Maximal slope of tensor product of Hermitian vector bundles

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 M ¯ \overline M of the tensor product. In the case where the generic fiber of M 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 M . Otherwise we use an explicit version of a result of Ramanan and Ramanathan to reduce the general case to the former one.