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2009 | 7 | 1 | 84-123
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Affine compact almost-homogeneous manifolds of cohomogeneity one

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EN
Abstrakty
EN
This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.
Twórcy
autor
  • Department of Mathematics, University of California at Riverside, California, USA, zguan@math.ucr.edu
Bibliografia
  • [1] Ahiezer D., Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom., 1983, 1, 49–78 http://dx.doi.org/10.1007/BF02329739[Crossref]
  • [2] Birkhoff G., Rota G., Ordinary Differential Equations, Fourth Edition, John Wiley & Sons, New York, 1989
  • [3] Calabi E., Extremal Kähler metrics, Seminar on Differential Geometry, Annals of Math. Stud., Princeton Univ. Press, Princeton, N.J., 1982, 102, 259–290
  • [4] Calabi E., Extremal Kähler metrics II, Differential Geometry and Complex Analysis, Springer-Verlag, Berlin, 1985, 95–114
  • [5] Ding W., Tian G., Kähler-Einstein metrics and the generalized Futaki invariant, Invent. Math., 1992, 110, 315–335 http://dx.doi.org/10.1007/BF01231335[Crossref]
  • [6] Dorfmeister J., Guan Z., Fine structure of reductive pseudo-Kählerian spaces, Geom. Dedicata, 1991, 39, 321–338 http://dx.doi.org/10.1007/BF00150759[Crossref]
  • [7] Dorfmeister J., Guan Z., Supplement to ‘Fine structure of reductive pseudo-Kählerian spaces’, Geom. Dedicata, 1992, 42, 241–242 http://dx.doi.org/10.1007/BF00147553[Crossref]
  • [8] Guan Z., On certain complex manifolds, PhD thesis, University of California at Berkeley, 1993
  • [9] Guan Z., Quasi-Einstein metrics, Internat. J. Math., 1995, 6, 371–379 http://dx.doi.org/10.1142/S0129167X95000110[Crossref]
  • [10] Guan D., Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends, Trans. Amer. Math. Soc., 1995, 347, 2255–2262 http://dx.doi.org/10.2307/2154938[Crossref]
  • [11] Guan D., Examples of holomorphic symplectic manifolds which are not Kählerian II, Invent. Math., 1995, 121, 135–145 http://dx.doi.org/10.1007/BF01884293[Crossref]
  • [12] Guan D., Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one II, J. Geom. Anal., 2002, 12, 63–79 [Crossref]
  • [13] Guan D., Classification of compact complex homogeneous spaces with invariant volumes, Trans. Amer. Math. Soc., 2002, 354, 4493–4504 http://dx.doi.org/10.1090/S0002-9947-02-03102-1[Crossref]
  • [14] Guan D., Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one III, Internat. J. Math., 2003, 14, 259–287 http://dx.doi.org/10.1142/S0129167X03001806[Crossref]
  • [15] Guan D., Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one IV, Ann. Glob. Anal. Geom., 2006, 30, 139–167 http://dx.doi.org/10.1007/s10455-006-9026-8[Crossref]
  • [16] Guan D., Extremal-solitons and exponential C ∞ convergence of the modified Calabi ow on certain CP 1 bundles, Pacific J. Math., 2007, 233, 91–124 http://dx.doi.org/10.2140/pjm.2007.233.91[Crossref]
  • [17] Guan D., Type I almost-homogeneous manifolds of cohomogeneity one, preprint
  • [18] Guan D., Type II almost-homogeneous manifolds of cohomogeneity one, preprint
  • [19] Guan D., Jacobi fields and geodesic stability, in preparation
  • [20] Guan D., Chen X., Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one, Asian J. Math., 2000, 4, 817–829
  • [21] Huckleberry A., Snow D., Almost homogeneous Kähler manifolds with hypersurface orbits, Osaka J. Math., 1982, 19, 763–786
  • [22] Humphreys J.E., Introduction to Lie algebras and representation theory, Second printing revised, Graduate Texts in Mathematics 9, Springer-Verlag, New York-Berlin, 1978
  • [23] Kobayashi S., Nomizu K., Foundations of differential geometry I. Reprint of the 1963 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996
  • [24] Koszul J.L., Sur la forme hermitienne canonique des espaces homogènes complexes, Canad. J. Math., 1995, 7, 562–576 (in French) [Crossref]
  • [25] Podesta F., Spiro A., New examples of almost homogeneous Kähler-Einstein manifolds, Indiana Univ. Math. J., 2003, 52, 1027–1074 http://dx.doi.org/10.1512/iumj.2003.52.2321[Crossref]
  • [26] Spiro A., The Ricci tensor of an almost homogeneous Kähler manifold, Adv. Geom., 2003, 3, 387–422 http://dx.doi.org/10.1515/advg.2003.022[Crossref]
  • [27] Steinsiek M., Transformation group on homogeneous rational manifolds, Math. Ann., 1982, 260, 423–435 http://dx.doi.org/10.1007/BF01457022[Crossref]
  • [28] Tian G., Kähler-Einstein metrics with positive scalar curvature, Invent. Math., 1997, 130, 1–37 http://dx.doi.org/10.1007/s002220050176[Crossref]
  • [29] Wan Z., Lie algebras, Translated from the Chinese by Che Young Lee, International Series of Monographs in Pure and Applied Mathematics, 104, Pergamon Press, Oxford-New York-Toronto, Ont., 1975
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-008-0055-3
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