Bergman kernel on Riemann surfaces and Kaehler metric on symmetric products

TL;DR

This paper studies the Bergman kernel on hyperbolic Riemann surfaces and its relation to Kähler metrics on symmetric products, providing estimates that connect complex analysis, geometry, and embeddings into Grassmannians.

Contribution

It provides new estimates of the Bergman metric in terms of the Poincaré metric and explores embeddings of symmetric products into Grassmannians with induced Kähler metrics.

Findings

Bergman metric estimates in terms of Poincaré metric

Embedding symmetric products into Grassmannians analyzed

Volume form estimates for symmetric products derived

Abstract

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincar\'e metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $\Omega^{\otimes k}_X$, where $\O$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding Bergman metric on $X$ in terms of the Poincar\'e metric. We then consider a certain natural embedding of the symmetric product of $X$ into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of $\Omega^{\otimes k}_X$. The Fubini-Study metric on the Grassmannian restricts to a K\"ahler metric on the symmetric product of $X$. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of $\Omega^{\otimes k}_X$ and the volume form for the orbifold K\"ahler form on the symmetric product given by the Poincar\'e metric on $X$.