# Rational Homotopy Type Of Complements Of Submanifold Arrangements

Alexander Zakharov111xaxa3217 ( •_•) gmail com

###### Abstract

In this work we will provide an explicit model for cdga controlling the
rational homotopy type of the complement to a smooth arrangement
$X-\cup_{i}Z_{i}$ in a smooth compact algebraic variety $X$. This generalizes
the corresponding result of J. Morgan in case of a divisor with normal
crossings to arbitrary smooth arrangements. The model is given in terms of the
arrangement $Z_{i}$.

## Introduction

###  0.1 Main results

######  Definition 0.1.1.

An arrangement $(X,Z_{*})$ is a smooth compact algebraic variety $X$ over
$\mathbb{C}$ and a collection $\\{Z_{i}\subset X,i\in N\\}$ of subvarieties
for an ordered index set $N$ such that the set theoretical intersections
$Z_{I}=\bigcap\limits_{i\in I}Z_{i},I\subset N$ and $Z_{\emptyset}=X$ are
smooth $C^{\infty}$-submanifolds. (In other words, the corresponding scheme
theoretic intersections with the reduced scheme structure are smooth
subschemes.) The complement of the arrangement $(X,Z_{*})$ is the variety
$U=X-\cup_{i}Z_{i}$.

We study the topology of the arrangement $(X,Z_{*})$.

######  Definition 0.1.2.

A collection $\\{D_{i}\subset X\\}$ is called an NCD if $D_{i}$ are smooth and
$\cup D_{i}$ is a divisor with normal crossings.

Consider an arrangement $(X,D_{*})$, where $D_{*}$ is an NCD. Let
$U=X-\cup_{i}D_{i}$ and $j:U\to X$ be the inclusion.

In order to simplify sign issues and making the product structure clear, it
will be convient to use variables $d\nu_{I}=d\nu_{i_{1}}\wedge\ldots\wedge
d\nu_{i_{k}},I=\\{i_{1},\ldots,i_{k}\\}$ for $i_{1}<\ldots<i_{k}$ viewed as
monomials in the Grassmann algebra of degree $\ -|I|$.

The default coefficient ring for cohomology groups is always $\mathbb{Q}$ and
will usually be ommited from notation.

The following theorem is due to J. Morgan [3]:

######  Theorem.

Consider the Leray spectral sequence $E_{r}$ associated with the inclusion
$j:U\to X$ and the constant sheaf $\underline{\mathbb{Q}}$. Then
$E^{pq}_{2}=H^{p}(X,R^{q}j_{*}\underline{\mathbb{Q}})\simeq\bigoplus\limits_{I:|I|=q}H^{p}(D_{I})$
with the differential $d_{2}$ equal to the Gysin map form a cdga. There is a
quasi-isomorphism of dg-algebras:

$E^{**}_{2}\simeq\bigoplus\limits_{I}H^{*}(X,X-D_{I})\cdot d\nu_{I}\sim
C^{*}_{sing}(U;\mathbb{Q})$

with the differential on the LHS written as $d_{2}=\sum\limits_{i,I:i\in
I}g_{I-\\{i\\},I}\otimes\iota_{i}$

Here $\iota_{i}d\nu_{j}=\delta_{ij}$ and $g_{I-\\{i\\},I}:H^{*}_{D_{I}}(X)\to
H^{*}_{D_{I-\\{i\\}}}(X)$ are the natural homomorphisms induced by the
inclusion $D_{I-\\{i\\}}\to D_{I}$ for $i\in I$.

The quasi-isomorphism of dg-algebras in the theorem is non-canonical, though
there is a natural one after tensoring with $\mathbb{C}$.

The theorem implies $E_{3}=E_{\infty}$. Note that if $(X,Z_{*})$ is a smooth
arrangement with $Z_{i}$ of codimension $>1$, then the Leray spectral sequence
does not degenerate at $E_{3}$. An example is $(\mathbb{C}P^{2},pt)$.
Remarkably the framework developed by Morgan allows us to prove the following
generalization for an arbitrary smooth compact algebraic arrangement:

######  Theorem 0.1.1 (On the homotopy type of the complement, theorem 3.4.1
in the main text).

For a smooth compact algebraic arrangement $(X,Z_{*})$ there is a second
quadrant multiplicative spectral sequence $E^{pq}_{r}$ such that:

  1. 1.

We have $E^{pq}_{1}=\bigoplus\limits_{|I|=-p}H^{q}_{Z_{I}}(X)\cdot d\nu_{I}$.

  2. 2.

The differential $d_{1}=\sum\limits_{i,I:i\in
I}g_{I-\\{i\\},I}\otimes\iota_{i}$, where $g_{I^{\prime}I},I^{\prime}\subset
I$ are the natural homomorphisms:

$H^{*}_{Z_{I}}(X)\to H^{*}_{Z_{I^{\prime}}}(X)$

induced by the inclusions $Z_{I}\to Z_{I^{\prime}}$.

  3. 3.

The multiplication

$H^{q}_{Z_{I}}(X)\cdot d\nu_{I}\otimes H^{q^{\prime}}_{Z_{J}}(X)\cdot
d\nu_{J}\to H^{q+q^{\prime}}_{Z_{IJ}}(X)\cdot d\nu_{I}\wedge d\nu_{J}$

with the usual sign rule makes $(E^{pq}_{1},d_{1})$ into a cdga.

  4. 4.

There is a natural quasi-isomorphism of cdga’s:

$E^{**}_{1}\otimes\mathbb{C}\sim C^{*}_{sing}(X-\cup_{i}Z_{i};\mathbb{C})$

In particular $E_{2}=E_{\infty}$.

  5. 5.

There is a non canonical quasi-isomorphism of the rational homotopy type of
$X-\cup_{i}Z_{i}$ with $E^{**}_{1}$.

In the case of an NCD-complement this spectral sequence is related to the
above Leray spectral sequence via décalage.

The proof is based on the notion of a mixed Hodge diagram introduced by Morgan
in [3]. It is a cdga version of the mixed Hodge complex invented by Deligne
[1]. The most useful property of a mixed Hodge diagram is that it has a
cofibrant replacement (see also [4]), which produces a mixed Hodge structure
on the minimal model of cdga forming the Hodge diagram. In particular its
weight filtration $W$ has a natural Deligne splitting (proposition 1.2.2) over
$\mathbb{C}$, thus providing a natural quasi-isomorphism of the corresponding
${}_{W}E_{1}$-term with the minimal algebra. Then a general result due to
Morgan and Sullivan (theorem 1.2.3) implies the existence of a non-canonical
quasi-isomorphism over $\mathbb{Q}$.

Let $j:X-\cup_{i}Z_{i}\to X$ be the inclusion of the complement. The sheaf
$j_{*}j^{*}\underline{\mathbb{Q}}$ has a natural resolution in terms of
arrangement, called the Mayer-Vietoris complex (definition 3.1.5). The initial
aim of this work is to construct a Hodge diagram structure on this resolution.
Once this is done, the main theorem follows.

It is natural to look for a similar model which better reflects the
combinatorics of the arrangement. One way to proceed is to consider for
example the corresponding intersection poset. The notion of a locally
geometric lattice was applied in [6] to describe the cohomology ring of some
arrangement complements in terms of the Orlik-Solomon algebra and the
cohomology of smooth intersections. We will prove here similar results without
requiring the intersections to be "clean", i.e. looking locally like a linear
subspace arrangement. Further we make weaker assumptions on the poset.

Let $(L,\leq)$ be an arrangement poset (definition 2.2.1) on $X$ and
$\mathcal{M}$ be an OS-algebra (definition 2.5.2). Let
$U=X-\bigcup\limits_{x>0\in L}L_{x}$ be the complement. An example of
$(L,\leq)$ is the cubical poset $\mathcal{C}$ of all intersections as in the
theorem above. The minimal element $L_{0}\in L$ corresponds to $X$ and $x\leq
y\in L$ implies $L_{x}\supset L_{y}$ in $X$. We require that $(L,\leq)$ is
graded by the rank function $r$ such that $r(0)=0$ and $r(x)=r(y)+1$ if $x$
lies over $y$. The last condition is denoted by $x:>y$

The following result is similar to theorem 1.3 in [6]:

######  Theorem 0.1.2 (On the cohomology of the complement, theorem 3.3.1).

There is a multiplicative spectral sequence of pure Hodge structures with
$E^{pq}_{1}$ of weight $q$ such that

  1. 1.

