On Kähler structures over symmetric products of a Riemann surface

Biswas, Indranil

doi:10.1090/S0002-9939-2012-11732-0

On Kähler structures over symmetric products of a Riemann surface
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Proc. Amer. Math. Soc. 141 (2013), 1487-1492 Request permission

Abstract:

Given a positive integer $n$ and a compact connected Riemann surface $X$, we prove that the symmetric product $S^n(X)$ admits a Kähler 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 Kähler form of nonpositive holomorphic sectional curvature. In particular, if $X$ is hyperelliptic, then $S^n(X)$ admits a Kähler form of nonpositive holomorphic sectional curvature if and only if $n = 1 \leq \text {genus}(X)$.

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