zbMATH — the first resource for mathematics

On rigid-analytic Picard varieties. (English) Zbl 1044.14007
Let \(R\) be a complete discrete valuation ring with field of fractions \(K\), and let \(X\) be a smooth, proper, connected rigid analytic space over \(K\) which possesses a \(K\)-rational point. Under the hypothesis that \(X_K\) has a strict semi-stable formal model over \(R\), the authors construct a rigid analytic variety, \(\text{Pic}_{X/K}\), representing the rigid analytic Picard functor of \(X\), and the Poincaré line bundle on \(\text{Pic}_{X/K} \times X\). They show that the connected component of \(\text{Pic}_{X/K}\) is, after a finite base extension, an extension of a smooth abeloid variety (a smooth proper rigid group variety) by an affine torus. They also show that the rigid Néron-Severi group is finitely generated.

14G22 Rigid analytic geometry
14K30 Picard schemes, higher Jacobians
14C22 Picard groups
Full Text: DOI
[1] [Ar 1] Artin, M., Algebraization of formal moduli I, Global Analysis, Papers in Honor of K. Kodaira, Princeton University Press (1969), 21-71.
[2] [Ar 2] Artin, M., The Implicit Function Theorem in Algebraic Geometry, Arithmetical Algebraic Geometry, ProceedingsPaper of the Conference on Arithmetic Algebraic Geometry at Purdue University in 1963, Harper and Row, New York (1965), 13-34.
[3] Bass H., Ann. Math. 86 pp 16– (1967)
[4] [B] Blanchard, A., Les varieAteAs analytiques complexes, E.N.S. 73 (1956), 157-202. · Zbl 0073.37503
[5] Bosch S., Springer Grundl. pp 261– (1984)
[6] Bosch S., Topology 30 (4) pp 653– (1991)
[7] Bosch S., Springer Ergeb. 3. Folge pp 21– (1990)
[8] Bosch S., Math. Ann. 295 pp 291– (1993)
[9] Bosch S., Math. Ann. 301 pp 339– (1995)
[10] [Bo] Bourbaki, N., Commutative Algebra, Elements of Mathematics, Hermann, Paris1972.
[11] de Jong A. J., Publ. Math. IHE 83 pp 51– (1996)
[12] Demazure M., Springer Lect. Notes Math. 151 pp 152– (1970)
[13] Freitag E., Folge pp 13– (1988)
[14] Ge, Compos. Math. 25 pp 23– (1977)
[15] Grothendieck A., Publ. Math. IHES 4 pp 8–
[16] [FGA] Grothendieck, A., Fondements de la GeAomeAtrie AlgeAbrique, SeAminaire Bourbaki 1957-62, SecreAtariat MatheAmatiques, Paris 1962.
[17] [TCGA] Grothendieck, A., Techniques de construction en geAomeAtrie analytique. I-X, SeAm.Cartan13 (1960/61), n07-16.
[18] Grothendieck A., Springer Lect. Notes Math. pp 224– (1971)
[19] Grothendieck A., Springer Lect. Notes Math. pp 225– (1971)
[20] [TE] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B., Toroidal embeddings I, Springer Lect. Notes Math.339, Berlin-Heidelberg-New York 1973.
[21] [Ki] Kiehl, R., Analytische Familien a noider Algebren, Heidelberger Sitzungsber., Springer-Verlag (1968), 25-49.
[22] LuEtkebohmert W., Math. Ann. 286 pp 341– (1990)
[23] LuEtkebohmert W., Math. 468 pp 167– (1995)
[24] [Mo] Moret-Bailly, L., Pinceaux de VarieAteAs AbeAliennes, AsteArisque 129 (1985).
[25] Ms, Funct. Anal. Appl. 11 pp 234– (1977)
[26] [Ra 1] Raynaud, M., VarieAteAs abeAliennes et geAomeAtrie rigide, Actes du congreAs international de Nice 1 (1970), 473-477.
[27] Ra, Soc. Math. France 39 pp 319– (1974)
[28] [Se] Serre, J.P., Groupes algeAbriques et corps de classes, Hermann, Paris 1959.
[29] Ta, Invent. Math. 12 pp 257– (1971)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.