On the Kähler metrics over Symd(X)

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