Continuity of volumes on arithmetic varieties

Abstract

We introduce the volume function for C ∞ C^{\infty } -hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic Hilbert-Samuel formula for a nef C ∞ C^{\infty } -hermitian invertible sheaf. We also give other applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker’s inequality, etc.