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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Central Kähler metrics

Author: Gideon Maschler
Journal: Trans. Amer. Math. Soc. 355 (2003), 2161-2182
MSC (2000): Primary 53C55, 53C25, 58E11
Published electronically: January 31, 2003
MathSciNet review: 1973986
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Abstract: The determinant of the Ricci endomorphism of a Kähler metric is called its central curvature, a notion well-defined even in the Riemannian context. This work investigates two types of Kähler metrics in which this curvature potential gives rise to a potential for a gradient holomorphic vector field. These metric types generalize the Kähler-Einstein notion as well as that of Bando and Mabuchi (1986). Whenever possible the central curvature is treated in analogy with the scalar curvature, and the metrics are compared with the extremal Kähler metrics of Calabi. An analog of the Futaki invariant is employed, both invariants belonging to a family described in the language of holomorphic equivariant cohomology. It is shown that one of the metric types realizes the minimum of an $L^2$ functional defined on the space of Kähler metrics in a given Kähler class. For metrics of constant central curvature, results are obtained regarding existence, uniqueness and a partial classification in complex dimension two. Consequently, on a manifold of Fano type, such metrics and Kähler-Einstein metrics can only exist concurrently. An existence result for the case of non-constant central curvature is stated, and proved in a sequel to this work.

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  • [A] T. Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), 63-95. MR 81d:53047
  • [AG] A. Andreotti and H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193-259. MR 27:343
  • [B] S. Bando, An obstruction for Chern class forms to be harmonic, unpublished.
  • [BM1] S. Bando and T. Mabuchi, On some integral invariants on complex manifolds, I. Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), 197-200. MR 88g:58036
  • [BM2] S. Bando and T. Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic Geometry, Sendai 1985, Advanced Studies in Pure Mathematics 10, North-Holland, Amsterdam-New York, 1987, 11-40. MR 89c:58029
  • [Bc1] S. Bochner, Vector fields on complex and real manifolds, Ann. of Math. (2) 52 (1950), 642-649. MR 12:2839
  • [Bc2] S. Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776-797. MR 8:230a
  • [BY] S. Bochner and K. Yano, Curvature and Betti Numbers, Annals of Mathematics Studies 32, Princeton University Press, Princeton, N.J., 1953. MR 15:989f
  • [Bt1] R. Bott, Vector fields and characteristic numbers, Michigan Math J. 14 (1967), 231-244. MR 35:2297
  • [Bt2] R. Bott, A residue formula for holomorphic vector-fields, J. Differential Geometry 1 (1967), 311-330. MR 38:730
  • [Br] M. Braverman, private communication.
  • [BU] D. Borthwick and A. Uribe, Almost Complex Structures and Geometric Quantization, preprint dg-ga/9608006, 1996; Math. Res. Lett. 3 (1996), 845-861; erratum, ibid. 5 (1998), 211-212. MR 98e:58084; MR 99c:58061
  • [C1] E. Calabi, Extremal Kähler metrics, Seminar on Differential Geometry (S. T. Yau, ed.), Annals of Mathematics Studies 102, Princeton University Press, Princeton, N.J., 1982, 259-290. MR 83i:53088
  • [C2] E. Calabi, Extremal Kähler metrics II, Differential Geometry and Complex Analysis (I. Chaval and H.M. Farkas, eds.), Springer, Berlin-New York, 1985, 95-114. MR 86h:53067
  • [Co] H.-D. Cao, Existence of gradient Kähler-Ricci solitons, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), 1-16, A. K. Peters, Wellesley, MA, 1996. MR 98a:53058
