A curvature formula associated to a family of pseudoconvex domains
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 269-313.

We shall give a definition of the curvature operator for a family of weighted Bergman spaces { t } associated to a smooth family of smoothly bounded strongly pseudoconvex domains {D t }. In order to study the “boundary term” in the curvature operator, we shall introduce the notion of geodesic curvature for the associated family of boundaries {D t }. As an application, we get a variation formula for the norms of Bergman projections of currents with compact support. A flatness criterion for { t } and its applications to triviality of fibrations are also given in this paper.

Nous définissons l’opérateur de courbure pour une famille d’espaces de Bergman pondérés { t } associés à une famille lisse de domaines lisses bornés strictement pseudoconvexes {D t }. Afin d’étudier le “terme au bord” dans l’opérateur de courbure, nous introduisons la notion de courbure géodésique pour la famille des bords associés. Comme application, nous obtenons une formule de variation pour les normes de projections de Bergman des courants à support compact. Un critère de platitude pour { t } et ses applications à la trivialité des fibrations sont également données dans cet article.

Published online:
DOI: 10.5802/aif.3082
Classification: 32A25, 32L25, 32G05
Keywords: Brunn–Minkowski theory, Prekopa theorem, $\overline{\partial }$-equation, Hörmander theory.
Mot clés : théorie de Brunn–Minkowski, théorème de Prekopa, $\overline{\partial }$-équation, théorie d’Hörmander
Wang, Xu 1

1 Fudan University School of Mathematical Sciences Shanghai, 200433 (China)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Wang, Xu. A curvature formula associated to a family of pseudoconvex domains. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 269-313. doi : 10.5802/aif.3082. https://aif.centre-mersenne.org/articles/10.5802/aif.3082/

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