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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Universal vector bundle over the reals


Authors: Indranil Biswas and Jacques Hurtubise
Journal: Trans. Amer. Math. Soc. 363 (2011), 6531-6548
MSC (2010): Primary 14F05, 14P99
Published electronically: July 25, 2011
MathSciNet review: 2833567
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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$   Pic$ ^d(X_{\mathbb{R}})$ if and only if the Euler characteristic $ \chi(L)$ is odd for $ L \in$   Pic$ ^d(X_{\mathbb{R}})$. There is a universal quaternionic algebraic line bundle over $ X\times$   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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05345-6
PII: S 0002-9947(2011)05345-6
Keywords: Real curve, moduli space, real universal bundle
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.