The Canonical Arithmetic Height of Subvarieties of an Abelian Variety over a Finitely Generated Field

This paper is the sequel of [2]. In [4], S. Zhang defined the canonical height of subvarieties of an abelian variety over a number field in terms of adelic metrics. In this paper, we generalize it to an abelian variety defined over a finitely generated field over Q. Our way is slightly different from his method. Instead of using adelic metrics directly, we introduce an adelic sequence and an adelic structure (cf. §§3.1). Let K be a finitely generated field over Q with d = tr. degQ(K), and B = (B;H1, . . . , Hd) a polarization of K, i.e., B is a projective arithmetic variety whose function field is K, and H1, . . . , Hd are nef C -hermitian line bundles on B. Let A be an abelian variety over K, and L a symmetric ample line bundle on A. Fix a projective arithmetic variety A over B and a nef C-hermitian Q-line bundle L on A such that A is the generic fiber of A → B and L is isomorphic to L on A. Then we can assign the naive height h (A,L) (X) to a subvariety