Leray spectral sequence for the complement of an arrangement of high-codimensional varieties Assume $X$ is a smooth complex varieties and $D_1,\dots,D_r$  are smooth irreducible divisors that "intersect like hyperplanes", i.e. their union is locally given by a product of linear functions. Then the equation (2.3-1) of Looijenga's paper Cohomology of M3 gives a spectral sequence that computes the cohomology of $X-D_1-\dots-D_r$.
Note that in the case $X=C^n$ where $C$ is a smooth curve, and the divisors in the list are the big diagonals $\Delta_{ij}=\{(x_1,\dots,x_n)\in C^n: x_i=x_j\}$, the spectral sequence above is consistent with Theorem 1.1 of Kriz 1994, i.e. they predict the same spectral sequence converging to $H^*(F(C,n))$, where $F(C,n)=X-\bigcup \Delta_{ij}$ is the unordered configuration space.
Is there a version of Looijenga's (2.3-1) for arrangements of smooth irreducible closed subvarieties that may have codimension higher than one? The examples of subvarieties with particular interest are the big diagonals in $X=V^n$ where $V$ is smooth variety of dimension at least $2$, and the preimages of coordinate projections (e.g. $\{P\}\times V^{n-1}\subseteq X$, where $P$ is a point of $V$).
 A: I think I figure out the answer to my question after seeing @PhilTosteson 's comments.
Short answer:
If an arrangement of smooth complex subvarieties intersect "like a hyperplane arrangement" and every multi-wise intersection is connected, then we have a spectral sequence whose $E_1$ page is the same as the $E_2$ in Bibby's "Cohomology of abelian arrangements" Theorem 4.1, except that the generator $g_Y$ (corresponding to a subvariety $Y$ of complex codimension $c$) now has bidegree $(2c,1)$ and Hodge type $(c,c)$. If the ambient variety is smooth projective (or more generally, such that $H^i$ is pure of weight $i$), then the spectral sequence degerates at $E_2$ page.
Long answer:
Consider an arrangement $\mathcal A$ of closed subspaces $Y_1,\dots,Y_r$ in a topological space $X$, and denote $M_\mathcal A = X - Y_1 - \dots Y_r$. Consider the poset $P$ consisting of all nonempty intersections of $Y_i$'s, ordered by inclusion. Form $\hat P:=P\cup \hat 1$ by adjoining a top element $\hat 1$ corresponding to $X$. Then Theorem 1.8 of Tosteson's "Lattice spectral sequences ..." gives an exact sequence converging to $H^*(M_\mathcal A;\mathbb Z)$, whose $E_1$ page is a direct sum over all $Z\in \hat P$ of terms involving
(1) the relative cohomology $H^i(X,X-Z;\mathbb Z)$ for $Z\in \hat P$.
(2) a combinatorial datum associated to $Z$ and $\hat P$, recorded as the reduced homology $\tilde H_{j-2}$ of the simplicial complex of the sub-poset $(Z,\hat 1):=\{Z':Z<Z'<\hat 1\}$.
In the case where $X$ and all $Z\in P$ are connected smooth complex varieties, tubular neighborhood + excision + Thom isomorphism implies a canonical isomorphism $H^i(X,X-Z;\mathbb Z) \cong H^{i-2\mathrm{codim}_{\mathbb C}Z}(Z;\mathbb Z)$. (The assumption of being complex varieties can be relaxed; it was to ensure that the normal bundle of Z in X has a canonical orientation.)
In the case where $P$ is isomorphic to a poset arising from a hyperplane arrangement, one can apply Folkman 1966 "The homology groups of a lattice" and get
$$ \tilde H_{j-2}((Z, \hat 1);\mathbb Z) = \mathbb Z^{\mu(Z,\hat 1)} = A_Z(\mathcal A)$$
if $j$ is the "rank" of $Z$ (i.e. the distance between $Z$ and $\hat 1$ in the poset), and 0 otherwise. Here, $\mu$ is the Moebius function on the poset, and $A_Z(\mathcal A)$ is the subgroup of the Orlik--Solomon algebra generated by $g_I$ with $\bigcap I = Z$.
At the end, if we apply this theorem to the $n$-th ordered configuration space of a $d$-dimensional complex variety, the $E_1$ page is the algebra given in Totaro 1996 Theorem 4, except that Totaro's $E_{2d}$ page corresponds to Tosteson's $E_1$ page, and the bidegree $E_{2d}^{0,2d-1}$ (where the Orlik--Solomon generator lives) corresponds to the bidegree $E_1^{2d,1}$. The only nontrivial differential is on page $2d$ in Totaro's sequence, and page $1$ in Tosteson's sequence.
