Let $X$ be a smooth complex variety (not necessarily compact) and let $D$ be a normal crossings divisors with components $D_1$, $D_2$, …, $D_N$. For a set of indices $I$, let $D_I = \bigcap_{i \in I} D_i$ and let $D^{\circ}_I = D_I \setminus \bigcup_{J \supsetneq I} D_J$. I would like to compute $H^{\ast}(X)$ in the case that I know all of the $H^{\ast}(D_I^{\circ})$.
It seems to me that there should be a spectral sequence whose first page is $\bigoplus_{\#(I)=p} H^{q-p}(D^{\circ}_I)$. If $I' \supset I$ with $\#(I') = \#(I)+1=p+1$, then the map $H^{q-p}(D^{\circ}_{I}) \to H^{q-(p+1)}(D^{\circ}_{I'})$ would given by the Gysin map (up to standard sign issues).
Does this sequence have a name, or is it a special case of something which has a name? Where can I read about it?
Best Answer
Let $X = T_n \supset T_{n-1} \supset \cdots \supset T_{-1} = \varnothing$ be a topological space filtered by closed subspaces, where for simplicity I assumed the filtration bounded. Then there is a spectral sequence $$ E_1^{pq} = H^{p+q}_c(T_p \setminus T_{p-1}) \implies H^{p+q}_c(X).$$ (Some mild point-set assumptions are required for this and what follows to be true, but let me disregard that.) As you noted in a comment, this gives the spectral sequence you want by Poincaré duality.
Usually, the spectral sequence of a filtered space is written in terms of ordinary cohomology rather than cohomology with compact support. One reason is that the compactly suppported spectral sequence is a special case of the usual one. Recall that if $U \subset \overline U$ is any compactification of a space $U$, then $H^\bullet_c(U) = H^\bullet(\overline U, \partial U)$, where $\partial U = \overline U \setminus U$. So if we choose an arbitrary compactification $X \subset \overline X$ and let $\overline T_p$ be the closure of $T_p$ in $\overline X$, then we get a filtration $$\overline X = \overline T_n \cup \partial X \supset \overline T_{n-1} \cup \partial X \supset \overline T_{n-2} \cup \partial X \supset \cdots \supset \overline T_{-1} \cup \partial X = \partial X.$$
Now the usual spectral sequence of a filtration reads $$ E_1^{pq} = H^{p+q}(\overline T_p \cup \partial X,\overline T_{p-1}\cup \partial X) = H^{p+q}(\overline T_p,\overline T_{p-1}) \implies H^{p+q}(\overline X, \partial X),$$ and this is the spectral sequence we wanted.
(Often one chooses $\overline X$ and $\overline T_p$ to be the one-point compactification of $X$ resp. $T_p$, and then the above is just the spectral sequence for the reduced cohomology a filtered based space. But allowing yourself to use any compactification is useful in the algebraic setting, which you are interested in.)
Here's an alternative "sheafy" derivation of the spectral sequence. The constant sheaf $\mathbf Z_X$ is filtered: $$\mathbf Z_X = \mathbf Z_{T_n} \supset \mathbf Z_{T_{n-1}} \supset \cdots \supset \mathbf Z_{T_{-1}} = 0$$ where I denote by $\mathbf Z_A$ the pushforward of the constant sheaf on the subspace $A$. The successive quotients in the associated graded for this filtration are of the form ${j_n}_!\mathbf Z$, where $j_n$ is the locally closed inclusion of $T_n \setminus T_{n-1}$. Taking compactly supported cohomology of this filtered object gives rise to a spectral sequence which is exactly the one we want. A useful reference for making sense of this is the section about filtered objects and spectral sequences in Lurie's "Stable $\infty$-categories" (Chapter 1 of Higher Algebra).
To answer a question you made in the comments, the spectral sequence of a filtered space is indeed compatible with mixed Hodge structure, when $X$ is an algebraic variety filtered by closed subvarieties. So the one you consider (by Poincaré duality) is compatible up to Tate twist. I don't know what is the canonical reference for this fact, but Sections 3 and 4 of Arapura's "The Leray spectral sequence is motivic" proves rather generally that the spectral sequence of a filtered algebraic variety is compatible with all kinds of extra "motivic" structure.