On the approximation of functions on a Hodge manifold

Alessandro Ghigi

doi:10.5802/afst.1350

Alessandro Ghigi ¹

¹ Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. 4, pp. 769-781.

Abstract
Abstract

If $(M, ω)$ 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.

Soit $(M, ω)$ 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.

EuDML MR Zbl

DOI: 10.5802/afst.1350

Author's affiliations:

Alessandro Ghigi ¹

¹ Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca

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     title = {On the approximation of functions on a {Hodge} manifold},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {769--781},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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     volume = {Ser. 6, 21},
     number = {4},
     year = {2012},
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Alessandro Ghigi. On the approximation of functions on a Hodge manifold. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. 4, pp. 769-781. doi : 10.5802/afst.1350. https://afst.centre-mersenne.org/articles/10.5802/afst.1350/

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