On Kaehler structures over symmetric products of a Riemann surface

Details On Kaehler structures over symmetric products of a Riemann surface On Kaehler structures over symmetric products of a Riemann surface

2011-06-19

arxiv.org/abs/1106.3733v3

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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