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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Homomorphisms of vector bundles on curves and parabolic vector bundles on a symmetric product


Authors: Indranil Biswas and Souradeep Majumder
Journal: Proc. Amer. Math. Soc. 140 (2012), 3017-3024
MSC (2010): Primary 14F05, 14H60
Published electronically: January 24, 2012
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Abstract: Let $ S^n(X)$ be the symmetric product of an irreducible smooth complex projective curve $ X$. Given a vector bundle $ E$ on $ X$, there is a corresponding parabolic vector bundle $ {\mathcal V}_{E*}$ on $ S^n(X)$. If $ E$ is nontrivial, it is known that $ {\mathcal V}_{E*}$ is stable if and only if $ E$ is stable. We prove that

$\displaystyle H^0(S^n(X),\, {\mathcal H}om_{\rm par}({\mathcal V}_{E*}\, , {\ma... ...\!=\! H^0(X,\, F\otimes E^\vee )\oplus (H^0(X,\, F)\otimes H^0(X,\, E^\vee )). $

As a consequence, the map from a moduli space of vector bundles on $ X$ to the corresponding moduli space of parabolic vector bundles on $ S^n(X)$ is injective.

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Additional Information

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

Souradeep Majumder
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
Email: souradip@math.tifr.res.in

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11227-4
PII: S 0002-9939(2012)11227-4
Keywords: Symmetric product, parabolic vector bundle, homomorphism, stability
Received by editor(s): March 29, 2011
Published electronically: January 24, 2012
Communicated by: Lev Borisov
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.