On the approximation of functions on a Hodge manifold

Soit $(M,\omega )$ une variété de Hodge et soit $f\in C^{\infty }(M,ℝ)$. Nous definissons une suite canonique de fonctions $f_N$ telle que $f_N\to f$ dans la topologie $C^{\infty }$. Cette construction admet une interprétation très simple du point de vue de l’application moment. En plus les fonctions $f_N$ sont algébriques réelles, c’est-à-dire qu’elles sont des fonctions régulières sur $M$ vue comme variété algébrique réelle. La définition des $f_N$ est inspirée de la quantification de Berezin-Toeplitz et s’appuie sur des idées de Donaldson. La preuve découle très vite de certains résultats dus à Fine, Liu et Ma.

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.

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