Global Adelic Space of an Arithmetic Variety

This chapter is devoted to the construction of the global adelic space of an arithmetic variety. This construction will be useful further in the study of the equidistribution of closed subvarieties. In Sect. 7.1 we establish a link between metric family on the trivial invertible sheaf and family of continuous functions on local analytifications. In Sect. 7.2 we prove the measurability of partial derivatives, which will be useful in the proof of Bogomolov type conjecture over an adelic curve with Archimedean places. In Sect. 7.3, we interpret the arithmetic \(\chi \) -volume by concave transform on the Newton-Okounkov body and show its convexity with respect to choices of metric families. In Sect. 7.4, we prove the Gâteaux differentiability of the arithmetic \(\chi \) -volume. In Sect. 7.5, we prove the measurability of fibre integrals of a measurable family of continuous functions. In Sects. 7.6 and 7.7, we construct the global adelic space of an arithmetic variety, which is a measure space fibred over the adelic curve, admitting the fibre integrals as the disintegration with respect to the base measure.