On the first order asymptotics of partial Bergman kernels
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 5, pp. 1193-1210.

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.

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.

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DOI : 10.5802/afst.1564
Classification : 32L10, 32A60, 32C20, 32U40, 81Q50
Mots clés : Bergman kernel function, singular Hermitian metric

Dan Coman 1 ; George Marinescu 2

1 Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, USA
2 Universität zu Köln, Mathematisches Institut, Weyertal 86-90, 50931 Köln, Deutschland & Institute of Mathematics ‘Simion Stoilow’, Romanian Academy, Bucharest, Romania
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Dan Coman; George Marinescu. On the first order asymptotics of partial Bergman kernels. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 5, pp. 1193-1210. doi : 10.5802/afst.1564. https://afst.centre-mersenne.org/articles/10.5802/afst.1564/

[1] Hugues Auvray; Xiaonan Ma; George Marinescu Bergman kernels on punctured Riemann surfaces (2016) (https://arxiv.org/abs/1604.06337) | Zbl

[2] Robert Berman Bergman kernels and equilibrium measures for ample line bundles (2007) (https://arxiv.org/abs/0704.1640)

[3] Bo Berndtsson Bergman kernels related to Hermitian line bundles over compact complex manifolds, Explorations in complex and Riemannian geometry (Contemp. Math.), Volume 332, American Mathematical Society, 2003, pp. 1-17 | DOI | MR | Zbl

[4] David Catlin The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata, 1997) (Trends in Mathematics) (1999), pp. 1-23 | MR | Zbl

[5] Dan Coman; Xiaonan Ma; George Marinescu Equidistribution for sequences of line bundles on normal Kähler spaces, Geom. Topol., Volume 21 (2017) no. 2, pp. 923-962 | DOI | Zbl

[6] Dan Coman; George Marinescu Equidistribution results for singular metrics on line bundles, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 3, pp. 497-536 | DOI | MR | Zbl

[7] Jean-Pierre Demailly Estimations L 2 pour l’opérateur ¯ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Éc. Norm. Supér., Volume 15 (1982) no. 3, pp. 457-511 | DOI | Zbl

[8] Chin-Yu Hsiao; George Marinescu Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles, Commun. Anal. Geom., Volume 22 (2014) no. 1, pp. 1-108 | DOI | MR | Zbl

[9] Chin-Yu Hsiao; George Marinescu Localization results for Bergman and Szegő kernels (2016) (in preparation)

[10] Xiaonan Ma; George Marinescu Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, 254, Birkhäuser, 2007, xiii+422 pages | MR | Zbl

[11] Florian T. Pokorny; Michael Singer Toric partial density functions and stability of toric varieties, Math. Ann., Volume 358 (2014) no. 3-4, pp. 879-923 | DOI | MR | Zbl

[12] Julius Ross; David Witt Nyström Analytic test configurations and geodesic rays, J. Symplectic Geom., Volume 12 (2014) no. 1, pp. 125-169 | DOI | MR | Zbl

[13] Julius Ross; Michael Singer Asymptotics of partial density functions for divisors, J. Geom. Anal., Volume 27 (2017), pp. 1803-1854 | DOI | MR | Zbl

[14] Julius Ross; Richard Thomas An obstruction to the existence of constant scalar curvature Kähler metrics, J. Differ. Geom., Volume 72 (2006) no. 3, pp. 429-466 | DOI | Zbl

[15] Bernard Shiffman; Steve Zelditch Random polynomials with prescribed Newton polytope, J. Am. Math. Soc., Volume 17 (2004) no. 1, pp. 49-108 | DOI | MR | Zbl

[16] Steve Zelditch Szegő kernels and a theorem of Tian, Int. Math. Res. Not., Volume 1988 (1998) no. 6, pp. 317-331 | DOI | Zbl

[17] Steve Zelditch; Peng Zhou Interface asymptotics of partial Bergman kernels on S 1 -symmetric Kaehler manifolds, J. Geom. Anal., Volume 27 (2017), pp. 1803-1854 | Zbl

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