Rigid analytic Picard varieties, Néron-Severi Group, linebundles on polystable fibrations of formal schemes
by
Urs T. Hartl
University of Ulm, Germany

For a smooth proper rigid analytic variety X over a complete discretely valued field, which has a semistable formal model, we construct the Picard variety, that represents the Picard functor on the category of smooth rigid analytic varieties. Its connected component is an extension of a smooth proper rigid group by an affine torus. The Néron-Severi group of X is finitely generated. In fact it is a subquotient of the Néron-Severi group of the special fiber of the formal model.

In the proof we reduce to the theorem on the representability of the Picard functor by M. Artin. This is done by extending line bundles on the rigid space to the formal model. Although not possible in general, we achieve this locally in the special situation of a product of two semistable formal schemes (``polystable fibration''). We can also control the obstruction to the global extension. It is given by the multiplicative part of the line bundle. This part is responsible for the torus in the Picard variety.

Date received: March 15, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacv-17.