Abstract
In this paper, we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincaré metric near the punctures and a holomorphic line bundle that 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 function, involving a fractional growth order of the tensor power.
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