Fubini-Study forms on punctured Riemann surfaces

TL;DR

This paper studies the behavior of Fubini-Study forms on punctured Riemann surfaces with Poincaré metrics, revealing polynomial growth patterns near punctures as tensor powers increase.

Contribution

It demonstrates the polynomial growth of Fubini-Study forms induced by high tensor powers of a polarized line bundle on punctured Riemann surfaces with Poincaré metrics.

Findings

Fubini-Study forms grow polynomially near punctures

Growth is uniform in neighborhoods of singularities

Application of methods from previous work [5]

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].