Bergman kernels on punctured Riemann surfaces

Abstract

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the punctured unit disc endowed with the standard Poincaré metric. As a consequence, we obtain an optimal uniform estimate of the supremum norm of the Bergman kernel, involving a fractional growth order of the tensor power.

arXiv comment42 pages, 2 figures; v.2 is a final update to agree with the published paper

References in corpus (9)

• On the asymptotic expansion of Bergman kernel
• Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles
• Berezin-Toeplitz quantization on Kaehler manifolds
• Emergent Conformal Symmetry of Quantum Hall States on Singular surfaces
• Extremal metrics and K-stability (PhD thesis)
• Asymptotic properties of extremal Kähler metrics of Poincaré type
• An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms
• On the first order asymptotics of partial Bergman kernels
• On the asymptotic behavior of Bergman kernels for positive line bundles

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• Fubini-Study forms on punctured Riemann surfaces
• Bergman kernel asymptotics for singular metrics on punctured Riemann surfaces