Maximal slope of tensor product of Hermitian vector bundles

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.