The Bochner–Hartogs dichotomy for bounded geometry hyperbolic Kähler manifolds
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, p. 239-270

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.

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.

Received : 2014-09-25
Revised : 2015-04-29
Accepted : 2015-05-20
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3011
Classification:  32E40
Keywords: Green’s function, pluriharmonic
@article{AIF_2016__66_1_239_0,
     author = {Napier, Terrence and Ramachandran, Mohan},
     title = {The Bochner--Hartogs dichotomy for bounded geometry hyperbolic K\"ahler manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     pages = {239-270},
     doi = {10.5802/aif.3011},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_1_239_0}
}
Napier, Terrence; Ramachandran, Mohan. The Bochner–Hartogs dichotomy for bounded geometry hyperbolic Kähler manifolds. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 239-270. doi : 10.5802/aif.3011. http://www.numdam.org/item/AIF_2016__66_1_239_0/

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

[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., Tome 68 (1992) no. 3, pp. 477-488 | Article

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

[5] Campana, Frédéric Remarques sur le revêtement universel des variétés kählériennes compactes, Bull. Soc. Math. France, Tome 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., Tome 28 (1975) no. 3, pp. 333-354

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

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

[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), Tome 15 (1982) no. 3, pp. 457-511

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

[11] Glasner, Moses; Katz, Richard Function-theoretic degeneracy criteria for Riemannian manifolds, Pacific J. Math., Tome 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., Tome 308 (1989) no. 3, pp. 67-70

[14] Gromov, Michel Kähler hyperbolicity and L 2 -Hodge theory, J. Differential Geom., Tome 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., Tome 62 (1906) no. 1, pp. 1-88 | Article

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

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

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

[20] Nakai, Mitsuru Green potential of Evans type of Royden’s compactification of a Riemann surface, Nagoya Math. J., Tome 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., Tome 5 (1995) no. 5, pp. 809-851 | Article

[22] Napier, Terrence; Ramachandran, Mohan The Bochner-Hartogs dichotomy for weakly 1-complete Kähler manifolds, Ann. Inst. Fourier (Grenoble), Tome 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., Tome 11 (1998) no. 2, pp. 375-396 | Article

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

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

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

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

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

[29] Stein, Karl Maximale holomorphe und meromorphe Abbildungen. I, Amer. J. Math., Tome 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), Springer, Berlin-New York (Lecture Notes in Math.) Tome 894 (1981), pp. 127-144

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