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On the first order asymptotics of partial Bergman kernels
Dan Coman; George Marinescu
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 5, p. 1193-1210

We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metric along the hypersurface. Finally, we study the asymptotics of the partial Bergman kernel function on a given compact set and near the vanishing locus.

Nous montrons, sous des hypothèses très générales, que le noyau de Bergman partiel des sections s’annulant sur une hypersurfaces analytique décroît exponentiellement dans un voisinage du lieu d’annulation. Pour un fibré ample, nous montrons une estimée uniforme du noyau de Bergman associé à une métrique singulière le long d’une hypersurface. Finalement nous étudions les asymptotiques du noyau de Bergman sur un compact près du lieu d’annulation.

Received : 2016-01-03
Accepted : 2016-08-18
Published online : 2017-12-15
DOI : https://doi.org/10.5802/afst.1564
Classification:  32L10,  32A60,  32C20,  32U40,  81Q50
Keywords: Bergman kernel function, singular Hermitian metric
@article{AFST_2017_6_26_5_1193_0,
     author = {Dan Coman and George Marinescu},
     title = {On the first order asymptotics of partial Bergman kernels},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {5},
     year = {2017},
     pages = {1193-1210},
     doi = {10.5802/afst.1564},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_5_1193_0}
}
Coman, Dan; Marinescu, George. On the first order asymptotics of partial Bergman kernels. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 5, pp. 1193-1210. doi : 10.5802/afst.1564. afst.centre-mersenne.org/item/AFST_2017_6_26_5_1193_0/

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