Shafarevich conjecture for abelian varieties

Question

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?

Answer 1

Let $B$ be a smooth projective curve over an algebraically closed field of characteristic zero. Let $K$ be the function field of $B$. Let $S$ be a finite set of closed points of $B$.

You might find the following reformulation of Faltings's theorem less confusing.

Theorem 1. (Faltings, geometric Shafarevich conjecture) Let $g$ be an integer. Then the set of $K$-isomorphism classes of $g$-dimensional principally polarized abelian varieties $A$ over $K$ with good reduction over $B-S$ and satisfying condition $(*)$ is finite.

If you're used to thinking about number fields, then you might be interested in reading Faltings's papers on the arithmetic analogue of the aforementioned finiteness result:

Theorem 2. (Faltings, arithmetic Shafarevich conjecture) Let $K$ be a number field and let $S$ be a finite set of closed points of $B = $ Spec $O_{K}[S^{-1}]$. Let $g$ be an integer. Then the set of $K$-isomorphism classes of $g$-dimensional principally polarized abelian varieties over $K$ with good reduction over $B-S$ is finite.

Remark 1. You can get rid of the words "principally polarized" via Zarhin's trick. See Szpiro's 1985 asterisque on the Mordell conjecture.

Remark 2. The reason you need condition * in Theorem 1 (i.e., the geometric Shafarevich conjecture) is because of the existence of non-rigid families of abelian varieties. This is an artifact of the function field realm which does not appear in the number field realm. If you don't want to use assumption * the best you can hope for is finiteness of the set of $K$-deformation classes (and not $K$-isomorphism classes).

Remark 3. Shafarevich didn't actually conjecture the above finiteness statements, as far as I know. He conjectured the analogous statements for curves of genus at least two (proven by Arakelov-Parshin in the function field case, and Faltings in the number field case).