The theta invariants and the volume function on arithmetic varieties

Abstract

We introduce a new arithmetic invariant for hermitian line bundles on arithmetic varieties. We use this invariant to measure the variation of the volume function with respect to the metric. We apply the theory developed here to the study of the arithmetic geometry of toric varieties. As an application, we obtain a generalized Hodge index theorem for hermitian line bundles which are not necessarily toric. When the metrics are toric, we recover some results due to Burgos, Phillippon, Sombra and Moriwaki.