On Compact Einstein Kähler Manifolds with Abundant Holomorphic Transformations

Abstract

Let (M,J,g) be a compact connected Kahler manifold and let Ric(g) denote the Ricci tensor. A compact Kahler 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 Kahler 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 Kahler manifold (M, J, g) is given by $${c_1}\left( M \right) = \frac{1}{{2\pi }}\left[ \gamma \right] \in {H^2}\left( {M,R} \right)$$ .