On Compact Einstein Kähler Manifolds with Abundant Holomorphic Transformations

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 % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB % LrhDaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa % WcbaGaaGymaaqabaGcdaqadaqaaiaad2eaaiaawIcacaGLPaaacqGH % 9aqpdaWcaaqaaiaaigdaaeaacaaIYaGaeqiWdahaamaadmaabaGaeq % 4SdCgacaGLBbGaayzxaaGaeyicI4SaamisamaaCaaaleqabaGaaGOm % aaaakmaabmaabaGaamytaiaacYcacaWGsbaacaGLOaGaayzkaaaaaa!4941! $${c_1}\left( M \right) = \frac{1}{{2\pi }}\left[ \gamma \right] \in {H^2}\left( {M,R} \right)$$ .