On Kaehler structures over symmetric products of a Riemann surface

Abstract

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

Final version; to appear in the Proc. of A.M.S

References in corpus (1)

• On the curvature of vortex moduli spaces