Kähler Manifolds Positive Ricci Curvature

Boudjemaa Anchouche

doi:10.24200/squjs.vol12iss2pp143-198

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Apr 26, 2017

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Jun 1, 2007

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Abstract

The aim of this paper is to give an overview of some results obtained in the field of Kähler manifolds of positive Ricci curvature.

Keywords

Kähler manifolds positive Ricci curvature kodaira dimension.

References

ANCHOUCHE, B. 1998. Sur la dimension logarithmique de Kodaira des varietes Kählcriennes com- pletes de courbure de Ricci positive. Math Zeit. 227: 403-421.
ANCHOUCHE, B. On the Asymptotic behavior of complete Kähler metrics of positive ricci curvature and of standard type. In preparation, 2007.
AUBIN, T. A course in differential geometry. Graduate studies in mathematics, 27. AMS, Providence, Rhode Island 2001.
BANDO, S. and MABUCHI, T. 1985. Uniqueness of einstein-Kähler metrics modulo connected group actions, Algebraic, Geometry, Sendai, Adv. Studies in Pure Math. 10 (1987).
BEAUVILLE, A. 2006. Riemannian holonomy and algebraic geometry, http://math.unice.fr/~ beauvill/pubs/rha.pdf.
BERGER, M. 2000. Riemannian geometry during the second half of the twentieth century. University Lecture Series. 17. AMS.
BERGER, M. 1953. Sur les groupes d'holonomie des varieties connections et des varieties Riemanninnes. Bul. Soc. Math. Franc 83: 279-330.
BIRKENHAK, C. and LANG, H. 1992. Complex abelian varieties. Grundlehren der mathematischen wissenschaften 302. Springer Verlag.
BISHOP, R.L. and CRITTENDEN, R. 1964. Geometry of manifolds, New York, Academic Press.
BOREL, A. 1991. Linear algebraic groups. Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York.
BOREL, A. and SERRE, J.P. 1951. Determination des puissance reduites de Steenrod dans la cohomoloie des groups classiques, Applications, C.R. Acad. Sci. Paris 233: 680-682.
BROCKER, T. and TOM DIECK, T. 1985. Representations of compact lie groups. Graduate texts in mathematics 98, Springer Verlag.
BRYANT, R. Manifolds with G2holonomy. http://euclid.ucc.ie/pages/Rtaff /Mckay /Talks/ g2. Pdf # Search= %22 bryant %20 holonomy %20 group %22.
CHAVEL, I. 1995. Riemannian geometry, A modern introduction. Cambridge University Press, lO8.
DEMAILLY, J.P. Complex analytic and algebraic geometry. http://www-fourier.ujf-grenoble.fr /~ demailly /books.html.
DE RHAM, G. 1952. Sur la reducibilite d'un espace de Riemann, Commm. Math. Helv. 26: 328-344.
FUTAKI, A. 1983. An Obstruction to the Existence of Einstein Klthler Metrics. Invent. Math.73: 437-443.
FUTAKI, A. 1988. Klthler-Einstein metrics and integral invariants. (Lect. Notes Math., vol. 1314) Berlin, Heidelberg New York: Springer.
GALLOT, S. HULIN, D. and LAFFONTAINE, J. 1990. Riemannian Geometry. Springer-Verlag, Second edition.
GREENBERG, M. 1967. Lectures on Algebraic Topology. Benjamin, New York.
GRIFFITHS, P. and HARRIS, J. 1978. Principles of algebraic geometry. John Wiley & Son's Inc.
HARTSHORNE, R. 1970. Ample subvarieties of algebraic varieties. Lect. Notes Math., Vol 156, Springer verlag.
HELGLTSON, S. 1978. Differential geometry, lie groups, and symmetric spaces. Academic Press., Inc.
HUMPHREYS, J. 1975. Linear Algebraic Groups Springer Verlag (Graduate Texts in Mathematics).
LITAKA, S. 1977. On logarithmic Kodaira dimension of algebraic varieties. Complex anal. and Alg. Geom. A collection of papers dedicated to Kodaira. Cambridge University Press,
JOYCE, D. 2000. Compact manifolds with special holonomy. Oxford mathematical monographs series, Oxford University Press,
