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

 

Essentially finite vector bundles on varieties with trivial tangent bundle


Authors: Indranil Biswas, A. J. Parameswaran and S. Subramanian
Journal: Proc. Amer. Math. Soc. 139 (2011), 3821-3829
MSC (2010): Primary 14L15, 14F05
Published electronically: March 15, 2011
MathSciNet review: 2823029
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Abstract: Let $ X$ be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle $ TX$ is trivial. Let $ F_X : X \longrightarrow X$ be the absolute Frobenius morphism of $ X$. We prove that for any $ n \geq 1$, the $ n$-fold composition $ F^n_X$ is a torsor over $ X$ for a finite group-scheme that depends on $ n$. For any vector bundle $ E \longrightarrow X$, we show that the direct image $ (F^n_X)_*E$ is essentially finite (respectively, $ F$-trivial) if and only if $ E$ is essentially finite (respectively, $ F$-trivial).


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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

A. J. Parameswaran
Affiliation: Kerala School of Mathematics, Kunnamangalam (PO), Kozhikode, Kerala 673571, India
Email: param_aj@yahoo.com

S. Subramanian
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
Email: subramnn@math.tifr.res.in

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10804-9
PII: S 0002-9939(2011)10804-9
Keywords: Essentially finite vector bundle, group–scheme, Frobenius morphism, tangent bundle
Received by editor(s): March 22, 2010
Received by editor(s) in revised form: September 15, 2010, and September 16, 2010
Published electronically: March 15, 2011
Communicated by: Lev Borisov
Article copyright: © Copyright 2011 American Mathematical Society