Let $\mathcal{A}$ be an Abelian scheme over a smooth curve $S^*\subset S$ and let $\mathcal{A_S}$ be the Neron model of $\mathcal{A}$ over $S$. Is it possible to describe the special fiber of the Neron model, $\mathcal{A_S}$, in terms of the limit Hodge structure of $\mathcal{A}$?
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Take a look at this classical paper of Clemens. Maybe this is what you are looking for? 