We have $E^{pq}_{1}=\bigoplus\limits_{x\in
L:r(x)=-p}H^{q}(X,X-L_{x})\otimes\mathcal{M}_{x}$.

  2. 2.

The differential $d_{1}=\sum\limits_{y<:x}g_{yx}\otimes\partial_{yx}$ is the
sum of its $L$-components, where $g$ is the natural homomorphism and
$\partial_{yx}$ are the structure maps on $\mathcal{M}$.

  3. 3.

We have $E_{2}=E_{\infty}$.

  4. 4.

There is a natural multiplicative isomorphism of Hodge structures:
$E^{pq}_{2}\simeq Gr^{Dec\ W}_{q}H^{p+q}(U)$.

  5. 5.

The above isomorphism provides a non-natural isomorphism of algebras
$E^{**}_{2}\simeq H^{*}(U;\mathbb{Q})$ and a natural isomorphism of the
complexifications $E^{**}_{2}\otimes\mathbb{C}\simeq H^{*}(U;\mathbb{C})$.

######  Remark 0.1.1.

Here $Gr^{Dec\ W}_{q}H^{n}(U,\mathbb{C})$ denotes the usual weight $q$-piece
of mixed Hodge structure on $H^{n}(U,\mathbb{C})$, see definition 1.2.1 for
$Dec\ W$.

The spectral sequence is called the Mayer-Vietoris spectral sequence of the
arrangement poset $(L,\leq)$ on $X$. We will refer to the spectral sequence in
the main theorem as the cubical Mayer-Vietoris spectral sequence.

######  Remark 0.1.2.

The above theorem is parallel to the fact that the Leray spectral sequence has
a mixed Hodge structure (see [13]). Both results agree in the mentioned case
of an open embedding of the complement to an NCD arrangement. In general case
the weight condition implies that the dg-algebra $E^{**}_{r},r>0$ can be
considered as a pure Hodge complex (a Hodge diagram in fact). Considering
$H^{*}(U)$ as a mixed Hodge complex one may rephrase the statement of the
theorem by saying that there is a natural isomorphism of pure Hodge complexes
$E^{**}_{2}\simeq Gr^{W}H^{*}(U)$. We emphasize, however, that $H^{*}(U)$
itself is rarely pure and one cannot, in general, recover the mixed Hodge
structure $H^{*}(U)$ by our model.

Further, if the arrangement poset $(L,\leq)$ is a geometric lattice we are
able to descend the cubical model of the main theorem to an $(L,\leq)$-model:

######  Theorem 0.1.3 (The lattice model for the homotopy type of the
complement, theorem 3.4.1).

If the arrangement poset $(L,\leq)$ is a geometric lattice, then the
$E_{1}$-term of the above spectral sequence naturally describes the
$\mathbb{C}$-homotopy type of the complement. As above, the rational homotopy
type is non canonically quasi-isomorphic to $E_{1}$.

We will prove this by providing an explicit quasi-isomorphism of the lattice
model $E_{1}(L)$ with the cubical model $E_{1}(\mathcal{C})$.

###  0.2 Applications

As an application we prove the following results. The first one is a
generalization of formality of the complement of a hyperplane arrangement
which can be traced back to works of Arnold, Brieskorn, Orlik-Solomon and
others, see [7],[8]. Consider an arrangement of vector subspaces in
$\mathbb{C}^{n}$. Let $(L,\leq)$ be the corresponding intersection poset.
Assume that $(L,\leq)$ is a geometric lattice. Here we give an alternative
proof of the following theorem by Feichter and Yuzvinsky [11].

######  Theorem 0.2.1 (theorem 4.1.1).

The complement $\mathbb{C}^{n}-\bigcup\limits_{x>0}L_{x}$ is formal.

The next theorem generalizes the Kriz-Totaro model (see [10]) for the rational
homotopy type of the configuration space of $n$-tuples of points on a smooth
compact algebraic variety $M$. Namely, for any finite graph $G$ one can
consider the chromatic configuration space $F(M,G)$ of points $x_{i}\in M$
labeled by vertices $i\in V(G)$ with the condition $x_{i}\neq x_{j}$ if
$(ij)\in E(G)$. Let $\Delta_{ab}\subset M^{n}$ be the diagonal $x_{a}=x_{b}$
and $[\Delta_{ab}]\in H^{2dim\ M}(M^{n})$ the corresponding cohomology class.

######  Theorem 0.2.2 (theorem 4.2.1).

The rational homotopy type of $F(M,G)$ is quasi-isomorphic to the cdga equal
to the quotient

$H^{*}(M^{n})\otimes\Lambda^{*}\langle\tilde{\Delta}_{ab},(ab)\in
E(G)\rangle/I$

where $deg(\tilde{\Delta}_{ab})=2dim\ M-1$ and $I$ is the differential graded
ideal generated by the relations

  1. 1.

$\partial(\tilde{\Delta}_{i_{1}i_{2}}\wedge\ldots\wedge\tilde{\Delta}_{i_{k-1}i_{k}})=0$,
for each cycle $(i_{1},\ldots,i_{k}=i_{1})$ in $G$.

  2. 2.

$p^{*}_{a}u\cdot\tilde{\Delta}_{ab}=p^{*}_{b}u\cdot\tilde{\Delta}_{ab},u\in
H^{*}(M)$.

The differential $d$ is given by $d\tilde{\Delta}_{ab}=[\Delta_{ab}]$.

###  0.3 Organization of the paper

The work is organized as follows. In the first section we recall basic facts
about Hodge complexes, define Hodge diagrams and state the theorem of Morgan
on the existence of the minimal model and the corresponding quasi-isomorphism
with $E_{1}$. Then, in the second section we develop the necessary machinery
to treat the general case of our theorem beyond the cubical lattice. There we
introduce $(L,\leq)$-chain algebras and define Orlik-Solomon algebras in this
context. In the third section we construct the Mayer-Vietoris resolution and
corresponding Hodge complexes and Hodge diagrams. In the last section we will
prove the stated applications.

###  0.4 Acknowlegements

I would like to thank my scientific advisor Alexey Gorinov for posing the
problem, inspiring support and fruitful suggestions, not to mention comments,
remarks and criticism, without which the final work would not be possible. The
author is also grateful to Dmitry Kaledin for reviewing a draft of the paper.

The study has been partially supported by the Laboratory Of Algebraic Geometry
And Its Applications funded within the framework of the HSE University Basic
Research Program.

##  1 Preliminaries on Hodge Complexes

If $W$ is a decreasing filtration on object $K$, then its $k$-th piece is
denoted by $W^{k}K=W_{-k}K$, where $W_{*}$ denotes the corresponding
increasing filtration. Likewise we agree to use this notation for associated
graded objects, e.g.
$Gr^{k}_{W}K={}_{W}Gr^{k}K=Gr^{W}_{-k}K=Gr^{k}(K,W)=Gr_{k}(K,W)$ etc. As
usual, all cohomological spectral sequences are written with respect to the
corresponding decreasing filtrations.

###  1.1 Filtrations

Here we recall some general preliminaries which can be found in [2],[1]. The
reader can skip this subsection and come back latter.

######  Lemma 1.1.1.

Let $A,B,C\subset E$ be a subobjects of $E$ in an abelian category, then
$(A+B)\cap(B+C)=(A+B)\cap C+B$

###### Proof.

Passing to a category of modules via Mitchells theorem, one can prove the
equality in an element-wise manner: if $a+b=b^{\prime}+c\in(A+B)\cap(B+C)$,
then $a+b=(a+b-b^{\prime})+b^{\prime}\in(A+B)\cap C+B$. ∎

Let $K^{*}$ be a complex with filtrations $W$, $F$. Define

$Z^{pq}_{r}:=Ker[W^{p}K^{*}\to K^{*}/W^{p+r}K^{*}]$
$B^{pq}_{r}:=dW^{p-r+1}+W^{p+1}$

We note that:

$dZ^{pq}_{r}\subset Z^{p+r,q-r+1}_{r}$ $dB^{pq}_{r}\subset B^{p+r,q-r+1}_{r}$

There is a corresponding spectral sequence

${}_{W}E^{pq}_{r}=\frac{Z^{pq}_{r}}{Z^{pq}_{r}\cap
B^{pq}_{r}}\simeq\frac{Z^{pq}_{r}+B^{pq}_{r}}{B^{pq}_{r}}$

with differential $d_{r}:E^{pq}_{r}\to E^{p+r,q-r+1}_{r}$. One has a natural
isomorphism:

$H^{*}(E_{r},d_{r})\simeq E_{r+1}$

The non trivial remark is that the realization of $E^{pq}_{r+1}$ as a sub-
quotient of $E^{pq}_{r}$ is neither consistent with the two given descriptions
of $E_{r}$. Namely, $Z_{r+1}\subset Z_{r}$ and $B_{r+1}\supset B_{r}$, but it
is not true that $Z_{r+1}\cap B_{r+1}\supset Z_{r}\cap B_{r}$ or
$Z_{r+1}+B_{r+1}\subset Z_{r}+B_{r}$.

