On Kaehler structures over symmetric products of a Riemann surface

TL;DR

This paper characterizes when symmetric products of Riemann surfaces admit Kähler forms with nonnegative or nonpositive holomorphic bisectional curvature, depending on genus and gonality.

Contribution

It provides necessary and sufficient conditions for the existence of certain Kähler forms on symmetric products of Riemann surfaces based on genus and gonality.

Findings

Kähler form with nonnegative holomorphic bisectional curvature exists iff genus ≤ 1

No Kähler form with nonpositive holomorphic sectional curvature if n ≥ gonality

Conditions depend on genus and gonality of the 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 $\text{genus}(X) \leq 1$. 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.