Bergman kernel asymptotics for singular metrics on punctured Riemann surfaces. (English) Zbl 1429.32004
The authors consider singular metrics on a punctured Riemann surface and on a line bundle, and study the behavior of the Bergman kernel in the neighborhood of the punctures. By adapting Berndtsson’s method, the authors show that the asymptotics depend on the distance to the singularities and on the parameters which encode the singularities of the metrics on the base manifold and on the bundle. The authors apply the results to some metrics for which the parameters can be explicitly given and to the Riemann sphere with two conical singularities later in the article. Also, these results have an interpretation in terms of the asymptotic profile of the density-of-states function of the lowest Landau level in quantum Hall effect.
Reviewer: Liwei Chen (Columbus)
MSC:
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
| 32A36 | Bergman spaces of functions in several complex variables |
| 30F99 | Riemann surfaces |
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