On Kaehler structures over symmetric products of a Riemann surface

Indranil Biswas

On Kaehler structures over symmetric products of a Riemann surface

Algebraic Geometry 2012-03-02 v3 Differential Geometry

Authors: Indranil Biswas

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.

Keywords

Kaehler structure symmetric product Riemann surface holomorphic bisectional curvature gonality

Cite

@article{arxiv.1106.3733,
  title  = {On Kaehler structures over symmetric products of a Riemann surface},
  author = {Indranil Biswas},
  journal= {arXiv preprint arXiv:1106.3733},
  year   = {2012}
}

Comments

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

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