Abel–Jacobi map and curvature of the pulled back metric

Abstract

Let [Formula: see text] be a compact connected Riemann surface of genus at least two. The Abel–Jacobi map [Formula: see text] is an embedding if [Formula: see text] is less than the gonality of [Formula: see text]. We investigate the curvature of the pull-back, by [Formula: see text], of the flat metric on [Formula: see text]. In particular, we show that when [Formula: see text], the curvature is strictly negative everywhere if [Formula: see text] is not hyperelliptic, and when [Formula: see text] is hyperelliptic, the curvature is nonpositive with vanishing exactly on the points of [Formula: see text] fixed by the hyperelliptic involution.