We shall give a definition of the curvature operator for a family of weighted Bergman spaces associated to a smooth family of smoothly bounded strongly pseudoconvex domains . 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 . As an application, we get a variation formula for the norms of Bergman projections of currents with compact support. A flatness criterion for 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 associés à une famille lisse de domaines lisses bornés strictement pseudoconvexes . 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 et ses applications à la trivialité des fibrations sont également données dans cet article.
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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
@article{AIF_2017__67_1_269_0, author = {Wang, Xu}, title = {A curvature formula associated to a family of pseudoconvex domains}, journal = {Annales de l'Institut Fourier}, pages = {269--313}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {67}, number = {1}, year = {2017}, doi = {10.5802/aif.3082}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3082/} }
TY - JOUR AU - Wang, Xu TI - A curvature formula associated to a family of pseudoconvex domains JO - Annales de l'Institut Fourier PY - 2017 SP - 269 EP - 313 VL - 67 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3082/ DO - 10.5802/aif.3082 LA - en ID - AIF_2017__67_1_269_0 ER -
%0 Journal Article %A Wang, Xu %T A curvature formula associated to a family of pseudoconvex domains %J Annales de l'Institut Fourier %D 2017 %P 269-313 %V 67 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3082/ %R 10.5802/aif.3082 %G en %F AIF_2017__67_1_269_0
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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