The two possible ways of identifying ${}_{W}E^{pq}_{r}$ result in two
different filtrations induces by $F$. The first is called $F_{dir}$, the
second is $F_{dir^{*}}$. On the other hand, since
$E_{r+1}=Ker[E_{r}\overset{d_{r}}{\to}E_{r}]/Im[d_{r}]$ one can induce a
filtration $F_{ind}$ on $E_{r+1}$ by that of $E_{r}$.

######  Lemma 1.1.2.

  1. a

$F_{dir}^{i}E_{r}K^{*}=Im[E_{r}F^{i}\to E_{r}K^{*}]$

  2. b

$F_{dir^{*}}^{i}E_{r}K^{*}=Ker[E_{r}K^{*}\to E_{r}K^{*}/F^{i}]$

###### Proof.

(a) By definition the LHS is

$Im[F^{i}\cap Z^{pq}_{r}\to E^{pq}_{r}]=\frac{Z^{pq}_{r}}{Z^{pq}_{r}\cap
B^{pq}_{r}}$

On the other hand $Im[E_{r}F^{i}\to E_{r}]$ is equal to

$Im[Z^{pq}_{r}(F^{i})\to Z^{pq}_{r}]$

but

$Z^{pq}_{r}(F^{i})=Ker(F^{i}\cap W^{p}\to K^{*}/W^{p+r})=Z^{pq}_{r}\cap F^{i}$

(b) We have:

$F_{dir^{*}}^{i}E_{r}=Im[F^{i}\cap(Z_{r}+B_{r})\to
E_{r}=\frac{Z_{r}+B_{r}}{B_{r}}]$

On the other hand:

$Ker[E_{r}K^{*}\to
E_{r}K^{*}/F^{i}]=Im[Ker(Z_{r}+B_{r}\to\frac{K^{*}/F^{i}}{B_{r}(K^{*}/F^{i})})\to
E_{r}]=$ $=Im[(Z_{r}+B_{r})\cap(B_{r}+F^{i})\to E_{r}]=Im[(Z_{r}+B_{r})\cap
F^{i}+B_{r}\to E_{r}]=$ $=F_{dir}^{i}E_{r}$

∎

######  Proposition 1.1.1.

For the filtrations introduced above the following holds:

  * i

$F_{dir}\subset F_{ind}\subset F_{dir^{*}}$

  * ii

On $E_{0},E_{1}$ all three filtrations coincide.

###### Proof.

Note that by definition the fitltrations are equal on $E_{0}K^{*}={}_{W}Gr\
K^{*}$. Further, since $B_{1}\subset Z_{1}$, one has
$F_{dir}^{i}E_{1}=F_{dir^{*}}^{i}E_{1}$.

a [$F_{dir}\subset F_{ind}$]. We will proceed by induction. Assume
$F_{dir}^{i}E_{r}\subset G^{i}E_{r}$, for a filtration $G$ on $E_{r}$. By the
previous

$F_{dir}^{i}E_{r+1}=Im[Ker(E_{r}F^{i}\overset{d_{r}}{\to}E_{r})\to E_{r+1}]$

and by definition

$G^{i}E_{r+1}=Im[Ker(G^{i}E_{r}\overset{d_{r}}{\to}E_{r})\to E_{r+1}]$

then, since $Im[E_{r}F^{i}\to E_{r}]\subset G^{i}E_{r}$, one gets:

$F_{dir}^{i}E_{r+1}\subset G^{i}E_{r+1}$

Putting $G=F_{ind}$ proves the claim.

b [$F_{ind}\subset F_{dir^{*}}$]. Assume $G^{i}E_{r}\subset F_{dir^{*}}E_{r}$.
Then:

$G^{i}E_{r+1}=Im[Ker[G^{i}E_{r}\overset{d_{r}}{\to}E_{r}]\to E_{r+1}]$
$F_{dir^{*}}^{i}E_{r+1}=Ker[E_{r+1}K^{*}\to E_{r+1}K^{*}/F^{i}]$

Since $E_{r+1}$ is a homology of $E_{r}$ and $E_{r}K^{*}\to E_{r}K^{*}/F^{i}$
is a morphism of complexes, from

$G^{i}E_{r}\subset F_{dir^{*}}^{i}E_{r}=Ker[E_{r}K^{*}\to E_{r}K^{*}/F^{i}]$

one immediatly have:

$G^{i}E_{r+1}\subset F_{dir*}^{i}E_{r+1}$

∎

######  Lemma 1.1.3 (on two filtrations).

If $\forall r\in 0,\ldots,r_{0}$ the differential $d_{r}$ is strict, then for
all $r^{\prime}\in 0,\ldots,r_{0}+1$:

  * i

$F_{dir}=F_{ind}=F_{dir^{*}}$ on $E_{r^{\prime}}$

  * ii

the sequence $0\to E_{r^{\prime}}F^{i}\to E_{r^{\prime}}K^{*}\to
E_{r^{\prime}}K^{*}/F^{i}\to 0$ is exact

  * iii

${}_{F}Gr^{i}E_{r^{\prime}}\simeq E_{r^{\prime}}\ {}_{W}Gr^{i}K^{*}$

###### Proof.

We will proceed by induction. Assume for a given $r$ the above sequence is
exact and all three filtrations coincide. Let us prove that $E_{r+1}F^{i}\to
E_{r+1}K^{*}$ is an inclusion with the image equal to $F_{ind}^{i}E_{r+1}$.
Consider:

$\frac{Ker[F^{i}E_{r}\overset{d_{r}}{\to}F^{i}E_{r}]}{Im[d_{r}]}=\frac{Ker[E_{r}F^{i}\overset{d_{r}}{\to}E_{r}F^{i}]}{Im[d_{r}]}\to\frac{Ker[E_{r}\overset{d_{r}}{\to}E_{r}]}{Im[d_{r}]}=E_{r+1}$

The first equality holds under induction hypothesis, hence the second map is
an inclusion since $d_{r}$ is $F$-strict. Its image in $E_{r+1}$ is
$Im[Ker[F^{i}_{ind}E_{r}\to E_{r}]\to E_{r+1}]$, which is by definition
$F^{i}_{ind}E_{r+1}$. Thus $E_{r+1}F^{i}\to E_{r+1}K^{*}$ is an inclusion with
image $F^{i}_{ind}E_{r+1}$, hence $F^{i}_{dir}=F^{i}_{ind}$ on $E_{r+1}$ as
required. To show surjectivity of $E_{r+1}\to E_{r+1}K^{*}/F^{i}$ note that

$0\to F^{i}E_{r}\to E_{r}K^{*}\to
E_{r}K^{*}/F^{i}=E_{r}K^{*}/F^{i}E_{r}K^{*}\to 0$

is exact, then taking cohomology w.r.t $d_{r}$ produces a long exact sequence,
hence

$F^{i}_{ind}E_{r+1}\to Ker[E_{r+1}\to E_{r+1}K^{*}/F^{i}]$

is onto, thus $F^{i}_{ind}=F^{i}_{dir^{*}}$ on $E_{r+1}$. Since
$F^{i}E_{r+1}\to E_{r+1}$ is into, the same long exact sequence implies also
that

$E_{r+1}\to E_{r+1}K^{*}/F^{i}$

is onto.

