Canonical extensions of N\'eron models of Jacobians

TL;DR

This paper extends the understanding of Néron models of Jacobians by identifying the canonical extension with a functor classifying line bundles with connections, and compares integral structures on de Rham cohomology.

Contribution

It generalizes Raynaud's theorem by describing the identity component of the canonical extension for Jacobians with regular models.

Findings

Identifies the identity component of the canonical extension with a functor of line bundles with connections.

Proves a comparison isomorphism between two canonical integral structures on de Rham cohomology.

Extends classical results to cases with regular models and partial degree conditions.

Abstract

Let A be the N\'eron model of an abelian variety A_K over the fraction field K of a discrete valuation ring R. Due to work of Mazur-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. In this paper, 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^{\natural,0}_{X/R} 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 N\'eron model J of J_K with the functor Pic^0_{X/R}. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of X_K.