On the Kähler metrics over Sym < sup > d < /sup > (X)

Abstract

Let X be a compact connected Riemann surface of genus g, with g≥2. For each d < η(X), where η(X) is the gonality of X, the symmetric product Symd(X) embeds into Picd(X) by sending an effective divisor of degree d to the corresponding holomorphic line bundle. Therefore, the restriction of the flat Kähler metric on Picd(X) is a Kähler metric on Symd(X). We investigate this Kähler metric on Symd(X). In particular, we estimate it is Bergman kernel. We also prove that any holomorphic automorphism of Symd(X) is an isometry.