Canonical extensions of Néron models of Jacobians

Cais, Bryden

doi:10.2140/ant.2010.4.111

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Canonical extensions of Néron models of Jacobians

Bryden Cais

Vol. 4 (2010), No. 2, 111–150

DOI: 10.2140/ant.2010.4.111

Abstract

Let $A$ be the Néron model of an abelian variety $A_{K}$ over the fraction field $K$ of a discrete valuation ring $R$ . By work of Mazur and Messing, there is a functorial way to prolong the universal extension of $A_{K}$ by a vector group to a smooth and separated group scheme over $R$ , called the canonical extension of $A$ . Here we study the canonical extension when $A_{K} = J_{K}$ is the Jacobian of a smooth, proper and geometrically connected curve $X_{K}$ over $K$ . Assuming that $X_{K}$ admits a proper flat regular model $X$ over $R$ that has generically smooth closed fiber, our main result identifies the identity component of the canonical extension with a certain functor ${Pic}_{X ∕ R}^{♮, 0}$ classifying line bundles on $X$ that have partial degree zero on all components of geometric fibers and are equipped with a regular connection. This result is a natural extension of a theorem of Raynaud, which identifies the identity component of the Néron model $J$ of $J_{K}$ with the functor ${Pic}_{X ∕ R}^{0}$ . As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of $X_{K}$ .

Keywords

canonical extensions, Néron models, Jacobians, relative Picard functor, group schemes, Grothendieck's pairing, Grothendieck duality, integral structure, de Rham cohomology, abelian variety, rigidified extensions

Mathematical Subject Classification 2000

Primary: 14L15

Secondary: 14F30, 14F40, 11G20, 14K30, 14H30

Milestones

Received: 28 October 2008

Revised: 18 July 2009

Accepted: 15 August 2009

Published: 26 January 2010

Authors

Bryden Cais
Department of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montréal, QC H3A 2K6 Canada
http://www.math.mcgill.ca/bcais/

Vol. 4, No. 2, 2010