Universal vector bundle over the reals
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- by Indranil Biswas and Jacques Hurtubise PDF
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Abstract:
Let $X_{\mathbb R}$ be a geometrically irreducible smooth projective curve, defined over $\mathbb R$, such that $X_{\mathbb R}$ does not have any real points. Let $X = X_{\mathbb R}\times _{\mathbb R} \mathbb C$ be the complex curve. We show that there is a universal real algebraic line bundle over $X_{\mathbb R}\times \text {Pic}^d(X_{\mathbb R})$ if and only if the Euler characteristic $\chi (L)$ is odd for $L \in \text {Pic}^d(X_{\mathbb R})$. There is a universal quaternionic algebraic line bundle over $X\times \text {Pic}^d(X)$ if and only if the degree $d$ is odd. (Quaternionic algebraic vector bundles are defined only on a complexification.)
Take integers $r$ and $d$ such that $r \geq 2$, and $d$ is coprime to $r$. Let ${\mathcal M}_{X_{\mathbb R}}(r,d)$ (respectively, ${\mathcal M}_{X}(r,d)$) be the moduli space of stable vector bundles over $X_{\mathbb R}$ (respectively, $X$) of rank $r$ and degree $d$. We prove that there is a universal real algebraic vector bundle over $X_{\mathbb R}\times {\mathcal M}_{X_{\mathbb R}}(r,d)$ if and only if $\chi (E)$ is odd for $E \in {\mathcal M}_{X_{\mathbb R}}(r,d)$. There is a universal quaternionic vector bundle over $X\times {\mathcal M}_X(r,d)$ if and only if the degree $d$ is odd.
The cases where $X_{\mathbb R}$ is geometrically reducible or $X_{\mathbb R}$ has real points are also investigated.
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Additional Information
- Indranil Biswas
- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
- MR Author ID: 340073
- Email: indranil@math.tifr.res.in
- Jacques Hurtubise
- Affiliation: Department of Mathematics, McGill University, Burnside Hall, 805 Sherbrooke Street W., Montreal, Québec, Canada H3A 2K6
- Email: jacques.hurtubise@mcgill.ca
- Received by editor(s): September 10, 2009
- Received by editor(s) in revised form: January 22, 2010, and January 25, 2010
- Published electronically: July 25, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 6531-6548
- MSC (2010): Primary 14F05, 14P99
- DOI: https://doi.org/10.1090/S0002-9947-2011-05345-6
- MathSciNet review: 2833567