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On Compact Einstein Kähler Manifolds with Abundant Holomorphic Transformations

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Manifolds and Lie Groups

Part of the book series: Progress in Mathematics ((PM,volume 14))

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Abstract

Let (M,J,g) be a compact connected Kähler manifold and let Ric(g) denote the Ricci tensor. A compact Kähler manifold (M,J,g) is said to be Einstein if Ric(g) = kg for some k ∈ R. If we denote by γ the Ricci form of (M,J,g) (γ(X,Y) = Ric(g)(X,JY)) and by ω the Kähler form, (M,J,g) is Einstein if and only if γ = kω (k ∈ R). Let H2 (M, ℝ) denote the 2nd cohomology group with the coefficients in R. It is known that the first Chern class c1 (M) of a compact Kähler manifold (M, J, g) is given by

$${c_1}\left( M \right) = \frac{1}{{2\pi }}\left[ \gamma \right] \in {H^2}\left( {M,R} \right)$$

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Authors and Affiliations

  1. Osaka University, Toyonaka, Osaka 560, Japan

    Yusuke Sakane

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  1. Yusuke Sakane
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Editors and Affiliations

  1. Department of Mathematics, Washington University, St. Louis, Missouri, 63130, USA

    Jun-ichi Hano

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© 1981 Springer Science+Business Media New York

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Sakane, Y. (1981). On Compact Einstein Kähler Manifolds with Abundant Holomorphic Transformations. In: Hano, Ji., Morimoto, A., Murakami, S., Okamoto, K., Ozeki, H. (eds) Manifolds and Lie Groups. Progress in Mathematics, vol 14. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-5987-9_18

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  • DOI: https://doi.org/10.1007/978-1-4612-5987-9_18

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-5989-3

  • Online ISBN: 978-1-4612-5987-9

  • eBook Packages: Springer Book Archive

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