The Continuity of Deligne’s Pairing

Theorem A. Let f : X → S be a flat and projective morphism of varieties over C with n = dim f . Let L0, . . . , Ln be C -hermitian line bundles on X. Then, the metric of Deligne’s pairing 〈L0, . . . , Ln〉(X/S) is continuous. Here we note that the C of hermitian line bundles in this note is slightly stronger than the sense of [11] and [8]. For simplicity, we explain it in terms of the C of functions. A function φ on X is said to be C at x ∈ X (in this note) if there are an open neighborhood U of x, a complex manifold V , and a C-function Φ on V such that U is a closed analytic subset of V and φ = Φ|U . As an application of the above theorem, we have the following result, which is, in some sense, a generalization of Kawaguchi’s Hodge index theorem [6].