Finally, since $E_{r}F^{i+1}\to E_{r}F^{i}$ is equivalent to
$F^{i+1}E_{r}\hookrightarrow F^{i}E_{r}$, one gets a natural isomorphism:

${}_{F}Gr^{i}E_{r}K^{*}=E_{r}{}_{F}Gr^{i}K^{*}$

∎

######  Corollary 1.1.1.

If $d_{r}$ is $F_{ind}-$strict for all $r\geq 0$, then
${}_{F}E_{1}={}_{F}E_{\infty}$

###### Proof.

By the previous ${}_{F}Gr^{i}E_{r}K^{*}=E_{r}\ {}_{F}Gr^{i}K^{*}$, in
particular for $r=\infty$ one has:

${}_{F}Gr^{i}F{}_{W}Gr\
H^{*}(K^{*})={}_{F}Gr^{i}{}_{W}E_{\infty}={}_{W}E_{\infty}\
{}_{F}Gr^{i}K^{*}={}_{W}Gr\ {}_{F}E_{1}K^{*}$

hence ${}_{F}E_{\infty}K^{*}={}_{F}E_{1}K^{*}$. ∎

###  1.2 Mixed Hodge Complexes and Hodge Diagrams

######  Definition 1.2.1.

A data
$(C_{\mathbb{Q}}^{*},(C_{\mathbb{C}}^{*},F),C_{\mathbb{Q}}\bigotimes\limits_{\mathbb{Q}}\mathbb{C}\sim
C_{\mathbb{C}})$, where

  * (i)

$C_{\mathbb{k}}^{*}$ is a complex over $\mathbb{k}$

  * (ii)

$C_{\mathbb{Q}}\bigotimes\limits_{\mathbb{Q}}\mathbb{C}\sim C_{\mathbb{C}}$ is
a quasi-isomorphism.

called a pure Hodge complex of weight $m$ if

  1. (i)

${}_{F}E_{1}C_{\mathbb{C}}^{*}={}_{F}E_{\infty}C_{\mathbb{C}}^{*}$

  2. (ii)

$(H^{n}(C^{*}),F)$ defines a pure Hodge structure of weight $n+m$. This
condition is phrased by saying that $F$ is $n+m$-conjugate.

The data will be denoted simply as $(C^{*},F)$.

Note that if $(C^{*},F)$ is a Hodge complex of weight $m$, then there is a
canonical identification:

${}_{F}E^{pq}_{1}=F^{p}H^{p+q}\cap\bar{F}^{q+m}H^{p+q}$

where $\bar{F}$ is the complex conjugate filtration to $F$.

######  Definition 1.2.2.

A data
$((K_{\mathbb{Q}}^{*},W_{\mathbb{Q}}),(K_{\mathbb{C}}^{*},W_{\mathbb{C}},F),(K_{\mathbb{Q}}\bigotimes\limits_{\mathbb{Q}}\mathbb{C},W_{\mathbb{Q}}\bigotimes\limits_{\mathbb{Q}}\mathbb{C})\sim(K_{\mathbb{C}},W_{\mathbb{C}}))$
where

  * (i)

$K_{\mathbb{k}}$ is a complex over $\mathbb{k}$.

  * (ii)

$(W_{\mathbb{Q}}\bigotimes\limits_{\mathbb{Q}}\mathbb{C},W_{\mathbb{Q}}\bigotimes\limits_{\mathbb{Q}}\mathbb{C})\sim(K_{\mathbb{C}},W_{\mathbb{C}})$
is a filtred quasi-isomorphism.

called a mixed Hodge complex, if $Gr^{W}_{p}K^{*}={}_{W}Gr^{-p}$ is a pure
Hodge complex of weight $p$. The data will be denoted by $(K^{*},W,F)$.

######  Corollary 1.2.1.

Assume that $(K^{*},F,W)$ is a mixed Hodge complex, then

  1. (a)

${}_{F}E_{1}K^{*}={}_{F}E_{\infty}K^{*}$

  2. (b)

${}_{W}E_{2}K^{*}={}_{W}E_{\infty}K^{*}$

###### Proof.

First we note that the filtration $F$ induces on
${}_{W}E^{pq}_{1}K^{*}=H^{p+q}{}_{W}Gr^{p}K^{*}$ the pure Hodge structure of
weight $(p+q)-p=q$. Since $d_{r}$ is compatible with filtration $F$ all terms
${}_{W}E^{pq}_{r}$ are mixed Hodge structures and $d_{r}$ is a morphism of
Hodge structures. The purity implies that $d_{>1}=0$, hence
${}_{W}E_{2}K^{*}={}_{W}E_{\infty}K^{*}$.

According to the above corollary, in order to show that
${}_{F}E_{1}={}_{F}E_{\infty}$ it is enough to prove that $d_{r}$ are
$F_{ind}$-strict for all $r\geq 0$. By the previous the only non-trivial
differentials are $d_{0},d_{1}$. Since ${}_{W}Gr\ K^{*}$ is a pure Hodge
complex, $d_{0}$ is $F$-strict. As we seen $d_{1}$ is a morphism of Hodge
structures, thus it is automatically $F$-strict on $E_{1}$. ∎

######  Definition 1.2.3.

We denote by $[A\to B]:=Cone(A\to B)[1]$.

######  Proposition 1.2.1 (Mixed cone construction).

Assume $A\to B$ is a morphism of mixed Hodge complexes, then $K:=[A\to
B],K^{n}=A_{n}\oplus B_{n-1}$ becomes a mixed Hodge complex if we put:

  1. a

$W^{p}K:=W^{p}A\oplus W^{p-1}B$

  2. b

$F^{p}K:=F^{p}A\oplus F^{p}B$

###### Proof.

It is enough to show that ${}_{W}Gr^{p}K^{*}$ is a pure Hodge complex of
weight $-p$. Note that ${}_{W}Gr^{p}K^{*}$ as a complex ignores $f$ and is
equal to the direct sum

${}_{W}Gr^{p}A\oplus{}_{W}Gr^{p-1}B[-1]$

hence $F$ induces a pure Hodge structure of weight $n-p$ on its $n$-th
cohomology equal to $H^{n}({}_{W}Gr^{p}A)\oplus H^{n-1}({}_{W}Gr^{p-1}B)$.
Hence $[A\to B]$ is a mixed Hodge complex. ∎

######  Definition 1.2.4.

If $(C^{*},F)$ is a Hodge complex of weight $m$, then $(C^{*}[a],F)$ is a
Hodge complex of weight $m+a$. If $(K^{*},F,W)$ is a mixed Hodge complex, then
$(K^{*}[a],F,W[a])$ is a mixed Hodge complex, where: $W[a]^{p}=W^{a+p}$.

Similarly one can define the Tate twists of mixed Hodge complexes.

The following will be called itered cone construction.

######  Example 1.2.1 (Iterated cone).

Let $K=[A_{0}\to A_{1}\to A_{2}\to\ldots]$ be a bicomplex formed by complexes
$A_{i}$. One can consider $K$ as:

$[A_{0}\to[A_{1}\to[A_{2}\to\ldots]]]$

If $A_{i}$ a mixed Hodge complexes and all differentials are morphisms of
mixed Hodge complexes, then $K$ is a mixed Hodge complex.

The following theorem is the main tool for construction mixed Hodge
structures.

######  Theorem 1.2.1 (Deligne [1]).

The cohomology of a mixed Hodge complex $(K,F,W)$ provides a mixed Hodge
structure.

######  Remark 1.2.1.

Since we work with weight filtrations on Hodge complexes it is necessary to do
the following remark. The mixed Hodge structure weight filtration on
$H^{n}:=H^{n}(K)$ is given by the décalage, i.e. $(Dec\
W)_{k}H^{n}=W_{k-n}H^{n}$ and thus:

$Gr^{Dec\ W}_{-k}H^{n}=Gr^{k}_{Dec\ W}H^{n}=Gr^{k+n}_{W}H^{n}$

For example, if the Hodge complex $(K,F,W)$ is pure of weight $0$, then
$H^{n}$ is pure of weight $n$. In general, since the Hodge filtration $F$ on
$E^{pq}_{2}(K,W)=Gr^{p}_{W}H^{n}$ is $q$-conjugate, we have $Gr^{Dec\
W}_{q}H^{n}={}_{W}E^{pq}_{2}$.

