CONTINUITY OF VOLUMES ON ARITHMETIC VARIETIES

We introduce the volume function for C(infinity)-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(infinity)-hermitian invertible sheaf. We also give other applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker's inequality, etc.