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 $$\varphi: {\rm Sym}^d(X) \rightarrow {\rm Pic}<sup>d(X)$$ is an embedding if $d$ is less than the gonality of $X$. We investigate the curvature of the pull-back, by $\varphi$, of the flat metric on ${\rm Pic}<sup>d(X)$. 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.