For a general complex $K$ the decalage $Dec\ W$ is defined by:

$(Dec\ W)^{k}K^{n}:=\\{x\in W^{k+n}K^{n}|dx\in W^{k+n+1}K^{n+1}\\}$

There is a natural quasi-isomorphism $E_{0}(K,Dec\ W)\sim E_{1}(K,W)$ and
natural isomorphisms
$E^{pq}_{r}(K,Dec\ W)\simeq E^{p+n,-p}_{r+1}(K,W),n=p+q$. See the original
paper [1] by Deligne for details.

######  Definition 1.2.5.

If $(V,F,W)$ is a mixed Hodge structure then $(Gr^{*}_{W}V,F,W)$ is a pure
Hodge structure called the associated pure Hodge structure.

######  Definition 1.2.6 (Hodge diagram, see [3][4]).

A (mixed) Hodge diagram is a mixed Hodge complex with a multiplication
compatible with filtrations and the differential making the rational and the
complex parts of the Hodge complex into connected cdga.

######  Definition 1.2.7.

A mixed Hodge dg-algebra $(A^{*},F,W)$ is a mixed Hodge diagram with
$A_{\mathbb{C}}=A_{\mathbb{Q}}\otimes\mathbb{C}$ such that $(A^{n},F,Dec\ W)$
is a mixed Hodge structure compatible with the differential and
multiplication.

For example, if $(K^{*},F,W)$ is a mixed Hodge diagram, then $E^{**}_{1}(K,W)$
is a mixed Hodge dg-algebra with pure terms. In particular the cohomology of a
mixed Hodge diagram is a mixed Hodge algebra.

######  Proposition 1.2.2 (Deligne splitting).

A mixed Hodge structure $(V,F,W)$ admits a canonical decomposition
$V=\bigoplus V^{p,q}$ such that $\bigoplus_{p+q=k}V^{p,q}\to Gr^{k}_{W}V$ is
an isomorphism onto Hodge components of the associated pure Hodge structure.
Hence there is a canonical isomorphism of vector spaces
$Gr^{*}_{W}V_{\mathbb{C}}\simeq V_{\mathbb{C}}$. In particular, if $V$ is a
mixed Hodge dg-algebra, then there is a natural isomorphism of dg-algebras
$V_{\mathbb{C}}\simeq Gr^{*}_{W}V_{\mathbb{C}}$.

######  Remark 1.2.2.

In general there is no such decomposition on the level of Hodge complex,
though one can lift the canonical splitting on its cohomology.

This implies a powerfull corollary:

######  Corollary 1.2.2.

If $(C,F,W)$ is a Hodge complex (diagram), then there is a natural isomorphsim
$H^{*}(C_{\mathbb{C}})\simeq E^{**}_{2}(C_{\mathbb{C}},W)$ of vector spaces
(algebras).

###### Proof.

The cohomology of $H^{*}(C)$ form a mixed Hodge structure. If $C$ in addition
is a Hodge diagram, then $H^{*}(C)$ is a mixed Hodge algebra. There a chain of
natural isomorphisms of vectors spaces (algebras):

$H^{*}(C_{\mathbb{C}})\simeq{}_{W}Gr^{*}H^{*}(C_{\mathbb{C}})\simeq
E^{**}_{\infty}(C_{\mathbb{C}},W)=E^{**}_{2}(C_{\mathbb{C}},W)$

∎

Within the framework introduced by Morgan in [3], then developed in
homotopical context by Joana Cirici in [4], the above corollary can be made
even stronger, it is the foundation of our work.

######  Theorem 1.2.2 (Morgan, Joana Cirici).

If $(K^{*},F,W)$ is a mixed Hodge diagram, then the minimal model
$\mathcal{N}$ of $K_{\mathbb{Q}}$ is equipped with a mixed Hodge dg-algebra
structure $(\mathcal{N},F,W)$ and a bifiltered quasi-siomorphism
$(\mathcal{N},F,W)\to(K^{*},F,W)$ of Hodge diagrams. Moreover,
$d:W_{k}\mathcal{N}\to W_{k-1}(\mathcal{N}^{+}\cdot\mathcal{N}^{+})$.

######  Corollary 1.2.3 (Morgan, theorem 9.6 in [3]).

There is a natural quasi-isomorphism $\mathcal{N}_{\mathbb{C}}\sim
E_{1}(K_{\mathbb{C}},W)$ of cdga’s, thus there is a natural cdga’s quasi-
isomorphism

$E^{**}_{1}(K_{\mathbb{C}},W)\sim K_{\mathbb{C}}$

which non canonically descents to an quasi-isomorphism of algebras over
$\mathbb{Q}$:

$E^{**}_{1}(K_{\mathbb{Q}},W)\sim K_{\mathbb{Q}}$

###### Proof.

The morphism $(\mathcal{N},W)\to(K^{*},W)$ is a filtered quasi-isomorphism,
thus $E_{1}(\mathcal{N},W)\simeq E_{1}(K^{*},W)$. Since $(\mathcal{N},F,W)$ is
a mixed Hodge dg-algebra, there is a canonical splitting over $\mathbb{C}$:

$\mathcal{N}\simeq Gr_{Dec\ W}^{*}\mathcal{N}=E_{0}(\mathcal{N},Dec\ W)\simeq
E_{1}(\mathcal{N},W)$

Then the Morgan-Sullivan theorem (theorem 10.1 in [3]) implies that such
splitting over $\mathbb{C}$ can be descended over $\mathbb{Q}$. This completes
the proof. ∎

For a general treatment of minimal models in mixed context see the paper [4]
by Joana Cirici.

##  2 Lattices

In this section we introduce the notion of $(L,\leq)$-chain algebra, show the
existence of basic operations such as pullbacks, pushforwards and the
projection formula; this will allow us to handle homological combinatorics
behind complexes of sheaves related to the arrangement.

###  2.1 Posets

Here $(L,\leq)$ denotes a poset. For a subset $S\subset L$ we write $S\leq
t,t\in L$ and call $t$ an upper bound of $S$ if $s\leq t,\forall s\in S$.

######  Definition 2.1.1.

The set of maximal elements $\max(S)\subset L$ consists of $t\in S$ such that
for all $t^{\prime}\in S$, $t^{\prime}\geq t$ implies $t^{\prime}=t$.

Dually we define $\min(S)\subset L$.

######  Definition 2.1.2.

If $(L,\leq)$ is a poset and $S\subset L$, then $t=sup\ {S}\in L$ is called
the supremum of $S$, if $t\geq S$ is an upper bound and if for any upper bound
$t^{\prime}\geq S$ we have $t^{\prime}\leq t$. In other words, $sup\ {S}$ is
the lowest upper bound of $S$.

######  Definition 2.1.3.

Denote $\overset{\circ}{sup}{S}\subset L$ to be the subset of all upper bounds
$t\geq S$ such that for all $t^{\prime}\geq S$, $t\geq t^{\prime}$ implies
$t^{\prime}=t$. In other words $\overset{\circ}{sup}S=\min\\{t\in L|t\geq
S\\}$ is the set of minimal upper bounds.

Dually we define $inf$ and $\overset{\circ}{inf}$.

######  Definition 2.1.4.

A poset $(L,\leq)$ is lattice if for any $S\subset L$ there is $sup(S)$ and
$inf(S)$. For any $a,b\in L$ the operations $a\vee b:=sup(a,b)$ (join) and
$a\wedge b:=inf(a,b)$ (meet) are well-defined.

Recall that the order of the lattice is recovered from operations
$\vee,\wedge$ by $a\leq b$ iff $a\wedge b=b$ etc.

######  Definition 2.1.5.

We say that $x$ covers $y$ if $x\geq y$ and there are no other elements
between $x$ and $y$. Denote this relation as $x:>y$

######  Definition 2.1.6.

A poset $(L,\leq)$ is graded, if there is an order-preserving function
$r:L\to\mathbb{Z}$ called grading such that $r(x)=r(y)+1$ if $x:>y$.

Clearly $r$, if exists, is determined by its values on $\min(L)$.

If the poset $(L,\leq)$ has an infinum we denote it by $0=inf(L)\in L$ and
call $(L,\leq)$ local. In this case we assume $r(0)=0$.

write $L[0,x]=\\{t\in L:t\leq x\\}$.

