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?

$\begingroup$ This is close to what I am looking for, but it appears as though Clemens assumes that the monodromy is unipotent. I would like to be able to answer this question in general. Specifically I would like to say that the cohomology of an irreducible component of the special fiber of the Neron model is given by the coinvariants of the limit Hodge structure of A with respect to the monodromy. $\endgroup$ – AJ Stewart Nov 4 '10 at 1:02

$\begingroup$ Since the monodromy is always quasiunipotent you can make a finite cyclic base change branched at zero, apply Clemens' description upstairs, and then analyze the effect of the cyclic group action. $\endgroup$ – Tony Pantev Nov 4 '10 at 3:33

$\begingroup$ Thank you. I was starting to think along these lines, but I wanted to check whether or not the result was in the literature before I started to prove anything. $\endgroup$ – AJ Stewart Nov 5 '10 at 1:54