JOYCE, D. 1996. Compact 8-manifolds with holonomy Spin(7), Inventiones mathematicae 123: 507-552.
JOYCE, D. 1996. Compact riemannian 7-manifolds with holonomy G2. Journal of differential geometry 43: 291-328.
KNAPP, A.W. 2001. Lie groups beyond an introduction. Progress in mathematics 140. Birkhauser.
KOBAYASHI, S. 1961. On compact Kähler manifolds with positive ricci tensor, Ann. of Math. 74: 570-574.
KOBAYASHI, S. and NOMIZU, K. 1969. Foundation of differential geometry, volumes I & II. John Wiley & Sons.
KOBAYASHI, S. 1995. Transformation groups in differential geometry. Classics in mathematics, Springer-Verlag.
KOBAYASHI, S. 1987. Differential geometry of complex vector bundles. Iwanami Shoten Publisher and Princeton University Press.
KODAIRA, K. 1986. Complex manifolds and deformation of complex structures. Classics in mathematics. Springer Verlag.
KODAIRA, K. 1954. On Klthler varieties of restricted type. Ann. of Math. 60: 28-48.
KOLLAR, J. and MORI, S. 1998. Birational Geometry of Algebraic Varieties, Cambridge University Press.
LOHKAMP, J. 1994. Metrics of negative Ricci Curvature. Ann. of Math. (2) 140(3): 655-683.
MABNCHI, T. 1986. K-energy Maps Integrating Futaki Invariants. Tohoku Math. J., 38: 245-257.
MATSUSHIMA, Y. 1957. Sur la structure du groupe d'homeomorphismes analytiques d'une certaine variete Kählerienne. Nagoya Math. J. 11: 145-150.
MIYANISHI, M. Open algebraic surfaces. CRM monograph series. AMS, volume 12, 2001.
MOK, N. 1984. An embedding Theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties. Dull. Soc. Math. de France. 112: 197-258.
MOK, N. 1990. An embedding Theorem of complete Kähler manifolds of positive Ricci curvature onto quasi-projective varieties. Math. Ann. 286: 373-408.
MOK, N., SIU, Y.T. and YAU, S.T. 1981. Poincare-Lelong equation on complete Kähler manifolds. Compo Math. 44: 183-218.
MORGAN, J.W. and TIAN, G. 2007. Ricci flow and the poincare conjecture, Clay Mathematics Monographs. American Mathematical Society.
MUMFORD, D. 1974. Abelian varieties, Second Edition, Tata Lecture Notes, Oxford University Press, London.
MUNKRES, J.R. 1975. Topology, a first course. Prentice-Hall Inc.
MYERS, S.D. 1941. Riemannian manifolds with positive mean curvature. Duke Mat. J. 8: 401-404.
NEWLANDER, A. and NIRENBERG, L. 1957. Complex analytic coordinates in almost complex manifolds, Ann. of Math., 65: 391-404.
OHSAWA, T. 2002. Analysis of several complex variables. Translations of mathematical Monographs, volume 211.
SHAFAREVICH, I.R. 1977. Basic algebraic geometry. Springer-Verlag.
SALAMON, S. 1989. Riemannian geometry and holonomy groups. Longman Scientific & Technical.
TIAN, G. and YAU, S.T. 1990. Complete Kähler manifolds with zero Ricci curvature. I.J. Amer. Math. Soc. 3 (3): 579-609.
TIAN, G. 1990. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math., 101 -172.
TIAN, G.1997. Kähler Einstein manifolds with positive scalar curvature. Inv. Math. 137: 1-37.
W.K.TO, 1991. Quasi-projective embedding o noncom pact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions. Duke Math. J. 63: 745-789.
TSUJI, H. 1988. Logarithmic Fano Manifolds are simply connected. Tokyo J. Math 11(2): 359-362.
YAN, S.T. 1978. On the Ricci curvature of a compact Kähler manifold and the Monge-Ampere equation, I. Comm. Pure Appl. Math. 31(3): 339-411.
YEUNG, S.K. 1990. Complete Kähler manifolds of positive Ricci curvature. Math. Z. 204: 187-208.
WARNER, F.W. 1983. Foundations of differentiable manifolds and lie groups. Graduate Texts in Mathematics 94, Springer verlag.

