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Using elementary convexity arguments involving the Legendre transformation and the Prékopa-Leindler inequality, we prove the sharp Moser-Onofri inequality, which says that 1/16π ∫|∇φ|² + 1/4π ∫ φ - log (1/4π ∫ e^φ) ≥ 0 for any funcion φ ∈ C^∞(S²).

If $(M,\omega )$ is a Hodge manifold and $f\in C^{\infty }(M,ℝ)$ we construct a canonical sequence of functions $f_N$ such that $f_N\to f$ in the $C^{\infty }$ topology. These functions have a simple geometric interpretation in terms of the moment map and they are real algebraic, in the sense that they are regular functions when $M$ is regarded as a real algebraic variety. The definition of $f_N$ is inspired by Berezin-Toeplitz quantization and by ideas of Donaldson. The proof follows quickly from known results of Fine, Liu and Ma.

In this note we prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kähler metric on a compact Hermitian symmetric spaces of ABCD–type.

We consider Fano manifolds $M$ that admit a collection of finite automorphism groups $G_1,...,G_k$ , such that the quotients $M/G_i$ are smooth Fano manifolds possessing a Kähler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that $M$ admits a Kähler-Einstein metric too.