The Bochner–Hartogs dichotomy for bounded geometry hyperbolic Kähler manifolds
[La dichotomie de Bochner–Hartogs pour les variétés kählériennes hyperboliques à géométrie bornée]
Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 239-270.

Le résultat principal est que pour une variété kählérienne complète hyperbolique connexe à géométrie bornée d’ordre deux qui a exactement un bout, soit la première cohomologie à valeurs dans le faisceau structural s’annule, ou alors la variété admet une application propre holomorphe dans une surface de Riemann.

The main result is that for a connected hyperbolic complete Kähler manifold with bounded geometry of order two and exactly one end, either the first compactly supported cohomology with values in the structure sheaf vanishes or the manifold admits a proper holomorphic mapping onto a Riemann surface.

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DOI : 10.5802/aif.3011
Classification : 32E40
Keywords: Green’s function, pluriharmonic
Mot clés : fonction de Green, pluriharmonique
Napier, Terrence 1 ; Ramachandran, Mohan 2

1 Department of Mathematics Lehigh University Bethlehem, PA 18015 (USA)
2 Department of Mathematics University at Buffalo Buffalo, NY 14260 (USA)
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Napier, Terrence; Ramachandran, Mohan. The Bochner–Hartogs dichotomy for bounded geometry hyperbolic Kähler manifolds. Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 239-270. doi : 10.5802/aif.3011. http://www.numdam.org/articles/10.5802/aif.3011/

[1] Amorós, J.; Burger, M.; Corlette, K.; Kotschick, D.; Toledo, D. Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, 44, American Mathematical Society, Providence, RI, 1996, xii+140 pages | DOI

[2] Andreotti, Aldo; Vesentini, Edoardo Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Études Sci. Publ. Math. (1965) no. 25, pp. 81-130

[3] Arapura, D.; Bressler, P.; Ramachandran, M. On the fundamental group of a compact Kähler manifold, Duke Math. J., Volume 68 (1992) no. 3, pp. 477-488 | DOI

[4] Bochner, S. Analytic and meromorphic continuation by means of Green’s formula, Ann. of Math. (2), Volume 44 (1943), pp. 652-673

[5] Campana, Frédéric Remarques sur le revêtement universel des variétés kählériennes compactes, Bull. Soc. Math. France, Volume 122 (1994) no. 2, pp. 255-284

[6] Cheng, S. Y.; Yau, S. T. Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., Volume 28 (1975) no. 3, pp. 333-354

[7] Cousin, P. Sur les fonctions triplement périodiques de deux variables, Acta Math., Volume 33 (1910) no. 1, pp. 105-232 | DOI

[8] Delzant, Thomas; Gromov, Misha Cuts in Kähler groups, Infinite groups: geometric, combinatorial and dynamical aspects (Progr. Math.), Volume 248, Birkhäuser, Basel, 2005, pp. 31-55 | DOI

[9] Demailly, Jean-Pierre Estimations L 2 pour l’opérateur ¯ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. (4), Volume 15 (1982) no. 3, pp. 457-511

[10] Geoghegan, Ross Topological methods in group theory, Graduate Texts in Mathematics, 243, Springer, New York, 2008, xiv+473 pages | DOI

[11] Glasner, Moses; Katz, Richard Function-theoretic degeneracy criteria for Riemannian manifolds, Pacific J. Math., Volume 28 (1969), pp. 351-356

[12] Grauert, Hans; Riemenschneider, Oswald Kählersche Mannigfaltigkeiten mit hyper-q-konvexem Rand, Problems in analysis (Lectures Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969), Princeton Univ. Press, Princeton, N.J., 1970, pp. 61-79

[13] Gromov, Michel Sur le groupe fondamental d’une variété kählérienne, C. R. Acad. Sci. Paris Sér. I Math., Volume 308 (1989) no. 3, pp. 67-70

[14] Gromov, Michel Kähler hyperbolicity and L 2 -Hodge theory, J. Differential Geom., Volume 33 (1991) no. 1, pp. 263-292 http://projecteuclid.org/euclid.jdg/1214446039

[15] 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

[16] Hartogs, Fritz Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann., Volume 62 (1906) no. 1, pp. 1-88 | DOI

[17] Harvey, F. Reese; Lawson, H. Blaine Jr. On boundaries of complex analytic varieties. I, Ann. of Math. (2), Volume 102 (1975) no. 2, pp. 223-290

[18] Kropholler, P. H.; Roller, M. A. Relative ends and duality groups, J. Pure Appl. Algebra, Volume 61 (1989) no. 2, pp. 197-210 | DOI

[19] Li, Peter On the structure of complete Kähler manifolds with nonnegative curvature near infinity, Invent. Math., Volume 99 (1990) no. 3, pp. 579-600 | DOI

[20] Nakai, Mitsuru Green potential of Evans type of Royden’s compactification of a Riemann surface, Nagoya Math. J., Volume 24 (1964), pp. 205-239

[21] Napier, Terrence; Ramachandran, Mohan Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems, Geom. Funct. Anal., Volume 5 (1995) no. 5, pp. 809-851 | DOI

[22] Napier, Terrence; Ramachandran, Mohan The Bochner-Hartogs dichotomy for weakly 1-complete Kähler manifolds, Ann. Inst. Fourier (Grenoble), Volume 47 (1997) no. 5, pp. 1345-1365

[23] Napier, Terrence; Ramachandran, Mohan The L 2 ¯-method, weak Lefschetz theorems, and the topology of Kähler manifolds, J. Amer. Math. Soc., Volume 11 (1998) no. 2, pp. 375-396 | DOI

[24] Napier, Terrence; Ramachandran, Mohan Hyperbolic Kähler manifolds and proper holomorphic mappings to Riemann surfaces, Geom. Funct. Anal., Volume 11 (2001) no. 2, pp. 382-406 | DOI

[25] Napier, Terrence; Ramachandran, Mohan Filtered ends, proper holomorphic mappings of Kähler manifolds to Riemann surfaces, and Kähler groups, Geom. Funct. Anal., Volume 17 (2008) no. 5, pp. 1621-1654 | DOI

[26] Napier, Terrence; Ramachandran, Mohan L 2 Castelnuovo-de Franchis, the cup product lemma, and filtered ends of Kähler manifolds, J. Topol. Anal., Volume 1 (2009) no. 1, pp. 29-64 | DOI

[27] Ramachandran, Mohan A Bochner-Hartogs type theorem for coverings of compact Kähler manifolds, Comm. Anal. Geom., Volume 4 (1996) no. 3, pp. 333-337

[28] Sario, L.; Nakai, M. Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970, xx+446 pages

[29] Stein, Karl Maximale holomorphe und meromorphe Abbildungen. I, Amer. J. Math., Volume 85 (1963), pp. 298-315

[30] Sullivan, Dennis Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and Hausdorff dimension two, Geometry Symposium, Utrecht 1980 (Utrecht, 1980) (Lecture Notes in Math.), Volume 894, Springer, Berlin-New York, 1981, pp. 127-144

[31] Tworzewski, P.; Winiarski, T. Continuity of intersection of analytic sets, Ann. Polon. Math., Volume 42 (1983), pp. 387-393

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