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Let $X$ be a smooth projective variety and $D$ a log-canonical divisor and let $U = X \setminus D$. I have heard the slogan "log-canonical is just as good as normal crossings for Hodge theory". This question concerns how to make that slogan precise as applied to the cohomology of $U$.

If $D$ were normal crossings, then $H^{\ast}(U, \mathbb{C})$ would be the hypercohomology of the complex $\Omega^p(\log D)$ on $X$. Moreover, the spectral sequence of that hypercohomology would degenerate at $E_1$. I could compute the weight filtration on $H^{\ast}(U, \mathbb{C})$ by filtering $\Omega^p(\log D)$ according to how many poles my $p$-forms have along $D$.

What are the analogous statements for a log-canonical divisor? Let me emphasize that I would like statements which are true on $X$, without passing to a resolution of singularities.

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By "$D$ a divisor with log canonical singularities" do you mean that the pair $(X,D)$ is log canonical? – ulrich Apr 10 '13 at 4:06
I want $D$ a log-canonical divisor (what I wrote was a case of not thinking.) Just to check that we are talking about the same thing, this means that there is a birational map $p: Y \to X$ such that $Y$ is smooth, $p^{-1}(X)$ is normal crossings and $p^{\ast}(K_X + D) = K_Y + \sum e_i E_i$ with $E_i$ the exceptional divisors of $p$ and $e_i \leq 1$. – David Speyer Apr 10 '13 at 16:43

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