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On Kähler structures over symmetric products of a Riemann surface


Author: Indranil Biswas
Journal: Proc. Amer. Math. Soc. 141 (2013), 1487-1492
MSC (2010): Primary 14C20, 32Q05, 32Q10
DOI: https://doi.org/10.1090/S0002-9939-2012-11732-0
Published electronically: September 26, 2012
MathSciNet review: 3020836
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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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Additional Information

Indranil Biswas
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
Email: indranil@math.tifr.res.in

DOI: https://doi.org/10.1090/S0002-9939-2012-11732-0
Keywords: Riemann surface, symmetric product, Kähler form, curvature
Received by editor(s): August 25, 2011
Published electronically: September 26, 2012
Communicated by: Varghese Mathai
Article copyright: © Copyright 2012 American Mathematical Society

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