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Multiplier Hermitian structures on Kähler manifolds

Published online by Cambridge University Press:  22 January 2016

Toshiki Mabuchi
Toshiki Mabuchi*
Affiliation:
Department of Mathematics, Osaka University, Toyonaka, Osaka, 560-0043, Japan, [email protected]
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Abstract

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The main purpose of this paper is to make a systematic study of a special type of conformally Kähler manifolds, called multiplier Hermitian manifolds, which we often encounter in the study of Hamiltonian holomorphic group actions on Kähler manifolds. In particular, we obtain a multiplier Hermitian analogue of Myers’ Theorem on diameter bounds with an application (see [M5]) to the uniquness up to biholomorphisms of the “Kähler-Einstein metrics” in the sense of [M1] on a given Fano manifold with nonvanishing Futaki character.

Keywords

53C5514J4514J5032J25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 170 , 2003 , pp. 73 - 115
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

References

[A1] Aubin, T., Réduction du cas positif de l’équation de Monge-Ampére sur les variét é Kahlériennes compactes à la démonstration d’une inégualité, J. Funct. Anal., 57 (1984), 143–153.Google Scholar
[B1] Bourguignon, J. P. et al., Preuve de la conjecture de Calabi, Astérisque 58, Soc. Math. France, 1978.Google Scholar
[Ba] Bando, S., The K-energy map, almost Einstein-Kähler metrics and an inequality of the Miyaoka-Yau type, 39 (1987), Tohoku Math. J., 231–235.Google Scholar
[BM] Bando, S. and Mabuchi, T., Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic Geometry, Sendai, 1985, Adv. Stud. in Pure Math. 10, Kinokuniya and North-Holland, Tokyo and Amsterdam (1987), pp. 11–40.Google Scholar
[F1] Futaki, A., Kähler-Einstein metrics and integral invariants, Lect. Notes in Math. 1314, Springer-Verlag, Heidelberg, 1988.Google Scholar
[FM] Futaki, A. and Mabuchi, T., Bilinear forms and extremal Kähler vector fields associated with Kähler classes, Math. Ann., 301 (1995), 199–210.CrossRefGoogle Scholar
[K1] Kobayashi, S., Transformation groups in differential geometry, Erg. der Math. 70, Springer-Verlag, Heidelberg, 1972.CrossRefGoogle Scholar
[L1] Lichnerowicz, A., Géométrie des groupes de transformations, Dunod, Paris, 1958.Google Scholar
[L2] Lichnerowicz, A., Variétés kahlériennes et premiére classe de Chern, J. Diff. Geom., 1 (1967), 195–224.Google Scholar
[LY] Li, P. and Yau, S. T., On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153–201.Google Scholar
[M1] Mabuchi, T., Kähler-Einstein metrics for manifolds with nonvanishing Futaki character, Tôhoku Math. J., 53 (2000), 171–182.Google Scholar
[M2] Mabuchi, T., Vector field energies and critical metrics on Kähler manifolds, Nagoya Math. J., 162 (2001), 41–63.Google Scholar
[M3] Mabuchi, T., Heat kernel estimates and the Green functions on multiplier Hermitian manifolds, Tôhoku Math. J., 54 (2002), 259–275.CrossRefGoogle Scholar
[M4] Mabuchi, T., A multiplier Hermitian manifold with a Hamiltonian holomorphic vector field, in preparation.Google Scholar
[M5] Mabuchi, T., A Theorem of Calabi-Matsushima’s type, Osaka J. Math., 39 (2002), 49–57.Google Scholar
[Mat] Matsushima, Y., Holomorphic vector fields on compact Kähler manifolds, Conf. Board Math. Sci. Regional Conf. Ser. in Math. 7, Amer. Math. Soc., 1971.Google Scholar
[Mil] Milnor, J., Morse Theory, Ann. of Math. Stud. 51, Princeton University Press, Princeton, 1963.Google Scholar
[N1] Nakagawa, Y., An isoperimetric inequality for orbifolds, Osaka J. Math., 30 (1993), 733–739.Google Scholar
[S1] Siu, Y.-T., Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, DMV seminar 8, Birkhäuser, Basel · Boston, 1987.CrossRefGoogle Scholar
[T1] Tian, G., On Kähler-Einstein metrics on certain Kähler manifolds with C1 (M) > 0, Invent. Math., 89 (1987), 225–246.CrossRefGoogle Scholar
[TZ1] Tian, G. and Zhu, X. H., Uniqueness of Kähler-Ricci solitons on compact Kähler manifolds, C. R. Acad. Sci. Paris, 329 (1999), 991–995.CrossRefGoogle Scholar
[TZ2] Tian, G. and Zhu, X. H., Uniqueness of Kähler-Ricci solitons, Acta Math., 184 (2000), 271–305.Google Scholar
[TZ3] Tian, G. and Zhu, X. H., A new holomorphic invariant and uniqueness of Kähler-Ricci solitons, Comm. Math. Helv., 77 (2002), 297–325.CrossRefGoogle Scholar
[Y1] Yau, S. T., On the Ricci curvature of a compact Kähler manifold and the Monge-Ampére equation I, Comm. Pure Appl. Math., 31 (1978), 339–441.Google Scholar