×

zbMATH — the first resource for mathematics

An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties. (English) Zbl 0536.53062
In this article we are interested in the complex structure of non-compact Kähler manifolds of positive holomorphic bisectional curvature. We prove that under suitable geometric growth conditions, they can be biholomorphically embedded onto affine algebraic varieties. In particular, they are Stein manifolds. In case of complex dimension 2, we also prove that under the stronger assumption of positive sectional curvature and similar geometric growth conditions, such manifolds are actually biholomorphic to \({\mathbb{C}}^ 2\).

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] ANDREOTTI (A.) and VESENTINI (E.) . - Carleman estimates for the Laplace-Beltrami operator on complex manifolds , Publ. Math. Inst. Hautes Études Sci. 25 ( 1965 ), 81-130. Numdam | MR 30 #5333 | Zbl 0138.06604 · Zbl 0138.06604 · doi:10.1007/BF02684398 · numdam:PMIHES_1965__25__81_0 · numdam:PMIHES_1965__27__153_0 · eudml:103855
[2] BISHOP (R. L.) and GOLDBERG (S. I.) . - On the second cohomology group of a Kähler manifold of positive curvature , Proc. Amer. Math. Soc. 16 (19765), 119-122. MR 30 #2441 | Zbl 0125.39403 · Zbl 0125.39403 · doi:10.2307/2034011
[3] BOMBIERI (E.) and GIUSTI (E.) . - Harnack’s inequality for elliptic differential equations on minimal surfaces , Inv. Math. 15 ( 1972 ), 24-46. MR 46 #8057 | Zbl 0227.35021 · Zbl 0227.35021 · doi:10.1007/BF01418640 · eudml:142118
[4] CHEEGER (J.) and EBIN (D. G.) . - Comparison Theorems in Riemannian Geometry , North-Holland Publishing Co., Amsterdam, 1975 . MR 56 #16538 | Zbl 0309.53035 · Zbl 0309.53035
[5] CHEEGER (J.) and GROMOLL (D.) . - The splitting theorem for manifolds of non-negative Ricci curvature , J. Diff. Geom., 6, ( 1971 ), 119-128. MR 46 #2597 | Zbl 0223.53033 · Zbl 0223.53033
[6] CHEEGER (J.) and GROMOLL (D.) . - On the structure of complete manifolds of non-negative curvature , Ann. of Math. 96 ( 1972 ), 413-443. MR 46 #8121 | Zbl 0246.53049 · Zbl 0246.53049 · doi:10.2307/1970819
[7] CHENG (S.-Y.) and YAU (S.-T.) . - Differential equations on Riemannian manifolds and their geometric applications , Comm. Pure Appl. Math. 28 ( 1975 ), 333-354. MR 52 #6608 | Zbl 0312.53031 · Zbl 0312.53031 · doi:10.1002/cpa.3160280303
[8] CROKE (C.) . - Some isoperimetric inequalities and consequences , Ann. Scient. Ec. Norm. Sup. 13 ( 1980 ). Numdam | MR 83d:58068 | Zbl 0465.53032 · Zbl 0465.53032 · numdam:ASENS_1980_4_13_4_419_0 · eudml:82059
[9] GREENE (R. E.) and WU (H.) . - Analysis on non-compact Kähler manifolds , Proc. Symp. Pure Math., Vol. 30, Part II, Amer. Math. Soc. ( 1977 ). Zbl 0383.32005 · Zbl 0383.32005
[10] GREENE (R. E.) and WU (H.) . - Function Theory on Manifolds which Possess a Pole , Lecture Notes in Math., Vol. 669, Springer-Verlag, Berlin-Heidelberg-New York, 1979 . MR 81a:53002 | Zbl 0414.53043 · Zbl 0414.53043
[11] GOODMAN (J. E.) . - Affine open subsets of algebraic varieties and ample divisors , Ann. of Math. 89 ( 1969 ), 160-183. MR 39 #4170 | Zbl 0159.50504 · Zbl 0159.50504 · doi:10.2307/1970814
[12] HÖRMANDER (L.) . - L2-estimates and existence theorems for the \partial -operator , Acta. Math. 113 ( 1965 ), 89-152. Zbl 0158.11002 · Zbl 0158.11002 · doi:10.1007/BF02391775
[13] LELONG (P.) . - Fonctions entières (n variables) et fonctions pluri-sousharmoniques d’ordre fini dans Cn , J. Anal. Math. 12 ( 1964 ), 365-407. MR 29 #3668 | Zbl 0126.29602 · Zbl 0126.29602 · doi:10.1007/BF02807441
[14] LELONG (P.) . - Sur la structure des courants positifs fermés , Séminaire Pierre Lelong (Analyse), Année 1975 / 1976 , Lectures Notes in Math., Vol. 578, pp. 136-156, Springer-Verlag, Berlin-Heidelberg-New York. MR 58 #6328 | Zbl 0354.32014 · Zbl 0354.32014
