The Bochner-Hartogs dichotomy for weakly 1-complete Kähler manifolds
Napier, Terence ; Ramachandran, Mohan
Annales de l'Institut Fourier, Tome 47 (1997), p. 1345-1365 / Harvested from Numdam
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On démontre que si M est une variété kählérienne faiblement 1-complète avec un seul bout, alors H c 1 (M,𝒪)=0 ou bien il existe une application holomorphe propre de M sur une surface de Riemann.

It is proved that if M is a weakly 1-complete Kähler manifold with only one end, then H c 1 (M,𝒪)=0 or there exists a proper holomorphic mapping of M onto a Riemann surface.

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     author = {Napier, Terence and Ramachandran, Mohan},
     title = {The Bochner-Hartogs dichotomy for weakly 1-complete K\"ahler manifolds},
     journal = {Annales de l'Institut Fourier},
     volume = {47},
     year = {1997},
     pages = {1345-1365},
     doi = {10.5802/aif.1602},
     mrnumber = {99e:32012},
     zbl = {0904.32008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1997__47_5_1345_0}
}

Napier, Terence; Ramachandran, Mohan. The Bochner-Hartogs dichotomy for weakly 1-complete Kähler manifolds. Annales de l'Institut Fourier, Tome 47 (1997) pp. 1345-1365. doi : 10.5802/aif.1602. http://gdmltest.u-ga.fr/item/AIF_1997__47_5_1345_0/

[AV] A. Andreotti and E. Vesentini, Carlemann estimates for the Laplace-Beltrami equation on complex manifolds, Publ. Math. Inst. Hautes Études Sci., 25 (1965), 81-130. | Numdam | Zbl 0138.06604

[ABR] D. Arapura, P. Bressler, and M. Ramachandran, On the fundamental group of a compact Kähler manifold, Duke Math. J., 64 (1992), 477-488. | MR 94e:57040 | Zbl 0783.57009

[B] S. Bochner, Analytic and meromorphic continuation by means of Green's formula, Ann. of Math., 44 (1943), 652-673. | MR 5,116f | Zbl 0060.24206

[C] M. Coltoiu, Complete locally pluripolar sets, J. reine angew. Math., 412 (1990), 108-112. | MR 91h:32010 | Zbl 0711.32008

[Co] P. Cousin, Sur les fonctions triplement périodiques de deux variables, Acta Math., 33 (1910), 105-232. | JFM 41.0492.02

[D1] J.-P. Demailly, Estimations L2 pour l'operateur ∂ d'un fibrée vectoriel holomorphe semi- positif au-dessus d'une variété Kählerienne complète, Ann. Scient. Éc. Norm. Sup., 15 (1982), 457-511. | Numdam | MR 85d:32057 | Zbl 0507.32021

[D2] J.-P. Demailly, Cohomology of q-convex spaces in top degrees, Math. Z., 204 (1990), 283-295. | MR 91e:32014 | Zbl 0682.32017

[G] M. Gaffney, A special Stokes theorem for Riemannian manifolds, Ann. of Math., 60 (1954), 140-145. | MR 15,986d | Zbl 0055.40301

[GR] H. Grauert and O. Riemenschneider, Kählersche Mannigfältigkeiten mit hyper-q-konvexen Rand, Problems in analysis (A Symposium in Honor of S. Bochner, Princeton 1969), Princeton University Press, Princeton (1970), 61-79. | Zbl 0211.10302

[GW] R. Greene and H. Wu, Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier, 25-1 (1975), 215-235. | Numdam | MR 52 #3583 | Zbl 0307.31003

[Gr] M. Gromov, Kähler hyperbolicity and L2-Hodge theory, J. Diff. Geom., 33 (1991), 263-292. | MR 92a:58133 | Zbl 0719.53042

[Gu] R. Gunning, Introduction to holomorphic functions of several variables, Vol. II, Wadsworth, Belmont, 1990.

[H] F. Hartogs, Zur Theorie der analytischen Functionen mehrener unabhangiger Veränderlichen insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann., 62 (1906) 1-88. | JFM 36.0483.02

[HL] F.R. Harvey and H.B. Lawson, Boundaries of complex analytic varieties I, Math. Ann., 102 (1975), 223-290. | MR 54 #13130 | Zbl 0317.32017

[HM] L.R. Hunt and J.J. Murray, Plurisubharmonic functions and a generalized Dirichlet problem, Michigan Math. J., 25 (1978), 299-316. | MR 80b:32018 | Zbl 0385.32015

[K] M. Kalka, On a conjecture of Hunt and Murray concerning q-plurisubharmonic functions, Proc. Amer. Math. Soc., 73 (1979), 30-34. | MR 80e:32014 | Zbl 0413.35023

[Ka] B. Kaup, Über offene analytische Äquivalenzrelationen auf komplexen Räumen, Math. Ann., 183 (1969), 6-16. | MR 40 #1600 | Zbl 0176.03805

[N] S. Nakano, Vanishing theorems for weakly 1-complete manifolds II, Publ. R.I.M.S., Kyoto, 10 (1974), 101-110. | MR 52 #3617 | Zbl 0298.32019

[NR] T. Napier and M. Ramachandran, Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems, Geom. and Func. Analysis, 5 (1995), 809-851. | MR 96g:32020 | Zbl 0860.53045

[Na] R. Narasimhan, The Levi problem for complex spaces II, Math. Ann., 146 (1962), 195-216. | MR 32 #229 | Zbl 0131.30801

[Ni] T. Nishino, L'existence d'une fonction analytique sur une variété analytique complexe à dimension quelconque, Publ. Res. Inst. Math. Sci., 19 (1983), 263-273. | MR 85c:32021 | Zbl 0529.32004

[O1] T. Ohsawa, Weakly 1-complete manifold and Levi problem, Publ. R.I.M.S., Kyoto, 17 (1981), 153-164. | MR 82j:32031 | Zbl 0465.32011

[O2] T. Ohsawa, Completeness of noncompact analytic spaces, Publ. R.I.M.S., Kyoto, 20 (1984), 683-692. | MR 86i:32017 | Zbl 0568.32008

[P] M. Peternell, Algebraische Varietäten und q-vollständige komplexe Räume, Math. Z., 200 (1989), 547-581. | MR 90c:32027 | Zbl 0675.32014

[R] M. Ramachandran, A Bochner-Hartogs type theorem for coverings of compact Kähler manifolds, Comm. Anal. Geom., 4 (1996), 333-337. | MR 98k:32039 | Zbl 0955.32018

[Ri] R. Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann., 175 (1968), 257-286. | MR 36 #5386 | Zbl 0153.15401

[S] Y.-T. Siu, Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Diff. Geom., 17 (1982), 55-138. | MR 83j:58039 | Zbl 0497.32025

[St] K. Stein, Maximale holomorphe und meromorphe Abbildungen, I, Amer. J. Math., 85 (1963), 298-315. | MR 27 #2651 | Zbl 0144.33901

[W] H. Wu, On certain Kähler manifolds which are q-complete, Complex Analysis of Several Variables, Proceedings of Symposia in Pure Mathematics, 41, Amer. Math. Soc., Providence (1984), 253-276. | MR 85j:32031 | Zbl 0552.32015