

[6] Griffiths, Phillip; Schmid, Wilfried Locally homogeneous complex manifolds, Acta Math., Tome 123 (1969), pp. 253-302 | Article | MR 259958 | Zbl 0209.25701

[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 419850 | Zbl 0355.14003

[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 | Article | Numdam | MR 1215595 | Zbl 0896.58024

[9] Jost, J.; Zuo, K. Harmonic maps into Bruhat-Tits buildings and factorizations of p-adically unbounded representations of π 1 of algebraic varieties. I, J. Algebraic Geom., Tome 9 (2000) no. 1, pp. 1-42 | MR 1713518 | Zbl 0984.14011

[10] Katzarkov, L. On the Shafarevich maps, Algebraic geometry—Santa Cruz 1995, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 62 (1997), pp. 173-216 | MR 1492537 | Zbl 0906.14007

[11] Kollár, János Shafarevich maps and plurigenera of algebraic varieties, Invent. Math., Tome 113 (1993) no. 1, pp. 177-215 | Article | MR 1223229 | Zbl 0819.14006

[12] Noguchi, Junjiro Meromorphic mappings of a covering space over C m into a projective variety and defect relations, Hiroshima Math. J., Tome 6 (1976) no. 2, pp. 265-280 | MR 422694 | Zbl 0338.32016

[13] Noguchi, Junjiro On the value distribution of meromorphic mappings of covering spaces over C m into algebraic varieties, J. Math. Soc. Japan, Tome 37 (1985) no. 2, pp. 295-313 | Article | MR 780664 | Zbl 0566.32019

[14] Noguchi, Junjiro; Ochiai, Takushiro Geometric function theory in several complex variables, American Mathematical Society, Providence, RI, Translations of Mathematical Monographs, Tome 80 (1990) (Translated from the Japanese by Noguchi) | MR 1084378 | Zbl 0713.32001

[15] Noguchi, Junjiro; Winkelmann, Jörg; Yamanoi, Katsutoshi Degeneracy of holomorphic curves into algebraic varieties, J. Math. Pures Appl. (9), Tome 88 (2007) no. 3, pp. 293-306 | MR 2355461 | Zbl 1135.32018

[16] Simpson, Carlos T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, pp. 5-95 | Article | Numdam | MR 1179076 | Zbl 0814.32003

[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., Tome 16 (2004) no. 5, pp. 749-788 | Article | MR 2096686 | Zbl 1073.32007

[19] Zimmer, Robert J. Ergodic theory and semisimple groups, Birkhäuser Verlag, Basel, Monographs in Mathematics, Tome 81 (1984) | MR 776417 | Zbl 0571.58015

[20] Zuo, Kang Kodaira dimension and Chern hyperbolicity of the Shafarevich maps for representations of π 1 of compact Kähler manifolds, J. Reine Angew. Math., Tome 472 (1996), pp. 139-156 | Article | MR 1384908 | Zbl 0838.14017

[21] Zuo, Kang Representations of fundamental groups of algebraic varieties, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1708 (1999) | MR 1738433 | Zbl 0987.14014