On the Kaehler metrics over ${mathrm{Sym}^{d}(X)$

Abstract

Let $X$ be a compact connected Riemann surface of genus $g$, with $g\ge 2$. For each $d<\eta (X)$, where $\eta (X)$ is the gonality of $X$, the symmetric product ${\text{Sym}}^d(X)$ embeds into ${\text{Pic}}^d(X)$ by sending an effective divisor of degree $d$ to the corresponding holomorphic line bundle. Therefore, the restriction of the flat Kähler metric on ${\text{Pic}}^d(X)$ is a Kähler metric on ${\text{Sym}}^d(X)$. We investigate this Kähler metric on ${\text{Sym}}^d(X)$. In particular, we estimate it's Bergman kernel. We also prove that any holomorphic automorphism of ${\text{Sym}}^d(X)$ is an isometry.

Final version; to appear in Journal of Geometry and Physics

References in corpus (1)

• On the L^2-metric of vortex moduli spaces