######  Definition 2.1.7.

A poset $(L,\leq)$ with the unique minimal element $0\in L$, is a local
(graded) lattice, if for any $x\in L$ $L[0,x]$ is a (graded) lattice. If $L$
is a graded local lattice, then $r(x):=r(L[0,x]),r(0)=0$ is called the grading
of $L$.

Clearly the grading $r$ of a local lattice $L$ satisfy $r(x)=r(y)+1$ if
$x:>y$.

######  Definition 2.1.8.

Elements $Atoms(L):=\\{x\in L:r(x)=1\\}$ are called atoms of the graded local
lattice $(L,\leq)$.

######  Definition 2.1.9.

A graded lattice $(L,\leq)$ is geometric, if (i) $x=a_{1}\vee\ldots\vee
a_{k},a_{i}\in Atoms(L)$ for any $x\in L$, (ii) $r(x\vee y)+r(x\wedge y)\leq
r(x)+r(y)$.

######  Definition 2.1.10.

Elements $x_{i}\in L,i\leq n$ in a graded posed are called independend, if
$r(x_{1}\vee\ldots\vee x_{n})=\sum r(x_{i})$. If the lattice is geometric,
then $x,y$ are independend implies $r(x\wedge y)=0$.

###  2.2 Arrangement posets

Consider the poset $2^{X}$ of subsets of $X$ ordered by inclusion.

Let $(L,\leq)$ be a local graded lattice.

######  Definition 2.2.1.

An arrangement poset is a $(L,\leq)$-subspace of $X$, i.e. a functor
$F:(L,\leq)^{op}\to{2^{X}}$ with values $F(p)=:L_{p}\subset X,p\in L$ such
that:

  1. 1.

$L_{0}=X$.

  2. 2.

for $I\subset Atoms(L)$ we have $L_{I}:=\bigcap\limits_{i\in
I}L_{i}=\bigsqcup\limits_{t\in\overset{\circ}{sup}(I)}L_{t}$

  3. 3.

For each $x\in X$ the full subposet $\\{p\in L:L_{p}\ni x\\}\subset L$ is a
graded lattice of the form $[0,t],t\in L$.

  4. 4.

$\bigcap\limits_{a\leq x:r(a)=1}L_{a}=L_{x}$

######  Definition 2.2.2.

The arrangement poset $(L,\leq)$ on $X$ is good if $L$ is geometric lattice.

If $(L,\leq)$ is good, then $L_{I}=L_{sup(I)}\subset X$.

######  Example 2.2.1.

(i) If $(X,Z_{*})$ is an arrangement, then $Z_{I}$ form the cubical geometric
lattice $\mathcal{C}_{I}=Z_{I}$ ordered by inclusion of indices. Then the
natural map $(\mathcal{C},\leq)^{op}\to 2^{X}$ defines a good arrangement
poset.

(ii) If $X$ is a smooth variety, then the configuration space of $n$-ordered
points on $X$ is $X^{n}-\Delta_{*}$, where $\Delta$ is the intersection poset
of all diagonals in $X^{n}$. Each diagonal
$\Delta_{\\{S_{1},\ldots,S_{k}\\}}=\\{i,j\in S_{t}\implies
x_{i}=x_{j}\\}\subset X^{n}$ is parametrized by an unordered collection of
disjoint subsets
$\\{S_{1},S_{2},\ldots,S_{k}\\},S_{i}\subset\\{1,\ldots,n\\}$. Then
$r(\Delta_{\\{S_{1},\ldots,S_{k}\\}})=\sum\limits_{i\leq k}|S_{i}|-1$. The
atoms a formed by big diagonals $\Delta_{ab}:=\Delta_{\\{a,b\\}}$. The poset
$\Delta$ defines a good arrangement poset.

(iii) Consider the arrangement given by two planes and one line
$\Pi_{1},\Pi_{2},l\subset\mathbb{C}^{3}$ meeting exactly at one point $p$ in
general position. The corresponding intersection poset consists of
$0=\mathbb{C}^{3},\Pi_{1},\Pi_{2},l,l_{12}=\Pi_{1}\cap\Pi_{2},p$ is not even
ranked. One can always pass to the corresponding cubical lattice to solve the
issue. Another way is to duplicate $p$ by adding $p^{\prime}$ to the poset,
then replace $p\leq l_{12}$ by $p^{\prime}\leq l_{2}$ and $p^{\prime}\leq p$.

###  2.3 $(L,\leq)$–objects

Let $(L,\leq)$ be a graded poset.

######  Definition 2.3.1 ($(L,\leq)$-object).

An $(L,\leq)$-object in a category $C$ is a functor $F:(L,\leq)^{op}\to C$.
Its values are denoted by $F_{s}\in C,s\in L$. Corresponding morphisms
$g^{F}_{yx}:F_{x}\to F_{y},x\geq y\in L$ are called the structure maps.

Assume $(L,\leq)$ is a local lattice. Note that for any $x^{\prime}\leq
x,y^{\prime}\leq y$ there is a unique map
$\gamma^{xy}_{x^{\prime}y^{\prime}}:\overset{\circ}{sup}(x,y)\to\overset{\circ}{sup}(x^{\prime},y^{\prime})$
such that $\gamma^{xy}_{x^{\prime}y^{\prime}}(t)\leq t$. It is defined by
taking $sup(x^{\prime},y^{\prime})$ in $L[0,t]$.

######  Definition 2.3.2.

If $C$ is a monoidal additive category, then a $(L,\leq)$-algebra in $C$ is a
functor $A:(L,\leq)^{op}\to C$ with morphisms:

$m_{x,y}:A_{x}\otimes A_{y}\to\bigoplus\limits_{t\in
x\overset{\circ}{\vee}y}A_{t}=:A_{x\overset{\circ}{\vee}y}$

with components $m^{t}_{x,y}$ such that

  1. 1.

$m$ is associative, i.e.:

$m^{t_{3}}_{t_{1},z}\circ(m^{t_{1}}_{x,y}\otimes
id)=m^{t_{3}}_{x,t_{2}}\circ(id\otimes m^{t_{2}}_{y,z})$

for all
$t_{1}\in\overset{\circ}{sup}(x,y),t_{2}\in\overset{\circ}{sup}(y,z),t_{3}\in\overset{\circ}{sup}(x,y,z)$.

  2. 2.

$m$ is functorial, i.e. the following square commutes:

${A_{x}\otimes A_{y}}$${A_{x\overset{\circ}{\vee}y}}$${A_{x^{\prime}}\otimes
A_{y^{\prime}}}$${A_{x^{\prime}\overset{\circ}{\vee}y^{\prime}}}$$\scriptstyle{m}$$\scriptstyle{g^{A}\otimes
g^{A}}$$\scriptstyle{\gamma}$$\scriptstyle{m}$

In particular objects $F[x,y]:=\bigoplus\limits_{t:x\leq t\leq y\in L}F_{t}$
are associative algebras in $C$. If $C$ is symmetric we say that $F$ is
commutative, if $F[0,x]$ are commutative.

Below $(L,\leq)$ is a graded poset.

######  Definition 2.3.3.

An $(L,\leq)$-filtered object $K$ in additive category of complexes $Comp(C)$
is a collection of complexes $(K_{x},d_{x})\in Comp(C),d_{x}^{2}=0,\forall
x\in L$ and morphisms $\partial_{yx}:K_{x}\to K_{y}[1],y<x$ such that
$\partial:=\sum\limits_{y<x}\partial_{yx}$: defines a differential in
$\bigoplus\limits_{t\leq x\in L}K_{t},\forall x\in L$, i.e. $\partial^{2}=0$.

The range of $K$ is $\max{(r(x)-r(y)|\partial_{yx}\neq 0;x,y\in
L)}\in\mathbb{Z}$.

Morphisms $\partial_{xy}$, the differential $\partial$ and $d=\sum_{x}d_{x}$
are called the structure maps of $K$, the combinatorial differential and the
inner differential respectively.

By definiton $[\partial,d]=0$. Thus a $(L,\leq)$-filtered object is a
$L$-graded complex with an additional combinatorial differential which
decreases $L$-grading. We are mostly interested in case when
$\partial_{yx}\neq 0$ implies $y<:x\in L$:

######  Definition 2.3.4 ($(L,\leq)$-chains).

$(L,\leq)$-chains are $(L,\leq)$-filtered objects of range $1$.

Assume $(L,\leq)$ is a local lattice.

