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.
Revised:
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Keywords: Green’s function, pluriharmonic
Mot clés : fonction de Green, pluriharmonique
@article{AIF_2016__66_1_239_0, author = {Napier, Terrence and Ramachandran, Mohan}, title = {The {Bochner{\textendash}Hartogs} dichotomy for bounded geometry hyperbolic {K\"ahler} manifolds}, journal = {Annales de l'Institut Fourier}, pages = {239--270}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {66}, number = {1}, year = {2016}, doi = {10.5802/aif.3011}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.3011/} }
TY - JOUR AU - Napier, Terrence AU - Ramachandran, Mohan TI - The Bochner–Hartogs dichotomy for bounded geometry hyperbolic Kähler manifolds JO - Annales de l'Institut Fourier PY - 2016 SP - 239 EP - 270 VL - 66 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3011/ DO - 10.5802/aif.3011 LA - en ID - AIF_2016__66_1_239_0 ER -
%0 Journal Article %A Napier, Terrence %A Ramachandran, Mohan %T The Bochner–Hartogs dichotomy for bounded geometry hyperbolic Kähler manifolds %J Annales de l'Institut Fourier %D 2016 %P 239-270 %V 66 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3011/ %R 10.5802/aif.3011 %G en %F 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/articles/10.5802/aif.3011/
[1] Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, 44, American Mathematical Society, Providence, RI, 1996, xii+140 pages | DOI
[2] Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Études Sci. Publ. Math. (1965) no. 25, pp. 81-130
[3] On the fundamental group of a compact Kähler manifold, Duke Math. J., Volume 68 (1992) no. 3, pp. 477-488 | DOI
[4] Analytic and meromorphic continuation by means of Green’s formula, Ann. of Math. (2), Volume 44 (1943), pp. 652-673
[5] Remarques sur le revêtement universel des variétés kählériennes compactes, Bull. Soc. Math. France, Volume 122 (1994) no. 2, pp. 255-284
[6] Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., Volume 28 (1975) no. 3, pp. 333-354
[7] Sur les fonctions triplement périodiques de deux variables, Acta Math., Volume 33 (1910) no. 1, pp. 105-232 | DOI
[8] Cuts in Kähler groups, Infinite groups: geometric, combinatorial and dynamical aspects (Progr. Math.), Volume 248, Birkhäuser, Basel, 2005, pp. 31-55 | DOI
[9] Estimations 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] Topological methods in group theory, Graduate Texts in Mathematics, 243, Springer, New York, 2008, xiv+473 pages | DOI
[11] Function-theoretic degeneracy criteria for Riemannian manifolds, Pacific J. Math., Volume 28 (1969), pp. 351-356
[12] Kählersche Mannigfaltigkeiten mit hyper--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] 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] Kähler hyperbolicity and -Hodge theory, J. Differential Geom., Volume 33 (1991) no. 1, pp. 263-292 http://projecteuclid.org/euclid.jdg/1214446039
[15] Harmonic maps into singular spaces and -adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. (1992) no. 76, pp. 165-246
[16] 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] On boundaries of complex analytic varieties. I, Ann. of Math. (2), Volume 102 (1975) no. 2, pp. 223-290
[18] Relative ends and duality groups, J. Pure Appl. Algebra, Volume 61 (1989) no. 2, pp. 197-210 | DOI
[19] On the structure of complete Kähler manifolds with nonnegative curvature near infinity, Invent. Math., Volume 99 (1990) no. 3, pp. 579-600 | DOI
[20] Green potential of Evans type of Royden’s compactification of a Riemann surface, Nagoya Math. J., Volume 24 (1964), pp. 205-239
[21] 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] The Bochner-Hartogs dichotomy for weakly -complete Kähler manifolds, Ann. Inst. Fourier (Grenoble), Volume 47 (1997) no. 5, pp. 1345-1365
[23] The -method, weak Lefschetz theorems, and the topology of Kähler manifolds, J. Amer. Math. Soc., Volume 11 (1998) no. 2, pp. 375-396 | DOI
[24] Hyperbolic Kähler manifolds and proper holomorphic mappings to Riemann surfaces, Geom. Funct. Anal., Volume 11 (2001) no. 2, pp. 382-406 | DOI
[25] 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] 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] A Bochner-Hartogs type theorem for coverings of compact Kähler manifolds, Comm. Anal. Geom., Volume 4 (1996) no. 3, pp. 333-337
[28] Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970, xx+446 pages
[29] Maximale holomorphe und meromorphe Abbildungen. I, Amer. J. Math., Volume 85 (1963), pp. 298-315
[30] 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] Continuity of intersection of analytic sets, Ann. Polon. Math., Volume 42 (1983), pp. 387-393
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