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Let $X$ be a smooth projective algebraic variety over a field of characteristic zero. Let $U$ be the complement in $X$ of a simple normal crossings divisor $D$. For each degree $k$, put

$h^{p, q}[H^k(U)]:=\dim Gr_F^p Gr_{p+q}^W H^k(U, \mathbb{C}) $

for the Hodge numbers of the mixed Hodge structure on $H^k(U, \mathbb{Q})$, as defined by Deligne.

Consider the polynomial

$H(u, v):=\sum_{p, q, k \geq 0} (-1)^k h^{p, q} [H^k(U)]u^p v^q$

I would like to know the relation between $H(u, v)$ and $$ \sum (-1)^k \sum_{p+q=k} \dim H^q(X, \Omega^p(\log D)) u^p v^q $$

Are they equal?

I guess the answer comes just from the degeneration of some spectral sequence, but I'm a beginner in this subject, so any help would be very appreciated.


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