Let \(\widehat{A}\) be the Néron model of the abelian variety \(\widehat{A}_K\) dual to the 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 \(\widehat{A}_K\) by a vector group to a smooth and separated group scheme \({\mathcal E}(\widehat{A})\) over \(R\), called the canonical extension of \(\widehat{A}\). In this paper the canonical extension is studied 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, the main result of the paper identifies the identity component of the canonical extension with a certain functor \(\text{Pic}_{X/R}^{\natural,0}\) classifying line bundles on \(X\) that have partial degree zero on all components of geometric fibres and are equipped with a regular connection. Namely, the main result is the following:
Theorem: Let \(X\) be a proper flat and normal model of \(X_K\) over \(S=\text{Spec}(R)\). Suppose that the closed fibre of \(X\) is geometrically reduced and that either \(X\) is regular or that the residue field \(k\) of \(R\) is perfect. Then there is a canonical homomorphism of short exact sequences of smooth group schemes over \(S\) from \[ 0\longrightarrow\omega_J\longrightarrow{\mathcal E}(\widehat{J})^0\longrightarrow\widehat{J}^0\longrightarrow0 \] to \[ 0\longrightarrow f_*\omega_{X/S}\longrightarrow\text{Pic}_{X/R}^{\natural,0}\longrightarrow\text{Pic}_{X/R}^{0}\longrightarrow0 \] which is an isomorphism of exact sequences if and only if \(X\) has rational singularities.
Here, \(\widehat{J}\) is the Néron model of \(\widehat{J}_K\) and \(\omega_{X/S}\) is the relative dualizing sheaf of \(X\) over \(S\), and \(f_*\omega_{X/S}\) is the vector group attached to this locally free \({\mathcal O}_S\)-module, and \(\text{Pic}_{X/R}^{\natural,0}\) is the fppf-sheaf associated to the functor on \(S\)-schemes that assigns to each \(S\)-scheme \(\varphi:T\to S\) the set of isomorphism classes of pairs \(({\mathcal L},\nabla)\), where \({\mathcal L}\) is a line bundle on \(X_T\) whose restriction to all components of each geometric fibre of \(X_T\) has degree zero and \(\nabla:{\mathcal L}\to{\mathcal L}\otimes\varphi^*\omega_{X/S}\) is a regular connection on \({\mathcal L}\) over \(T\) (this last concept, explained in the text, coincides with the familiar notion of connection if \(f:X\to S\) is smooth).
The proof involves perfectness of Grothendieck’s pairing on component groups (which holds true in the cases under discussion) and, as already indicated, a suitable modification of the traditional notion of connection on a line bundle to a possibly non smooth context.
As an application a very interesting comparison isomorphism is proved between two canonical integral structures on the de Rham cohomology of \(X_K\). The paper is very clearly written.