Abel–Jacobi map and curvature of the pulled back metric

Abstract

Let X be a compact connected Riemann surface of genus at least two. The Abel–Jacobi map φ:Symd(X)→Picd(X) is an embedding if d is less than the gonality of X. We investigate the curvature of the pull-back, by φ, of the flat metric on Picd(X). In particular, we show that when d = 1, the curvature is strictly negative everywhere if X is not hyperelliptic, and when X is hyperelliptic, the curvature is nonpositive with vanishing exactly on the points of X fixed by the hyperelliptic involution.

Keywords

• gonality
• curvature
• symmetric product
• abel–jacobi map