######  Definition 2.3.5 ($(L,\leq)$-filtered algebra).

An $(L,\leq)$-filtered algebra in monoidal additive category of complexes
$Comp(C)$ is a $(L,\leq)$-filtered complex $K$ with a collection of morphisms:

$m^{t}_{x,y}:K_{x}\otimes K_{y}\to\bigoplus\limits_{t\in
x\overset{\circ}{\vee}y}K_{t}$

with components $m^{t}_{x,y}$ such that $m$ is associative and obey the
Leybniz rule, i.e. the objects $F[0,x]:=\bigoplus\limits_{t\leq x\in L}K_{t}$
are dg-algebras in $Comp(C)$ with differential $\partial+d$.

######  Definition 2.3.6.

We say that $(L,\leq)$-filtered algebra or $(L,\leq)$-algebra is well-graded,
if the multiplication satisfy $m^{t}_{xy}=0,t\in x\overset{\circ}{\vee}y$ if
$r(t)<r(x)+r(y)$.

######  Definition 2.3.7.

If $K$ is $(L,\leq)$-filtered complex and $A$ is $(L,\leq)$-complex, we define
the tensor product $A\otimes K$ as $(L,\leq)$-filtered complex given by:

  1. 1.

$(A\otimes K)_{x}=A_{x}\otimes K_{x},x\in L$

  2. 2.

$\partial^{A\otimes K}_{yx}:=g^{A}_{yx}\otimes\partial^{K}_{yx},y<:x\in L$ as
a morphism $(A\otimes K)_{x}\to(A\otimes K)_{y}$.

Similarly one can define a tensor product of $(L,\leq)$-objects. One can see,
that if $K$ is $(L,\leq)$-filtered algebra and $A$ is $(L,\leq)$-algebra, then
$A\otimes K$ is $(L,\leq)$-filtered algebra of the same range as $K$. If $K$
or $A$ is well-graded,the so is $A\otimes K$.

######  Definition 2.3.8.

A sublattice $L^{\prime}\subset L$ is called complete if for all $x,y\in
L^{\prime}$ and $t\in L^{\prime}:x\leq t\leq y$ implies $t\in L$.

Hence the preimage of complete sublattice under an order preserving map is
complete. The following notion is clear.

######  Proposition 2.3.1.

If $L\subset L^{\prime}$ is complete and $K^{\prime}$ is a
$(L^{\prime},\leq)$-filtered complex, then the restriction $K^{\prime}|_{L}$
is a $(L,\leq)$-filtered complex.

######  Definition 2.3.9.

An order preserving map $f:(L,\leq)\to(L^{\prime},\leq)$ is called
homomorphism of local lattices if

  1. 1.

$f(x\vee y)=f(x)\vee f(y)\in L^{\prime}[0,f(t)]$, for any $x,y\in L[0,t]$

  2. 2.

$f^{-1}(L^{\prime}[0,t])$ is a sublattice in $L$

######  Definition 2.3.10 (Totalization).

The totalization of $(L,\leq)$-filtered complex $K$ is the complex
$Tot(K)=\oplus_{x\in L}K_{x}$ with differential equal to the sum $d+\partial$.

######  Definition 2.3.11.

An order preserving map $(L,\leq)\to(L^{\prime},\leq)$ is called contraction,
if for any $y<:x\in L$, $f(y)<:f(x)$ of $f(y)=f(x)$.

######  Definition 2.3.12 (Pushforward).

Let $K$ be a $(L,\leq)$-chains complex in $Comp(C)$ and
$f:(L,\leq)\to(L^{\prime},\leq)$ is a contraction map.

Then the direct image of $K$ under $f$ is an $(L^{\prime},\leq)$-chains object
$f_{*}K$ with components:

$(f_{*}K)_{x^{\prime}}=Tot(K|_{f^{-1}x^{\prime}})$

and the structure maps:

$\partial^{f_{*}K}_{y^{\prime}x^{\prime}}:(f_{*}K)_{x^{\prime}}\to(f_{*}K)_{y^{\prime}}$

for $y^{\prime}<x^{\prime}$ equal to

$\sum\limits_{y<x|f(x)=x^{\prime},f(y)=y^{\prime}}\partial_{yx}$

######  Proposition 2.3.2.

The above definition is correct and there is natural isomorphism
$Tot(f_{*}K)\simeq Tot(K)$. In addition, if $f$ is a homomorphism and $K$ is
$(L,\leq)$-chains algebra, then $f_{*}K$ is $(L^{\prime},\leq)$-chains
algebra.

###### Proof.

Objects $(f_{*}K)_{x^{\prime}}$ are totalizations and thus complexes with its
own inner differential $d^{\prime}_{x^{\prime}}=\sum\limits_{x\in
f^{-1}x^{\prime}}d_{x}+\sum\limits_{y<x\in
f^{-1}x^{\prime}}\partial_{yx}=:d_{x^{\prime}}+\partial_{x^{\prime}}$.

Let $\partial^{\prime}=\partial^{f_{*}K}$ and $d^{\prime}=\sum
d^{\prime}_{x^{\prime}}$. By the construction

$(\bigoplus\limits_{x^{\prime}\in
L^{\prime}}(f_{*}K)_{x^{\prime}},d^{\prime}+\partial^{\prime})=(\bigoplus\limits_{x\in
L}K_{x},d+\partial)$

Thus $(d^{\prime}+\partial^{\prime})^{2}=0$ and it remains to show
$\partial^{\prime}_{y^{\prime}x^{\prime}}\partial_{x^{\prime}}+\partial_{y^{\prime}}\partial^{\prime}_{y^{\prime}x^{\prime}}$
for all pairs $y^{\prime}<x^{\prime}\in L^{\prime}$. Take $x\in
f^{-1}(x^{\prime}),y\in f^{-1}(y^{\prime})$, then $yx$-component of the last
expression is equal to:

$(\partial^{2})_{xy}-\sum\limits_{\begin{subarray}{c}z:\\\ x<z<y\\\
f(x)<f(z)<f(y)\\\ \end{subarray}}\partial_{xz}\partial_{zy}$

Since $K$ of range $1$ and $f$ is contraction
$\partial^{\prime}_{y^{\prime}x^{\prime}}\neq 0$ only for
$y^{\prime}<:x^{\prime}$. Thus the set $\\{z:f(x)<f(z)<f(y)\\}$ is empty,
hence $\partial^{\prime}_{y^{\prime}x^{\prime}}$ is a morphism of complexes
and the claim follows.

If $K$ is $(L,\leq)$-chains algebra and $f$ is a contraction homomorphism of
local lattices, then $Tot(f_{*}(K)|_{L[0,t]})=Tot(K|_{f^{-1}L[0,t]})$ is dg-
algebra. This defines the $(L^{\prime},\leq)$-chains algebra structure on
$f_{*}K$. ∎

Since the composition of two contraction maps is a contraction the pushforward
map satisfy $f_{*}\circ g_{*}=(f\circ g)_{*}$. If $\pi:(L,\leq)\to(pt,\leq)$
is the projection to the point, then $Tot(K)=\pi_{*}K$.

######  Definition 2.3.13 (Pullback).

If $f:(L,\leq)\to(L^{\prime},\leq)$ is an order preserving map and
$A^{\prime}$ is $(L^{\prime},\leq)$-object, then its pull-back under $f$ is
$(L,\leq)$-object $f^{*}A^{\prime}$ with values
$(f^{*}A^{\prime})_{x}:=A^{\prime}_{f(x)}$ and obvious structure maps.

The following is clear.

######  Proposition 2.3.3.

If $f$ above is a homomorphism and $A^{\prime}$ is
$(L^{\prime},\leq)$-algebra, then $f^{*}A^{\prime}$ has a natural structure of
$(L,\leq)$-algebra.

Note that pushing forward well-graded $(L,\leq)$-chains algebra or pulling
back well-gradd $(L,\leq)$-algebra need not be well-graded.

######  Remark 2.3.1.

Note that there is no general recipe for pulling back $(L,\leq)$-chains
objects even for homomorphism of lattices, e.g. the restriction is well-
defined only for complete sublattice.

######  Lemma 2.3.1.

If $\phi:L\to L$ is an order preserving map such that $\phi(x)\geq x,\forall
x\in L$ and $A$ is $(L,\leq)$-object, then there is a natural homomorphism
$u_{\phi}:\phi^{*}A\to A$.

