On Kähler structures over symmetric products of a Riemann surface

Abstract

Given a positive integer n n and a compact connected Riemann surface X X , we prove that the symmetric product S n ( X ) S^n(X) admits a Kähler form of nonnegative holomorphic bisectional curvature if and only if genus ( X ) ≤ 1 \text {genus}(X)\, \leq \, 1 . If n n is greater than or equal to the gonality of X X , we prove that S n ( X ) S^n(X) does not admit any Kähler form of nonpositive holomorphic sectional curvature. In particular, if X X is hyperelliptic, then S n ( X ) S^n(X) admits a Kähler form of nonpositive holomorphic sectional curvature if and only if n = 1 ≤ genus ( X ) n\,=\,1\, \leq \, \text {genus}(X) .