Illinois Journal of Mathematics

On the Kähler structures over Quot schemes

Indranil Biswas and Harish Seshadri

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Let $S^{n}(X)$ be the $n$-fold symmetric product of a compact connected Riemann surface $X$ of genus $g$ and gonality $d$. We prove that $S^{n}(X)$ admits a Kähler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if $n<d$. Let $\mathcal{Q}_{X}(r,n)$ be the Quot scheme parametrizing the torsion quotients of $\mathcal{O}^{\oplus r}_{X}$ of degree $n$. If $g\geq 2$ and $n\leq 2g-2$, we prove that $\mathcal{Q}_{X}(r,n)$ does not admit a Kähler structure such that all the holomorphic bisectional curvatures are nonnegative.

Article information

Illinois J. Math., Volume 57, Number 4 (2013), 1019-1024.

First available in Project Euclid: 1 December 2014

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Zentralblatt MATH identifier

Primary: 14C20: Divisors, linear systems, invertible sheaves 32Q05: Negative curvature manifolds 32Q10: Positive curvature manifolds


Biswas, Indranil; Seshadri, Harish. On the Kähler structures over Quot schemes. Illinois J. Math. 57 (2013), no. 4, 1019--1024. doi:10.1215/ijm/1417442560.

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