Slopes of tensor product

Abstract

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 \(\overline E_1,\ldots ,\overline E_d\) of adelic vector bundles over a proper adelic curve S, we give a lower bound of \(\widehat {\mu }_{\min }(\overline E_1\otimes _{\varepsilon ,\pi }\cdots \otimes _{\varepsilon ,\pi }\overline {E}_d)\) in terms of the sum of the minimal slopes of \(\overline E_i\) 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 \(\mathbb R\)-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.