  • [Cr] James B. Carrell, A remark on the Grothendieck residue map, Proc. Amer. Math. Soc. 70 (1978), 43-48. MR 58:11528
  • [Dm] J. P. Demailly, Holomorphic Morse inequalities, Proceedings of Symposia in Pure Mathematics 52, Part 2 (1991), 93-114. MR 93b:32048
  • [Dr] A. Derdzinski, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), 405-433. MR 84h:53060
  • [Fj] A. Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), 225-258. MR 58:1285
  • [Ft1] A. Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983), 437-443. MR 84j:53072
  • [Ft2] A. Futaki, Kähler-Einstein Metrics and Integral Invariants, Lecture Notes in Mathematics 1314, Springer-Verlag, Berlin and New York 1988. MR 90a:53053
  • [Ft3] A. Futaki, On compact Kähler manifolds of constant scalar curvature, Proc. Japan Acad., Ser. A, Math. Sci. 59 (1983), 401-402. MR 85i:53070
  • [Ft4] A. Futaki, Integral invariants in Kähler geometry, Amer. Math. Soc. Transl. Ser. 2, 160 (1994), 63-77. MR 93h:53044 (Japanese original)
  • [FM] A. Futaki and T. Mabuchi, Bilinear forms and extremal Kähler vector fields associated with Kähler classes, Math. Ann. 301 (1995), 199-210. MR 95m:32039
  • [FMS] A. Futaki and T. Mabuchi and Y. Sakane, Einstein-Kähler metrics with positive Ricci curvature, Kähler Metrics and Moduli Spaces, Advanced Studies in Pure Mathematics 18- II, Academic Press, Boston, MA, 1990, 11-83. MR 94c:32019
  • [FR1] A. Futaki and S. Morita, Invariant polynomials on compact complex manifolds, Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), 369-372. MR 86f:32033
  • [FR2] A. Futaki and S. Morita, Invariant polynomials of the automorphism group of a compact complex manifold, J. Differential Geom. 21 (1985), 135-142. MR 87h:32061
  • [FT] A. Futaki and K. Tsuboi, Eta invariants and automorphisms of compact complex manifolds, Advanced Studies in Pure Mathematics, 18-I, Academic Press, Boston, MA, 1990, 251-270. MR 93b:58137
  • [GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York 1978. MR 80b:14001
  • [Gu] D. Z.-D. Guan, Quasi-Einstein metrics, Internat. J. Math. 6 (1995), 371-379. MR 96e:53060
  • [H1] A. D. Hwang, Extremal Kähler metrics and the Calabi energy, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), 128-129. MR 96e:58035
  • [H2] A. D. Hwang, On the Calabi energy of extremal Kähler metrics, Internat. J. Math. 6, (1995), 825-830. MR 96i:58032
  • [H3] A. D. Hwang, On existence of Kähler metrics with constant scalar curvature, Osaka J. Math. 31 (1994), 561-595. MR 96a:53061
  • [HM] A. D. Hwang and G. Maschler, Central Kähler metrics with non-constant central curvature, Trans. Amer. Math. Soc. 355 (2003), 2183-2203.
  • [HS1] A. D. Hwang and S. R. Simanca, Distinguished Kähler metrics on Hirzebruch surfaces, Trans. Amer. Math. Soc. 347 (1995), 1013-1021. MR 95e:58045
  • [HS2] A. D. Hwang and S. R. Simanca, Extremal Kähler metrics on Hirzebruch surfaces which are locally conformally equivalent to Einstein metrics, Math. Ann. 309 (1997), 97-106. MR 98f:58056
  • [Kb1] S. Kobayashi, Transformation Groups in Differential Geometry, reprint of the 1972 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. MR 96c:53040
  • [Kb2] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Iwanami Shoten, Publishers, and Princeton University Press, 1987. MR 89e:53100
  • [Kb3] S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker, Inc., New York 1970. MR 43:3503
  • [Ki] N. Koiso, On rotationally symmetric Hamilton's equation for Kähler-Einstein metrics, Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics 18-I, Academic Press, Boston, MA, 1990, 327-337. MR 93d:53057