References

ANCHOUCHE, B. 1998. Sur la dimension logarithmique de Kodaira des varietes Kählcriennes com- pletes de courbure de Ricci positive. Math Zeit. 227: 403-421.

ANCHOUCHE, B. On the Asymptotic behavior of complete Kähler metrics of positive ricci curvature and of standard type. In preparation, 2007.

AUBIN, T. A course in differential geometry. Graduate studies in mathematics, 27. AMS, Providence, Rhode Island 2001.

BANDO, S. and MABUCHI, T. 1985. Uniqueness of einstein-Kähler metrics modulo connected group actions, Algebraic, Geometry, Sendai, Adv. Studies in Pure Math. 10 (1987).

BEAUVILLE, A. 2006. Riemannian holonomy and algebraic geometry, http://math.unice.fr/~ beauvill/pubs/rha.pdf.

BERGER, M. 2000. Riemannian geometry during the second half of the twentieth century. University Lecture Series. 17. AMS.

BERGER, M. 1953. Sur les groupes d'holonomie des varieties connections et des varieties Riemanninnes. Bul. Soc. Math. Franc 83: 279-330.

BIRKENHAK, C. and LANG, H. 1992. Complex abelian varieties. Grundlehren der mathematischen wissenschaften 302. Springer Verlag.

BISHOP, R.L. and CRITTENDEN, R. 1964. Geometry of manifolds, New York, Academic Press.

BOREL, A. 1991. Linear algebraic groups. Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York.

BOREL, A. and SERRE, J.P. 1951. Determination des puissance reduites de Steenrod dans la cohomoloie des groups classiques, Applications, C.R. Acad. Sci. Paris 233: 680-682.

BROCKER, T. and TOM DIECK, T. 1985. Representations of compact lie groups. Graduate texts in mathematics 98, Springer Verlag.

BRYANT, R. Manifolds with G2holonomy. http://euclid.ucc.ie/pages/Rtaff /Mckay /Talks/ g2. Pdf # Search= %22 bryant %20 holonomy %20 group %22.

CHAVEL, I. 1995. Riemannian geometry, A modern introduction. Cambridge University Press, lO8.

DEMAILLY, J.P. Complex analytic and algebraic geometry. http://www-fourier.ujf-grenoble.fr /~ demailly /books.html.

DE RHAM, G. 1952. Sur la reducibilite d'un espace de Riemann, Commm. Math. Helv. 26: 328-344.

FUTAKI, A. 1983. An Obstruction to the Existence of Einstein Klthler Metrics. Invent. Math.73: 437-443.

FUTAKI, A. 1988. Klthler-Einstein metrics and integral invariants. (Lect. Notes Math., vol. 1314) Berlin, Heidelberg New York: Springer.

GALLOT, S. HULIN, D. and LAFFONTAINE, J. 1990. Riemannian Geometry. Springer-Verlag, Second edition.

GREENBERG, M. 1967. Lectures on Algebraic Topology. Benjamin, New York.

GRIFFITHS, P. and HARRIS, J. 1978. Principles of algebraic geometry. John Wiley & Son's Inc.

HARTSHORNE, R. 1970. Ample subvarieties of algebraic varieties. Lect. Notes Math., Vol 156, Springer verlag.

HELGLTSON, S. 1978. Differential geometry, lie groups, and symmetric spaces. Academic Press., Inc.

HUMPHREYS, J. 1975. Linear Algebraic Groups Springer Verlag (Graduate Texts in Mathematics).

LITAKA, S. 1977. On logarithmic Kodaira dimension of algebraic varieties. Complex anal. and Alg. Geom. A collection of papers dedicated to Kodaira. Cambridge University Press,

JOYCE, D. 2000. Compact manifolds with special holonomy. Oxford mathematical monographs series, Oxford University Press,

JOYCE, D. 1996. Compact 8-manifolds with holonomy Spin(7), Inventiones mathematicae 123: 507-552.

