Maximal slope of tensor product of Hermitian vector bundles

TL;DR

This paper establishes an upper bound for the maximal slope of tensor products of Hermitian vector bundles over algebraic integer rings, using invariant theory and reduction techniques.

Contribution

It introduces a new method to estimate the maximal slope by combining classical invariant theory with reduction strategies for semistability.

Findings

Provides an explicit upper bound for the maximal slope.

Connects geometric invariant theory with Arakelov geometry.

Reduces complex cases to semistable cases using Ramanan-Ramanathan results.

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 theorem, we need to estimate the Arakelov degree of an arbitrary Hermitian line subbundle $\bar M$ 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 explicte version of a result of Ramanan and Ramanathan to reduce the general case to the former one.