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L 2 -theory for the ¯-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.

The present paper is a complement to the L 2 -theory for the ¯-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:MX to obtain a “regular” model of the L 2 -cohomology.

First, we show how the representation of the L loc 2 -cohomology of X on the level of (n,q)-forms in terms of “regular” L loc 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:MX 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 ¯ 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:MX pour obtenir un modèle « régulier » de la L 2 -cohomologie.

Tout d’abord, nous montrons comment la représentation de la L loc 2 -cohomologie de X au niveau de (n,q)-formes en termes de la L loc 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:MX 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:
Accepted:
Published online:
DOI: 10.5802/afst.1599
Classification: 32J25,  32C35,  32W05
Keywords: Cauchy–Riemann equations, L 2 -theory, singular complex spaces
Jean Ruppenthal 1

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