On the Kaehler metrics over ${mathrm{Sym}^{d}(X)$

TL;DR

This paper studies the Kähler metric on symmetric products of a Riemann surface, estimating its Bergman kernel and proving automorphisms are isometries, revealing geometric properties of these complex manifolds.

Contribution

It provides new estimates for the Bergman kernel and establishes that all holomorphic automorphisms are isometries on symmetric products of Riemann surfaces.

Findings

Bergman kernel estimates for the Kähler metric

Automorphisms of symmetric products are isometries

Embedding of symmetric products into Picard varieties

Abstract

Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$. For each $d <\eta(X)$, where $\eta(X)$ is the gonality of $X$, the symmetric product $\text{Sym}^d(X)$ embeds into $\text{Pic}^d(X)$ by sending an effective divisor of degree $d$ to the corresponding holomorphic line bundle. Therefore, the restriction of the flat K\"ahler metric on $\text{Pic}^d(X)$ is a K\"ahler metric on $\text{Sym}^d(X)$. We investigate this K\"ahler metric on $\text{Sym}^d(X)$. In particular, we estimate it's Bergman kernel. We also prove that any holomorphic automorphism of $\text{Sym}^d(X)$ is an isometry.