In this paper we study fundamental equations of holomorphically projective mappings from manifolds with equiaffine connection onto (pseudo-) Kähler manifolds with respect to the smoothness class of connection and metrics. We show that holomorphically projective mappings preserve the smoothness class of connections and metrics.

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.

We build on work by Z. Pasternak-Winiarski [PW2], and study a-Bergman kernels of bounded domains $\Omega \subset {ℂ}^N$ for admissible weights $a\in L¹\left(\Omega \right)$.

This article gives a description, by means of functorial intrinsic fibrations, of the geometric structure (and conjecturally also of the Kobayashi pseudometric, as well as of the arithmetic in the projective case) of compact Kähler manifolds. We first define special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure on the base being given by the divisor of multiple fibres. We next show that rationally connected Kähler...

For any compact Kähler manifold $X$ and for any equivalence relation generated by a symmetric binary relation with compact analytic graph in $X\times X$, the existence of a meromorphic quotient is known from Inv. Math. 63 (1981). We give here a simplified and detailed proof of the existence of such quotients, following the approach of that paper. These quotients are used in one of the two constructions of the core of $X$ given in the previous paper of this fascicule, as well as in many other questions.