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KAWA lecture notes on complex hyperbolic geometry
Erwan Rousseau
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 2, p. 421-443

These lecture notes are based on a mini-course given at the fifth KAWA Winter School on March 24-29, 2014 at CIRM, Marseille. They provide an introduction to hyperbolicity of complex algebraic varieties namely the geometry of entire curves, and a description of some recent developments.

Ces notes sont issues d’un mini-cours donné à la cinquième École d’Hiver KAWA du 24 au 29 Mars 2014, au CIRM à Marseille. Elles donnent une introduction à l’hyperbolicité des variétés algébriques complexes à savoir la géométrie des courbes entières, ainsi qu’une description de certains développements récents.

Published online : 2018-06-18
@article{AFST_2018_6_27_2_421_0,
     author = {Erwan Rousseau},
     title = {KAWA lecture notes on complex hyperbolic geometry},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {2},
     year = {2018},
     pages = {421-443},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2018_6_27_2_421_0}
}
Rousseau, Erwan. KAWA lecture notes on complex hyperbolic geometry. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 2, pp. 421-443. afst.centre-mersenne.org/item/AFST_2018_6_27_2_421_0/

[1] Ingrid Bauer; Fabrizio Catanese; Fritz Grunewald; Roberto Pignatelli Quotients of products of curves, new surfaces with p g =0 and their fundamental groups, Am. J. Math., Tome 134 (2012) no. 4, pp. 993-1049 | Article | MR 2956256 | Zbl 1258.14043

[2] Fedor A. Bogomolov Families of curves on a surface of general type, Dokl. Akad. Nauk SSSR, Tome 236 (1977) no. 5, pp. 1041-1044 | MR 0457450 | Zbl 0415.14013

[3] Damian Brotbek On the hyperbolicity of general hypersurfaces, Publ. Math., Inst. Hautes Étud. Sci., Tome 126 (2017), pp. 1-34 | Zbl 06827883

[4] Damian Brotbek; Lionel Darondeau Complete intersection varieties with ample cotangent bundles, Invent. Math., Tome 212 (2018) no. 3, pp. 913-940 | Article

[5] Marco Brunella Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. Éc. Norm. Supér., Tome 30 (1997) no. 5, pp. 569-594 | Article | MR 1474805 | Zbl 0893.32019

[6] Marco Brunella Courbes entières et feuilletages holomorphes, Enseign. Math., Tome 45 (1999) no. 1-2, pp. 195-216 | MR 1703368 | Zbl 1004.32011

[7] Marco Brunella Birational geometry of foliations, Publicações Matemáticas do IMPA, Instituto de Matemática Pura e Aplicada, 2004, iv+138 pages | MR 2114696 | Zbl 1082.32022

[8] Jean-Pierre Demailly Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry (Santa Cruz 1995) (Proceedings of Symposia in Pure Mathematics) Tome 62, American Mathematical Society, 1997, pp. 285-360 | MR MR1492539 | Zbl 0919.32014

[9] Jean-Pierre Demailly Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q., Tome 7 (2011) no. 4, pp. 1165-1207 | MR 2918158 | Zbl 1316.32014

[10] Jean-Pierre Demailly Towards the Green-Griffiths-Lang conjecture (2014) (https://arxiv.org/abs/1412.2986)

[11] Ya Deng Effectivity in the Hyperbolicity-related problem (2016) (https://arxiv.org/abs/1606.03831)

[12] Simone Diverio; Joël Merker; Erwan Rousseau Effective algebraic degeneracy, Invent. Math., Tome 180 (2010) no. 1, pp. 161-223 | Article | MR 2593279 | Zbl 1192.32014

[13] Simone Diverio; Erwan Rousseau The exceptional set and the Green-Griffiths locus do not always coincide, Enseign. Math., Tome 61 (2015) no. 3-4, pp. 417-452 | Zbl 1359.32020

[14] Julien Duval Sur le lemme de Brody, Invent. Math., Tome 173 (2008) no. 2, pp. 305-314 | Article | MR 2415309 | Zbl 1155.32017

[15] Eberhard Freitag Ein Verschwindungssatz für automorphe Formen zur Siegelschen Modulgruppe, Math. Z., Tome 165 (1979) no. 1, pp. 11-18 | Article | MR 521517 | Zbl 0393.10027

[16] Mark Green; Phillip Griffiths Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Berkeley 1979), Springer, 1980, pp. 41-74 | MR MR609557 | Zbl 0508.32010

[17] Jean-Pierre Jouanolou Hypersurfaces solutions d’une équation de Pfaff analytique, Math. Ann., Tome 232 (1978) no. 3, pp. 239-245 | MR 0481129 | Zbl 0354.34007

[18] János Kollár Singularities of pairs, Algebraic geometry (Santa Cruz 1995) (Proceedings of Symposia in Pure Mathematics) Tome 62, American Mathematical Society, 1997, pp. 221-287 | MR 1492525 | Zbl 0905.14002

[19] Serge Lang Hyperbolic and Diophantine analysis, Bull. Am. Math. Soc., Tome 14 (1986) no. 2, pp. 159-205 | Article | MR 828820 | Zbl 0602.14019

[20] Steven Shin-Yi Lu; Shing-Tung Yau Holomorphic curves in surfaces of general type, Proc. Nat. Acad. Sci. U.S.A., Tome 87 (1990) no. 1, pp. 80-82 | Article | MR 1031949 | Zbl 0702.32015

[21] Michael McQuillan Diophantine approximations and foliations, Publ. Math., Inst. Hautes Étud. Sci., Tome 87 (1998), pp. 121-174 | MR MR1659270 | Zbl 1006.32020

[22] Michael McQuillan Canonical models of foliations, Pure Appl. Math. Q., Tome 4 (2008) no. 3, pp. 877-1012 | MR 2435846 | Zbl 1166.14010

[23] Ngaiming Mok Metric rigidity theorems on Hermitian locally symmetric manifolds, Series in Pure Mathematics, Tome 6, World Scientific, 1989, xiv+278 pages | MR 1081948 | Zbl 0912.32026

[24] David Mumford Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math., Tome 42 (1977), pp. 239-272 | Article | MR 471627 | Zbl 0365.14012

[25] Alan Michael Nadel The nonexistence of certain level structures on abelian varieties over complex function fields, Ann. Math., Tome 129 (1989) no. 1, pp. 161-178 | Article | MR 979604 | Zbl 0675.14018

[26] Erwan Rousseau Hyperbolicity, automorphic forms and Siegel modular varieties, Ann. Sci. Éc. Norm. Supér., Tome 49 (2016) no. 1, pp. 249-255 | Zbl 1361.32032

[27] Goro Shimura Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, Tome 11, Princeton University Press, 1994, xiv+271 pages (Reprint of the 1971 original) | MR 1291394 | Zbl 0872.11023

[28] Jörg Winkelmann On Brody and entire curves, Bull. Soc. Math. Fr., Tome 135 (2007) no. 1, pp. 25-46 | MR 2430197 | Zbl 1159.32015

[29] Song-Yan Xie On the ampleness of the cotangent bundles of complete intersections, Invent. Math., Tome 212 (2018) no. 3, pp. 941-996