Abstract
Off-diagonal upper bounds are established for Bergman kernels associated to powers \(L^\lambda \) of holomorphic line bundles L over compact complex manifolds, asymptotically as the power \(\lambda \) tends to infinity. The line bundle is assumed to be equipped with a Hermitian metric with positive curvature form, which is \(C^\infty \) but not necessarily real analytic. The bounds are of the form \(\exp (-h(\lambda )\sqrt{\lambda \log \lambda })\) where h tends to infinity at a non-universal rate. This form is best possible.
The author was supported in part by NSF grants DMS-0901569 and DMS-1363324.
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Notes
- 1.
The other alternative asserts that u is \(C^\omega \), microlocally outside a conic neighborhood of one of the two ray bundles whose union is the characteristic variety of \({\bar{\partial }}_b\). This implies holomorphic extendibility to an appropriate wedge, and the above reasoning may then be repeated to gain the factor \(\exp (-c\lambda )\).
- 2.
Subtraction of the pluriharmonic second degree terms is natural, but is inessential here.
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Christ, M. (2018). Upper Bounds for Bergman Kernels Associated to Positive Line Bundles with Smooth Hermitian Metrics. In: Hitrik, M., Tamarkin, D., Tsygan, B., Zelditch, S. (eds) Algebraic and Analytic Microlocal Analysis. AAMA 2013. Springer Proceedings in Mathematics & Statistics, vol 269. Springer, Cham. https://doi.org/10.1007/978-3-030-01588-6_8
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