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Let $\mathcal{X}\rightarrow B$ be a family of projective varities ($B$ is DVR say) whose generic fibre is smooth, but the closed fibre is divisor with normal crossing singulairty. Is there some special cases known where the limiting Hodge structure on the closed fibre is pure? I am asking is there a known sufficient condition on the family or the closed fibre which ensures that the limiting hodge structure is pure?

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  • $\begingroup$ What about compact type curves? $\endgroup$ – Ja ok Oct 22 '17 at 10:33
  • $\begingroup$ Actually, with closed fiber of compact type, the monodromy is trivial. Plus what Arapura said, you get that. $\endgroup$ – Feng Hao Oct 22 '17 at 13:12
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Sure. Let $H$ denote the limit MHS in degree $i$ associated to $\mathcal{X}\to B$, where I will assume that it's semistable. $H$ is isomorphic as a vector space to $H^i(X_t)$ with $t\not=0$. Let $N$ denote the logarithm of monodromy which is unipotent by semi stability. The weight filtration of $H$ is determined by the rules $N W_k\subseteq W_{k-2}$ and $N^k:Gr^W_{i+k}\cong Gr^W_{i-k}$. A little bit of thought shows that

Lemma. $H$ is pure of weight $i$ if and only if $N=0$.

This would hold when $\mathcal{X}\to B$ is smooth, but there are other examples as well.

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