Chern scalar curvature and symmetric products of compact Riemann surfaces

Abstract

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