Journal of Algebraic Geometry

Author(s): Huayi Chen
Journal: J. Algebraic Geom. 18 (2009), 575-603.
Posted: December 3, 2008
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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

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

. 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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