Pure and Applied Mathematics Quarterly

Volume 12 (2016)

Number 4

Chern scalar curvature and symmetric products of compact Riemann surfaces

Pages: 463 – 471

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

Authors

Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India)

Harish Seshadri (Department of Mathematics, Indian Institute of Science, Bangalore, India)

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

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Received 28 June 2017