Chern scalar curvature and symmetric products of compact Riemann surfaces

TL;DR

This paper characterizes when symmetric products of compact Riemann surfaces admit Hermitian metrics with positive or negative Chern scalar curvature based on the genus and the symmetric product degree.

Contribution

It provides a complete classification of the sign of Chern scalar curvature on symmetric products of Riemann surfaces depending on genus and degree.

Findings

Negative Chern scalar curvature exists iff genus g ≥ 2.

Positive Chern scalar curvature exists iff degree d > g.

Symmetric products of genus 0 or 1 surfaces do not admit such metrics.

Abstract

Let $X$ be a compact connected Riemann surface of genus $g\geq 0$, and let ${\rm Sym}^d(X)$, $d \ge 1$, denote the $d$-fold symmetric product of $X$. We show that ${\rm Sym}^d(X)$ admits a Hermitian metric with negative Chern scalar curvature if and only if $g \geq 2$, and positive Chern scalar curvature if and only if $d > g$.