Gabber’s Lemma

For our study of Faltings’ heights on abelian varieties over number fields, it will be convenient if we can compactify the “moduli space” of g-dimensional abelian varieties (over Z) such that the universal abelian scheme extends to a semi-abelian scheme over the compactification. The reason for this wish is that ideally we’d like to set up a theory for which the Faltings height of an abelian variety over a number field K (viewed as a K-point on the moduli space, which in turn sits in the sought-after compactification) is related to the height of a point on a projective variety in the sense of Weil’s original conception of a height function (so we can apply finiteness properties of such “Weil heights” to the study of finiteness problems for abelian varieties.) There are several serious problems that arise in this plan. First of all, to have a meaningful notion of “moduli space” we need to fix the degree of a polarization, yet as we vary even though an isogeny class of abelian varieties over a fixed number field it is not at all clear what degrees are possible for polarization. In particular, it might not happen that the abelian variety admits a principal polarization, nor is even isogenous to one. Over an algebraically closed field the latter can always be achieved, yet this comes at a cost: understanding the effect of isogenies on the Faltings height will be a serious issue to be confronted later, not something we want to deal with right at the start of the theory. A second problem is that even if we focus on a fixed polarization degree, say d, the theory only works nicely over Z[1/d] (not Z, if d > 1) and the “moduli space” is actually a Deligne– Mumford stack and not a scheme. Compactifying a stack (especially over Spec Z[1/d]) is a highly nontrivial matter. In the case g = 1 there is a natural compactification via the Deligne–Rapoport theory of generalized elliptic curves, but beyond dimension 1 matters get very complicated. We’d like to avoid such things. Gabber’s result, to be stated a bit later, provides a substitute which will be sufficient for the purpose of developing properties of Faltings’ heights to prove the Mordell and Tate and Shafarevich conjectures. (For other purposes, such as integral models of Shimura varieties, one has to get into the work of Faltings–Chai and its generalizations on the compactification of moduli stacks.) Gabber’s Lemma (really a theorem) will provide a higher-dimensional generalization of the semi-stable reduction theorems for curves and abelian varieties. Let’s recall the statements of those two fundamental results, which have been discussed in earlier lectures (by Christian and me).