$L^2$-theory for the $\protect \overline{\partial }$-operator on complex spaces with isolated singularities

Jean Ruppenthal

doi:10.5802/afst.1599

L^{2}

-theory for the

\bar{\partial}

-operator on complex spaces with isolated singularities

Jean Ruppenthal ¹

¹ Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 2, pp. 225-258.

Abstract
Abstract

The present paper is a complement to the $L^{2}$ -theory for the $\bar{\partial}$ -operator on a Hermitian complex space $X$ of pure dimension $n$ with isolated singularities, presented in [17] and [13]. The general philosophy is to use a resolution of singularities $π : M \to X$ to obtain a “regular” model of the $L^{2}$ -cohomology.

First, we show how the representation of the $L_{l o c}^{2}$ -cohomology of $X$ on the level of $(n, q)$ -forms in terms of “regular” $L_{l o c}^{2}$ -cohomology on $M$ , given in [17], can be made explicit in terms of differential forms, if a certain reasonable extra condition is satisfied. Second, we prove the analogous statement for $L^{2}$ -cohomology, which is a new result. Finally, we use this in combination with duality observations to give a new proof of the main results from [13], where the resolution $π : M \to X$ is used to express the $L^{2}$ -cohomology of $X$ on the level of $(0, q)$ -forms in terms of “regular” $L^{2}$ -cohomology on $M$ .

Le présent article est un complément à la théorie de $L^{2}$ pour l’opérateur $\bar{\partial}$ sur un espace complexe hermitien $X$ de dimension pure $n$ avec des singularités isolées, présenté dans [17] et [13]. La philosophie générale est d’utiliser une résolution de singularités $π : M \to X$ pour obtenir un modèle « régulier » de la $L^{2}$ -cohomologie.

Tout d’abord, nous montrons comment la représentation de la $L_{l o c}^{2}$ -cohomologie de $X$ au niveau de $(n, q)$ -formes en termes de la $L_{l o c}^{2}$ -cohomologie « réguliere » sur $M$ , donnée dans [17], peut être fait explicite en termes de formes différentielles, si une certaine condition supplémentaire raisonnable est remplie. Deuxièmement, nous prouvons la déclaration analogue pour la $L^{2}$ -cohomologie, ce qui est un nouveau résultat. Enfin, nous l’utilisons en combinaison avec des observations de dualité pour donner une nouvelle preuve des principaux résultats de [13], où la résolution $π : M \to X$ est utilisée pour exprimer la $L^{2}$ -cohomologie de $X$ sur le niveau de $(0, q)$ -formes en termes de $L^{2}$ -cohomologie « régulière » sur $M$ .

Received: 2015-09-10
Accepted: 2017-04-27
Published online: 2019-05-02

MR Zbl

DOI: 10.5802/afst.1599

Classification: 32J25, 32C35, 32W05
Keywords: Cauchy–Riemann equations, $L^2$-theory, singular complex spaces

Author's affiliations:

Jean Ruppenthal ¹

¹ Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Jean Ruppenthal},
     title = {$L^2$-theory for the $\protect \overline{\partial }$-operator on complex spaces with isolated singularities},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {225--258},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Jean Ruppenthal. $L^2$-theory for the $\protect \overline{\partial }$-operator on complex spaces with isolated singularities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 2, pp. 225-258. doi : 10.5802/afst.1599. https://afst.centre-mersenne.org/articles/10.5802/afst.1599/

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