Chern scalar curvature and symmetric products of compact Riemann surfaces

Abstract

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

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