[15] MOK (N.) . - Courbure bisectionnelle positive et variétés algébriques affines , Comptes Rendus, Acad. Sc. Paris, Vol. 296, Série A ( 1983 ), 473-476. MR 85a:32015 | Zbl 0579.53043 · Zbl 0579.53043
[16] MOK (N.) . - Complete non-compact Kähler manifolds of positive curvature , Complex Analysis of Several Variables, Proc. of Sym. in Pure Math., Vol. 41, Amer. Math. Soc., 1984 . MR 85i:53071 | Zbl 0537.53054 · Zbl 0537.53054
[17] MOK (N.) , SIU (Y.-T.) and YAU (S.-T.) . - The Poincaré-Lelong equation on complete Kähler manifolds , Comp. Math., Vol. 44, Fasc. 1-3 ( 1981 ), 183-218. Numdam | MR 84g:32011 | Zbl 0531.32007 · Zbl 0531.32007 · numdam:CM_1981__44_1-3_183_0 · eudml:89516
[18] MOSER (J.) . - On Harnack’s theorem for elliptic differential equations , Comm. Pure and Appl. Math., 14 ( 1961 ), 577-591. MR 28 #2356 | Zbl 0111.09302 · Zbl 0111.09302 · doi:10.1002/cpa.3160140329
[19] RAMANUJAM (C. P.) . - A topological characterization of the affine plane as an algebraic variety , Ann. of Math. 94 ( 1971 ), 69-88. MR 44 #4010 | Zbl 0218.14021 · Zbl 0218.14021 · doi:10.2307/1970735 · www.jstor.org
[20] SERRE (J.-P.) . - Faisceaux algébriques cohérents , Ann. of Math. 61 ( 1955 ), 197-278. MR 16,953c | Zbl 0067.16201 · Zbl 0067.16201 · doi:10.2307/1969915
[21] SERRE (J.-P.) . - Géométrie algébrique et géométrie analytique , Ann. Inst. Fourier 6 ( 1956 ), 1-42. Numdam | MR 18,511a | Zbl 0075.30401 · Zbl 0075.30401 · doi:10.5802/aif.59 · numdam:AIF_1956__6__1_0 · eudml:73726
[22] SIU (Y.-T.) . - Pseudoconvexity and the problem of Levi , Bull. Amer. Math. Soc. 84 ( 1978 ), 481-512. Article | MR 57 #16648 | Zbl 0423.32008 · Zbl 0423.32008 · doi:10.1090/S0002-9904-1978-14483-8 · minidml.mathdoc.fr
[23] SIU (Y.-T.) . - Analyticity of sets associated to Lelong numbers and the extension of closed positive currents , Invent. Math. 27 ( 1874 ), 53-156. MR 50 #5003 | Zbl 0289.32003 · Zbl 0289.32003 · doi:10.1007/BF01389965 · eudml:142308
[24] SIU (Y.-T.) and YAU (S.-T.) . - Complete Kähler manifolds with non-positive curvature of faster than quadratic decay , Ann. Math. 105 ( 1977 ), 225-264. MR 55 #10719 | Zbl 0358.32006 · Zbl 0358.32006 · doi:10.2307/1970998
[25] SKODA (H.) . - Application des tachniques L2 à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids , Ann. Scient. Ec. Norm. Sup. 5 ( 1972 ), 548-580. Numdam | MR 48 #11571 | Zbl 0254.32017 · Zbl 0254.32017 · numdam:ASENS_1972_4_5_4_545_0 · eudml:81906
[26] STAMPACHIA (G.) . - Equations elliptiques du second ordre à coefficients discontinus , Séminaire de Mathématiques Supérieures 16, 1966 . MR 40 #4603 | Zbl 0151.15501 · Zbl 0151.15501
[27] THIE (P.) . - The Lelong number in a point of a complex analytic set , Math. Ann. 172 ( 1967 ), 269-312. MR 35 #5661 | Zbl 0158.32804 · Zbl 0158.32804 · doi:10.1007/BF01351593 · eudml:161592
[28] YAU (S.-T.) . - Harmonic functions on complete Riemannian manifolds , Comm. Pure Appl. Math. 28 ( 1975 ), 201-228. MR 55 #4042 | Zbl 0291.31002 · Zbl 0291.31002 · doi:10.1002/cpa.3160280203
[29] BLANC (C.) and FIALA (F.) . - Le type d’une surface et sa courbure totale , Comment. Math. Helv. 14 ( 1941 - 1942 ), 230-233. MR 4,53d | Zbl 0026.01501 · Zbl 0026.01501 · doi:10.1007/BF02565620 · eudml:138786
[30] GROMOLL (D.) and MEYER (W.) . - On complete open manifolds of non-negative curvature , Ann. of Math. 90 ( 1969 ), 75-90. MR 40 #854 | Zbl 0191.19904 · Zbl 0191.19904 · doi:10.2307/1970682
[31] POOR (W.) . - Some results on nonnegatively curved manifolds , J. Diff. Geom. 9 ( 1974 ), 583-600. MR 51 #11351 | Zbl 0292.53037 · Zbl 0292.53037
[32] WU (H.) . - An elemetary method in the study of nonnegative curvature , Acta Math. 142 ( 1979 ), 57-78. MR 80c:53054 | Zbl 0403.53022 · Zbl 0403.53022 · doi:10.1007/BF02395057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.