Extending abelian schemes and their polarizations from an open subset

Question

Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$. Suppose that $S\backslash U$ has codimension at least $2$ and that for every point $s \in S$, the ramification index at the stalk $\mathcal{O}_{S,s}$ is $1$, so that the hypotheses for the Faltings-Chai and Vasiu-Zink criteria hold at each stalk. Then

1. Can $A$ be extended to an abelian scheme over $S$?
2. If $A$ comes equipped with a principal polarization $\lambda$, and $A$ extends to an abelian scheme $B$ over $S$, can we extend $\lambda$ to a principal polarization on $B$?

Take a generic point $\eta$ of an irreducible component of $S \backslash U$. The stalk $\mathcal{O}_{S,\eta}$ has a map $f:\textrm{Spec}(\mathcal{O}_{S,\eta}) \to S$ and we have an abelian scheme $f^*A$ defined outside the maximal point of $\mathcal{O}_{S,\eta}$. We are in a position to apply either Faltings-Chai or Vasiu-Zink to extend $f^*A$ uniquely to an abelian scheme $A'_{\eta}$ over the stalk. We can then spread this out to an abelian scheme $A'_{U_{\eta}}$ over some open $U_{\eta}$ containing $\eta$.

For any other point $\zeta$ so that $\eta \in \overline{\zeta}$, we know $\zeta \in U \cap U_{\eta}$ and if $g: \textrm{Spec}(\mathcal{O}_{S,\zeta}) \to S$ is the natural map then $g^*A \cong g^*A'_{U_{\eta}}$ (abusing notation slightly). We can spread this out to an isomorphism of abelian schemes $g_{\zeta} : A |_{W_{\zeta}} \to A'_{U_{\eta}}|_{W_{\zeta}}$ over some open $W_{\zeta}$ around $\zeta$.

As long as the cocycle conditions are satisfied, we would be able to glue $A$ and $A_{U_{\eta}}'$, after potentially shrinking $U_{\eta}$, to an abelian scheme over a strictly larger open $U \cup U_{\eta}$. The logic behind the previous paragraph ensures that $\eta$ remains in $U_{\eta}$ after shrinking. Now a Zorn's lemma type argument allows us to conclude (1).

The cocycle conditions hold at least generically (after taking generic fibre above $S$), so by rigidity of abelian schemes, I suspect that they hold, and the question (1) is indeed true.

As for (2), the line bundle $L$ on $A$ which induces the principal polarization can be extended over $B$ to some $\overline{L}$, and we obtain a morphism $\phi_{\overline{L}}:B \to B^{\vee}$. I suspect that relative ampleness can be verified pointwise on $S$, and supposing that $\overline{L}$ is a polarization, it must be principal since it is so generically.

Answer 1

Here is a general outline. We can even extend over codimension $1$ points if we can verify good reduction.

$\textbf{Proposition}:$ Let $U \subset S$ be an open subset of a smooth scheme over $\mathbb{Z}$, with complement of codimension at least $2$. Let $A_U$ be an abelian scheme over $U$. Then $A_U$ extends to an abelian scheme $A$ over $S$.

$\textit{proof}:$ Consider the poset of extensions $(U \subset W,A_W)$ of $A_U$. It suffices to show that the maximal element is defined over $S$. Take a maximal element $(U \subset W,A_W)$ and suppose $W \neq S$. Let $\eta \in S \backslash W$ be a generic point of an irreducible component of the complement, and consider the stalk $S_{(\eta)}$. If this local ring is of mixed characteristic, then it is a quasi-healthy local ring, since the ramification index is $1$, and so we are able to use the Vasiu-Zink criterion. If it has residue characteristic $0$, then we are able to use the Faltings-Chai criterion. The conclusion is that since $A_W \big |_{S_{(\eta)}}$ is defined everywhere outside the maximal point, it extends uniquely to an abelian scheme $A_{(\eta)}/ S_{(\eta)}$.

We now proceed to spread $A_{(\eta)}$ out to an abelian scheme $A_V'$ defined over an open neighbourhood $V$ of $\eta$. The next step is to glue this abelian scheme with $A_W$, after possibly shrinking $V$, to a strictly larger abelian scheme.

Let $\zeta$ be the generic point of some closed subscheme of $S$ containing $\overline{\eta}$. Then $\phi_{\zeta}: (A_V')_{(\zeta)} \xrightarrow{\sim} (A_W)_{(\zeta)}$ as abelian schemes over the stalk, and this spreads out to an isomorphism of abelian schemes $\phi_{W'_{\zeta}}$ in some open subset $W'_{\zeta} \subset W \cap V$ which contains $\zeta$. Consider $W' = \cup_{\zeta}W'_{\zeta} \subset W \cap V$. By removing some closed subsets, we can shrink $V$ so that $W \cap V = W'$, and $\eta$ still lies in $V$ since by construction, we cannot have removed any closed subset containing $\eta$.

Now we are in a position to glue to an abelian scheme $A_{V' \cup W}$ over $V' \cup W$ which is strictly larger. We just need to verify the cocycle conditions for $\{\phi_{W'_{\zeta}}\}_{\zeta}$. More precisely, for any $\zeta_1,\zeta_2$ we must verify $$\phi_{W'_{\zeta_1}} \big |_{W'_{\zeta_1} \cap W'_{\zeta_2}} = \phi_{W'_{\zeta_2}} \big |_{W'_{\zeta_1} \cap W'_{\zeta_2}}$$ If $\eta_S$ is the generic point of $S$, then actually $(\phi_{W'_{\zeta_1}})_{(\eta_S)} = (\phi_{W'_{\zeta_2}})_{(\eta_S)}$, and due to rigidity of abelian schemes, the two maps above are indeed equal. This gives us a contradiction and so in fact $W = S$ as required.

As for (2), the line bundle $L$ on $A$ which induces the principal polarization can be extended over $B$ to some line bundle $L_B$. By the MO post below, the property of being a polarization is open and closed on the base, so $L_B$ is a polarization. Moreover it is principal because the degree of an isogeny of abelian schemes is constant on the base.

Is a polarization on an abelian scheme an open condition?