  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 $d > g$.
