Abeloid Varieties

Abstract

Every connected compact complex Lie group of dimension $g$ can be presented as a quotient $\mathbb{C}^{g}/\varLambda$ of the affine vector group $\mathbb{C}^{g}$ by a lattice $\varLambda$ of rank $2g$. From the multiplicative point of view, it can be presented as a quotient $\mathbb{G}_{m,\mathbb{C}}^{g} /M$ of the affine torus $\mathbb{G}_{m,\mathbb{C}}^{g}$ by a multiplicative lattice $M$ of rank $g$. In the rigid analytic case the situation is more complicated because of the phenomena of good and multiplicative reduction, which in general occur in a twisted form. For example look at the rigid analytic uniformization of abelian varieties of Theorem 5.6.5.

The fundamental example of a proper rigid analytic group $A_{K}$ is the analytic quotient $A_{K} = E_{K} / M_{K}$ in Raynaud representation; cf. Definition 6.1.5, where $E_{K}$ is an extension of a proper rigid analytic group $B_{K}$ with good reduction by an affine torus $T_{K}$, where $M_{K}$ is a lattice in $E_{K}$ of rank equal to $\dim T_{K}$; cf. Proposition 6.1.4. The main result of this chapter is that every smooth rigid analytic group, which is proper and connected, is of the form $E_{K} / M_{K}$ after a suitable extension of the base field. This is a generalization of Grothendieck’s Stable Reduction Theorem (Grothendieck in Groupes de Monodromie en Géométrie Algébrique, vols. 288, 340, Springer, Berlin/Heidelberg/New York, 1972; I, Exp. IX, 3.5) as well as of the rigid analytic uniformization of abelian varieties.

The proof requires advanced techniques; it mainly relies on the stable reduction theorem for smooth curve fibrations which are not necessarily proper. In Sect. 7.5 we compactify such a curve fibration by using the Relative Reduced Fiber Theorem 3.4.8 and approximation techniques provided in Sect. 3.6. Then we can apply the moduli space of marked stable curves. Therefore, one can cover the given group $A_{K}$ by a finite family of smooth curve fibrations with semi-stable reduction.

In a second step one deduces from such a covering the largest open subgroup $\overline{A}_{K}$ which admits a smooth formal $R$-model $\overline{A}$ by well-known techniques on group generation dating back to A. Weil; cf. Sect. 7.2. The formal group $\overline{A}$ is a formal torus extension of a formal abelian $R$-scheme $B$. The prolongation of the embedding $\overline{T}\hookrightarrow\overline{A}$ of the formal torus to a group homomorphism $T_{K}\to A_{K}$ of the associated affine torus $T_{K}$ follows by the approximation theorem and a discussion on the convergence of group homomorphisms; cf. Sect. 7.3.

Thus, the group homomorphism $\overline{A}_{K}\to A_{K}$ extends to a group homomorphism from the push-out $\widehat{A}_{K}:=T_{K}\amalg_{\overline {T}}\overline {A}$ to $A_{K}$. The surjectivity of the map $\widehat{A}_{K}\to A_{K}$ is shown by an analysis of the map from the curve fibration to $A_{K}$. In fact, the whole torus part is induced by the double points in the reduction of the stable curve fibration; cf. Sect. 7.4.

So far we are concerned only with the case, where the base field is algebraically closed. But it is not difficult to see that the whole approach can be done after a suitable finite separable field extension if one starts with a non-Archimedean field which is not algebraically closed.

If the non-Archimedean field in question has a discrete valuation, there is a notion of a formal Néron model. Then our result implies a semi-abelian reduction theorem for such Néron models. As a further application one can deduce that every abeloid variety has a dual; i.e., the Picard functor of translation invariant line bundles on $A_{K}$ is representable by an abeloid variety.