Slopes of tensor product

The purpose of this chapter is to study the minimal slope of the tensor product of a finite family of adelic vector bundles on an adelic curve. More precisely, give a family (E) over bar (1),..., (E) over bar (d) of adelic vector bundles over a proper adelic curve S, we give a lower bound of (mu) over cap (min)((E) over bar (1)circle times(epsilon),(pi) ... circle times(epsilon),(pi) (E) over bar (d)) in terms of the sum of the minimal slopes of (E) over bar (i) minus a term which is the product of three half of the measure of the infinite places and the sum of ln(dim(K)(E-i)), see Corollary 5.6.2 for details. This result, whose form is similar to the main results of [64, 22, 38], does not rely on the comparison of successive minima and the height proved in [155], which des not hold for general adelic curves. Our method inspires the work of Totaro [143] on p-adic Hodge theory and relies on the geometric invariant theory on Grassmannian. The chapter is organised as follows. In the first section, we regroup several fundamental properties of R-filtrations. We then recall in the second section some basic notions and results of the geometric invariant theory, in particular the Hilbert-Mumford criterion of the semistability. In the third section we give an estimate for the slope of a quotient adelic vector bundle of the tensor product adelic vector bundle, under the assumption that the underlying quotient space, viewed as a rational point of the Grassmannian (with the Plucker coordinates), is semistable in the sense of geometric invariant theory. In the fifth section, we prove a non-stability criterion which generalises [143, Proposition 1]. Finally, we prove in the sixth section the lower bound of the minimal slope of the tensor product adelic vector bundle in the general case.