Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Maximal slope of tensor product of Hermitian vector bundles


Author: Huayi Chen
Journal: J. Algebraic Geom. 18 (2009), 575-603
DOI: https://doi.org/10.1090/S1056-3911-08-00513-4
Published electronically: December 3, 2008
MathSciNet review: 2496458
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Abstract: We give an upper bound for the maximal slope of the tensor product of several non-zero Hermitian vector bundles on the spectrum of an algebraic integer ring. By Minkowski's First Theorem, we need to estimate the Arakelov degree of an arbitrary Hermitian line subbundle $ \overline M$ of the tensor product. In the case where the generic fiber of $ M$ is semistable in the sense of geometric invariant theory, the estimation is established by constructing, through the classical invariant theory, a special polynomial which does not vanish on the generic fibre of $ M$. Otherwise we use an explicit version of a result of Ramanan and Ramanathan to reduce the general case to the former one.


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Huayi Chen
Affiliation: CMLS, Ecole Polytechnique, Palaiseau 91120, France
Address at time of publication: Institut de Mathématiques de Jussieu, Université Paris 7 Diderot, 175, rue du Chevaleret 75013 Paris, France
Email: huayi.chen@polytechnique.org

DOI: https://doi.org/10.1090/S1056-3911-08-00513-4
Received by editor(s): June 1, 2007
Received by editor(s) in revised form: January 24, 2008
Published electronically: December 3, 2008

American Mathematical Society