The classical Néron model gives an extension of a family of abelian varieties over a punctured disc $\triangle^*$ to a family of analytic Lie groups over the whole disc $\triangle$ whose fibre over the origin is an extension of a semi-abelian variety by a finite group. Given a family of projective algebraic varieties.
$$
f: X\longrightarrow \triangle
$$
\noindent whose total space $X$ and fibres $X_s=f^{-1}(s)$ are smooth for $s\not= \{0\}$, and given a relative divisor
$$
Z \in Z^1 (X/\triangle)
$$
whose fundamental class $[Z_s] \in H^2 (X_s, \mathbb Z)$ in zero for $s\not= \{0\}$, we may define the Néron model for the family Pic $(X_s)$ of Picard varieties, and then $Z$ gives a section $\nu_Z$ of the Néron model.
Three extensions of the theory are possible:
(i) to families of intermediate Jacobians over 1-dimensional base spaces;
(ii) to families of abelian varieties over higher dimensional base spaces, and
(iii) the amalgamation of (i) and (ii).
Both (i) and (ii) have been done, and in the talk we shall report on (i) (joint work with Mark Green and Matt Kerr) and (ii) (thesis of Andrew Young). Both of these extensions exhibit interesting, non-classical phenomena.
The extension of (iii) and the ``algebraization'' of the story remain to be done.
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