  • [La] J. Lafontaine, Courbure de Ricci et fonctionelles critiques, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 12, 687-690. MR 84g:58035
  • [Le] C. LeBrun, Einstein metrics on complex surfaces, Geometry and Physics (Aarhus, 1995), Lecture Notes in Pure and Applied Mathematics, 184, Marcel Dekker, New York, 1997, 167-176. MR 97j:53048
  • [LSm] C. LeBrun and S. R. Simanca, Extremal Kähler metrics and complex deformation theory, Geom. Funct. Anal. 4 (1994), 298-336. MR 95k:58041
  • [LSn] C. LeBrun and M. Singer, Existence and deformation theory for scalar-flat Kähler metrics on compact complex surfaces, Invent. Math. 112 (1993), 273-313. MR 94e:53070
  • [Lc1] A. Lichnerowicz, Isométries et transformations analytiques d'une variété kählérienne compacte, Bull. Soc. Math. France 87 (1959), 427-437. MR 22:5012
  • [Lc2] A. Lichnerowicz, Variétés kählériennes et première classe de Chern, J. Differential Geom. 1 (1967), 195-223. MR 37:2150
  • [Lu] K. Liu, Holomorphic equivariant cohomology, Math. Ann. 303 (1995), 125-148. MR 97f:32041
  • [Mb] T. Mabuchi, An algebraic character associated with the Poisson Brackets, Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics 18-I, Academic Press, Boston, MA, 1990, 339-358. MR 93a:32047
  • [Ms] G. Maschler, Distinguished Kähler metrics and Equivariant Cohomological Invariants, thesis, State University of New York at Stony Brook, August 1997.
  • [Mt] Y. Matsushima, Remarks on Kähler-Einstein manifolds, Nagoya Math J. 46 (1972), 161-173. MR 46:2615
  • [Pt] J. Petean, Indefinite Kähler-Einstein metrics on compact complex surfaces, Comm. Math. Phys. 189 (1997), 227-235. MR 98i:32049
  • [Sm] S. R. Simanca, Precompactness of the Calabi energy, Internat. J. Math. 7 (1996), 245-254. MR 96m:58050
  • [Su] Y. T. Siu, The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group, Ann. of Math. (2) 127 (1988), 585-627. MR 89e:58032
  • [T] G. Tian, Kähler-Einstein metrics on algebraic manifolds, Transcendental methods in algebraic geometry (Cetraro, 1994), 143-185, Lecture Notes in Math., 1646, Springer, Berlin, 1996. MR 98j:32035
  • [Tf] C. W. Tønnesen-Friedman, Extremal Kähler Metrics on Ruled Surfaces, Institut for Matematik og Datalogi Odense Universitet Preprint Nr. 36, 1997.
  • [Ts1] K. Tsuboi, The lifted Futaki invariants and the ${\rm Spin}\sp c$-Dirac operators, Osaka J. Math. 32 (1995), 207-225. MR 97a:53109
  • [Ts2] K. Tsuboi, On the integral invariants of Futaki-Morita and the determinant of elliptic operators, Far East J. Math. Sci. 5 (1997), 305-319. MR 98h:58196
  • [TY] G. Tian and S. T. Yau, Kähler-Einstein metrics on complex surfaces with $C_1>0$, Comm. Math. Phys. 112 (1987), 175-203. MR 88k:32070
  • [W] E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692. MR 84b:58111
  • [Yn] K. Yano, Sur un théorème de M. Matsushima, Nagoya Math J. 12 (1957), 147-150. MR 20:2476
  • [Yo] M. Yotov, On the generalized Futaki invariant, electronic preprint math/9907055.
  • [Yu] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure Appl. Math. 31 (1978), 339-411. MR 81d:53045
  • [Z] W. P. Zhang, A remark on a residue formula of Bott, Acta Math. Sinica (N.S.) 6 (1990), 306-314. MR 91j:58153

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Additional Information

Gideon Maschler
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

Received by editor(s): November 1, 1999
Published electronically: January 31, 2003
Additional Notes: Partially supported by the Edmund Landau Center for research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).
Article copyright: © Copyright 2003 American Mathematical Society

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