Fubini-Study forms on punctured Riemann surfaces

Abstract: In this paper we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincar\'e metric near the punctures, and a holomorphic line bundle that polarizes the metric. We show that the quotient of the induced Fubini-Study forms by Kodaira maps of high tensor powers of the line bundle and the Poincar\'e form near the singularity grows polynomially uniformly on a neighborhood of the singularity as the tensor power tends to infinity, as an application of the method in [5].