Arithmetic Volumes over a General Adelic Curve

In this chapter, we use the results of Chap. 3 to study the volume functions of a normed graded algebra over a general adelic curve. The main idea is to cast to the trivial valuation case by Harder-Narasimhan filtration. In Sect. 4.1, we introduce the notion of graded algebra of adelic vector bundles and its casting, which is a normed graded algebra over a trivial valued field. In Sect. 4.2, we show that the sequence appearing on the left-hand side of the arithmetic Hilbert-Samuel formula actually converges, by using the convergence result established in Chap. 3 . In Sect. 4.3, we discuss normed graded modules, which are used in Sect. 4.4 to obtain bounds of \(\chi \) -volume in the case where the tensor product with a metrized torsion-free sheaf appears.