Positivity of Heights of Semi-stable Varieties

Let K be a number field and OK its ring of integers. Let E be an OK-module of rank N + 1 in P(E), the projective space representing lines in E. For all closed subvarieties X ⊆ P(E K) of dimension d let deg(X) be its degree with respect to the canonical line bundle O(1) of P(E K). If E is endowed with the structure of hermitian vector bundle over Spec(OK) we can define the Arakelov degree d̂eg ( E ) and the Faltings height hE(X) (see 3.1). Let V be the direct limit of the set VK of OK-modules E with an identity EK ∼= K as K varies in the set of number fields. Let X ⊆ P be a closed irreducible projective variety of dimension d defined over Q. We define