Canonical extensions of néron models of jacobians

Let A be the néron model of an abelian variety A_κ 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 _κ 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_κ = J_κ is the Jacobian of a smooth, proper and geometrically connected curve X_κ over K. Assuming that X_κ 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 ^b,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 Néron model J of J _k with the functor Pic⁰_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_κ.