Raynaud Extensions

Abstract

In the last chapter we presented the uniformization $J_{K}=\widehat {J}_{K}/M$ of the Jacobian variety $J_{K}$ of a connected smooth projective curve. The universal covering $\widehat{J}_{K}$ is a Raynaud extension; i.e. an affine torus extension of the generic fiber of a formal abelian $R$-scheme. The new topic in this chapter is the algebraization result for $\widehat{J}_{K}$; i.e., that $\widehat{J}_{K}$ is an algebraic torus extension of an abelian variety with good reduction.

We study this in the more general setting of uniformized abeloid varieties; i.e., of rigid analytic groups in Raynaud representation $E_{K}/M$, where $E_{K}$ is a Raynaud extension and where $M\subset E_{K}$ is a lattice of rank equal to the torus part of $E_{K}$. This requires a systematic study of Raynaud extensions and their line bundles with $M$-action. Thus, one is led to the construction of the dual of a uniformized abeloid variety. The algebraization of a uniformized abeloid variety is related to the existence of a polarization.

Of special interest are the polarizations of Jacobians $\operatorname{Jac}(X)$. There are two, the usual theta polarization and the canonical polarization which is related to a pairing on the homology group $H_{1}(X,\mathbb{Z})$ of the curve $X$. In Sect. 6.5 we discuss these polarizations. This is related to Riemann’s vanishing theorem Corollary 2.9.16 for Mumford curves.

In Sect. 6.6, following the article (Bosch and Lütkebohmert in Topology 30:653–698, 1991) we discuss the results of this chapter on the degeneration data of abelian varieties and compare them with the ones established in Faltings and Chai (Degeneration of Abelian Varieties, vol. 22, Springer, Berlin/Heidelberg/New York, 1990). Prerequisites on torus extensions and cubical structures are surveyed in the Appendix.