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# Pure and Applied Mathematics Quarterly

## Volume 12 (2016)

### Number 4

### Chern scalar curvature and symmetric products of compact Riemann surfaces

Pages: 463 – 471

DOI: https://dx.doi.org/10.4310/PAMQ.2016.v12.n4.a1

#### Authors

#### Abstract

Let $X$ be a compact connected Riemann surface of genus $g \geq 0$, and let $\mathrm{Sym}^d (X), d \geq 1$, denote the $d$-fold symmetric product of $X$. We show that $\mathrm{Sym}^d (X)$ admits a Hermitian metric with

(1) negative Chern scalar curvature if and only if $g \geq 2$, and

(2) positive Chern scalar curvature if and only if $d \gt g$.

#### Keywords

Gauduchon metric, Chern scalar curvature, symmetric product, pseudo-effectiveness

#### 2010 Mathematics Subject Classification

14H60, 32Q05, 32Q10

Received 28 June 2017

Published 26 July 2018