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On Compact Einstein Kähler Manifolds with Abundant Holomorphic Transformations

  • Yusuke Sakane
Part of the Progress in Mathematics book series (PM, volume 14)

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

Keywords

Chern Class Isotropy Subgroup Hermitian Manifold Holomorphic Line Bundle Compact Complex Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • Yusuke Sakane
    • 1
  1. 1.Osaka UniversityToyonaka, Osaka 560Japan

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