9
$\begingroup$

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.

Thanks!

$\endgroup$
0

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.