In the paper "Arakelov's theorem for abelian varieties" Faltings proves the Shafarevich conjecture for abelian varieties.
The statement is the following:
Let B be smooth projective a curve, S a finite set in B. There exist only finitely many families of principally polarized abelian varieties of dimension $g$ over $B$, with good reduction outside S and satisfying $(*)$
where $(\ast)$ is some technical condition not important in this post.
My concern is about the ground field $K$ on which the abelian varieties of the theorem are defined. It seems that $K$ should be a number field, but in his paper Faltings says that $K$ is any field (but on the other hand at the beginning he fixes the curve $B$ on an algebraically closed field $k$ of char, $0$). I need a clarification about the ground fields because I'm confused.
What happens when $K=\mathbb C$? Here there is no notion of good reduction since we don't have discrete valuations. Is still the theorem true? Does in this case "good reduction outside S" mean that the maps $f:A\to B$ attain non-singular values outside $S$, like in the classical one dimensional Shafarevich conjecture?