JOYCE, D. 1996. Compact riemannian 7-manifolds with holonomy G2. Journal of differential geometry 43: 291-328.

KNAPP, A.W. 2001. Lie groups beyond an introduction. Progress in mathematics 140. Birkhauser.

KOBAYASHI, S. 1961. On compact Kähler manifolds with positive ricci tensor, Ann. of Math. 74: 570-574.

KOBAYASHI, S. and NOMIZU, K. 1969. Foundation of differential geometry, volumes I & II. John Wiley & Sons.

KOBAYASHI, S. 1995. Transformation groups in differential geometry. Classics in mathematics, Springer-Verlag.

KOBAYASHI, S. 1987. Differential geometry of complex vector bundles. Iwanami Shoten Publisher and Princeton University Press.

KODAIRA, K. 1986. Complex manifolds and deformation of complex structures. Classics in mathematics. Springer Verlag.

KODAIRA, K. 1954. On Klthler varieties of restricted type. Ann. of Math. 60: 28-48.

KOLLAR, J. and MORI, S. 1998. Birational Geometry of Algebraic Varieties, Cambridge University Press.

LOHKAMP, J. 1994. Metrics of negative Ricci Curvature. Ann. of Math. (2) 140(3): 655-683.

MABNCHI, T. 1986. K-energy Maps Integrating Futaki Invariants. Tohoku Math. J., 38: 245-257.

MATSUSHIMA, Y. 1957. Sur la structure du groupe d'homeomorphismes analytiques d'une certaine variete Kählerienne. Nagoya Math. J. 11: 145-150.

MIYANISHI, M. Open algebraic surfaces. CRM monograph series. AMS, volume 12, 2001.

MOK, N. 1984. An embedding Theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties. Dull. Soc. Math. de France. 112: 197-258.

MOK, N. 1990. An embedding Theorem of complete Kähler manifolds of positive Ricci curvature onto quasi-projective varieties. Math. Ann. 286: 373-408.

MOK, N., SIU, Y.T. and YAU, S.T. 1981. Poincare-Lelong equation on complete Kähler manifolds. Compo Math. 44: 183-218.

MORGAN, J.W. and TIAN, G. 2007. Ricci flow and the poincare conjecture, Clay Mathematics Monographs. American Mathematical Society.

MUMFORD, D. 1974. Abelian varieties, Second Edition, Tata Lecture Notes, Oxford University Press, London.

MUNKRES, J.R. 1975. Topology, a first course. Prentice-Hall Inc.

MYERS, S.D. 1941. Riemannian manifolds with positive mean curvature. Duke Mat. J. 8: 401-404.

NEWLANDER, A. and NIRENBERG, L. 1957. Complex analytic coordinates in almost complex manifolds, Ann. of Math., 65: 391-404.

OHSAWA, T. 2002. Analysis of several complex variables. Translations of mathematical Monographs, volume 211.

SHAFAREVICH, I.R. 1977. Basic algebraic geometry. Springer-Verlag.

SALAMON, S. 1989. Riemannian geometry and holonomy groups. Longman Scientific & Technical.

TIAN, G. and YAU, S.T. 1990. Complete Kähler manifolds with zero Ricci curvature. I.J. Amer. Math. Soc. 3 (3): 579-609.

TIAN, G. 1990. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math., 101 -172.

TIAN, G.1997. Kähler Einstein manifolds with positive scalar curvature. Inv. Math. 137: 1-37.

W.K.TO, 1991. Quasi-projective embedding o noncom pact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions. Duke Math. J. 63: 745-789.

TSUJI, H. 1988. Logarithmic Fano Manifolds are simply connected. Tokyo J. Math 11(2): 359-362.

YAN, S.T. 1978. On the Ricci curvature of a compact Kähler manifold and the Monge-Ampere equation, I. Comm. Pure Appl. Math. 31(3): 339-411.

YEUNG, S.K. 1990. Complete Kähler manifolds of positive Ricci curvature. Math. Z. 204: 187-208.

WARNER, F.W. 1983. Foundations of differentiable manifolds and lie groups. Graduate Texts in Mathematics 94, Springer verlag.

Kähler Manifolds Positive Ricci Curvature

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