###### Proof.

Define $u_{\phi}:(\phi^{*}A)_{x}=A_{\phi(x)}\to A_{x}$ as the corresponding
structure map. Then for any pair $x^{\prime}\leq x$ the following diagram
commutes:
${\phi^{*}(A)_{x}}$${A_{x}}$${\phi^{*}(A)_{x^{\prime}}}$${A_{x^{\prime}}}$$\scriptstyle{u_{\phi}}$$\scriptstyle{u_{\phi}}$
∎

######  Lemma 2.3.2 (Projection formula).

Let $p:(L,\leq)\to(L^{\prime},\leq)$ be a homomorphism of lattices and
$A^{\prime}$ is $(L^{\prime},\leq)$-complex and $D$ is $(L,\leq)$-chains
complex, then there is a natural isomorphism of $(L,\leq)$-chains complexes:

$p_{*}(p^{*}A^{\prime}\otimes D)\simeq A^{\prime}\otimes p_{*}D$

###### Proof.

We have

$(p_{*}(p^{*}A^{\prime}\otimes D))_{x^{\prime}}=\bigoplus\limits_{x\in
p^{-1}x^{\prime}}(p^{*}A)_{x}\otimes D_{x}=\bigoplus\limits_{x\in
p^{-1}x^{\prime}}A_{x^{\prime}}\otimes D_{x}=(A^{\prime}\otimes
p_{*}D)_{x^{\prime}}$

Then it is straighforward to check that this identification commutes with
structure maps:

$\partial^{\prime}_{y^{\prime}x^{\prime}}:(p_{*}(p^{*}A^{\prime}\otimes
D))_{x^{\prime}}\to(p_{*}(p^{*}A^{\prime}\otimes D))_{y^{\prime}}$

and

$\partial^{\prime\prime}_{y^{\prime}x^{\prime}}:(A^{\prime}\otimes
p_{*}D)_{x^{\prime}}\to(A^{\prime}\otimes p^{*}D)_{y^{\prime}}$

∎

######  Definition 2.3.14 (Convolution).

Let $K$ be a $(L,\leq)$-chains object (algebra) and $A$ is a $(L,\leq)$-object
(algebra) in category of complexes $Comp(C)$ (vector spaces,sheaves etc).
Define the convolution

$A*K:=\pi_{*}(A\otimes K)=Tot(A\otimes K)$

where $\pi:(L,\leq)\to(pt,\leq)$ is a morphism to the trivial lattice with one
vertice.

Thus $Tot(K)$ is the convolution of $K$ with the trivial algebra (i.e. with
all vertices equal to $\mathbb{Q}$) and

$K*A=Tot(K\otimes A)=(\bigoplus\limits_{x\in L}K_{x}\otimes A_{x},d+\partial)$

######  Definition 2.3.15 (Rank filtration).

If $K$ is $(L,\leq)$-chains complex, then the complex $Tot(K)$ admits an
increasing filtration $W^{\prime}_{p}Tot(K):=\bigoplus\limits_{x\in L:r(x)\leq
p}K_{x}$. The filtration $W^{\prime}$ is called the rank filtration.

If $K$ is $(L,\leq)$-chains algebra, then the rank filtration $W^{\prime}$ is
multiplicative. If $K$ is well-graded, then $Tot(K)$ becomes a bicomplex with
$W^{\prime}$ corresponding to columns-wise filtration.

###  2.4 $(L,\leq)$–mixed Hodge complexes

Here we prove two lemmas which allow to produce Hodge complexes from
$(L,\leq)$-Hodge complexes.

First we note the existence of the natural Hodge structure on convolution.

######  Lemma 2.4.1.

Assume $(A,F,W)_{\bullet}$ is a $(L,\leq)$-Hodge complex of sheaves and $K$ is
$(L,\leq)$-chains complex of vector spaces. Then $A\otimes K$ is a
$(L,\leq)$-chains of Hodge complexes and the convolution $(A*K,F,W)$ has a
natural structure of a Hodge complex.

###### Proof.

The convolution $A*K=\oplus_{k}\bigoplus\limits_{x:r(x)=k}A_{x}\otimes K_{x}$
can be viewed as a bicomplex with the vertical differential $d_{v}$ leaving
the rank filtration the same and the horizontal $d_{h}$ which decreases rank
by $1$. Note that each summond $A_{x}\otimes K_{x}$ is a Hodge complex, since
it can be viewed as an iterated cone (definition 1.2.1). Thus they are endowed
with filtrations $F_{x},W_{x}$. Then we simply take the direct sum of
filtrations to obtain $F,W$. After $A_{x}\otimes K_{x}$ are endowed with Hodge
complex structures there are no additional degree shifts, since $d_{v}$ is
already of degree $1$. ∎

On the other hand the rank filtration (definition 2.3.15) can be used to
construct an other Hodge complex structure on the convolution:

######  Lemma 2.4.2.

Assume $(A,F,W)_{\bullet}$ is a $(L,\leq)$-pure Hodge complex of weight $0$
and $K$ is $(L,\leq)$-chains complex of vector spaces such that $H^{*}(K_{x})$
is concentrated in degree $-r(x)$, then there is the rank Hodge complex
$(A*K,F,W^{\prime})$. Moreover, both the rank and the natural Hodge mixed
Hodge structures $Dec\ W$ and $Dec\ W^{\prime}$ coincide on
$\mathbb{H}^{*}(A*K)$.

###### Proof.

(1) First we shall prove that $Gr^{p}(A*K,W^{\prime})$ is a pure Hodge complex
of weight $-p$. Note that $Gr^{p}(A*K,W^{\prime})=\bigoplus\limits_{x\in
L:r(x)=-p}A_{x}\otimes K_{x}$ is a sum of complexes, each term is the tensor
product $A_{x}\otimes K_{x}$ of complexes. Since $K_{x}\sim
H^{-r(x)}(K_{x})[r(x)]$, we obtain: $H^{n}(A_{x}\otimes
K_{x})=H^{n-p}(A_{x})\otimes H^{p}(K_{x})$. Since $A_{x}$ has weight $0$, the
Hodge filtration $F$ is $n-p$-opposed on $H^{n}(Gr^{p}(A*K,W^{\prime}))$, thus
$(A*K,F,W^{\prime})$ is a Hodge complex.

(2) Unfortunately we do not have a direct comparsion map between two
filtrations. To prove that $W$ and $W^{\prime}$ on
$H^{*}:=\mathbb{H}^{*}(A*K)$ are the same, is to prove
${}_{W}Gr^{p}{}_{W^{\prime}}Gr^{p^{\prime}}H^{*}=0$ unless $p=p^{\prime}$. We
will use lemma 1.1.3 on two filtrations of the first section. Namely,
$W_{dir}$ on
$E^{p^{\prime},n-p^{\prime}}_{2}(A*K,W^{\prime})={}_{W^{\prime}}Gr^{p^{\prime}}(H^{n})$
is equal to $W_{ind}$ if $d_{1}$ is $W_{dir}$-strict on
$E_{1}(A*K,W^{\prime})$. Thus it is enough to show that
${}_{W}Gr^{p}E^{p^{\prime},n-p^{\prime}}_{1}(A*K,W^{\prime})=0$ unless
$p=p^{\prime}$. The Hodge filtration $F$ on
$V^{n}:=E^{p^{\prime},n-p^{\prime}}_{1}(A*K,W^{\prime})=H^{n}(Gr^{p^{\prime}}(A*K,W^{\prime}))$
is $(n-p^{\prime})$-conjugate by the previous, thus $(F,W)$-Hodge complex
$Gr^{p^{\prime}}(A*K,W^{\prime})$ has weight $-p^{\prime}$, i.e. the
corresponding $(F,W)$-mixed Hodge structure on $V^{n}$ admits non trivial
weights only in degree $n-p^{\prime}$, thus (remark 1.2.1) $Gr^{Dec\
W^{\prime}}_{n-p}V^{n}=Gr^{W}_{-p}V^{n}={}_{W}Gr^{p}V^{n}\neq 0$ only if
$p^{\prime}=p$. ∎

######  Definition 2.4.1.

The Hodge complex structure provided by the lemma above is called the rank
Hodge complex.

###  2.5 Orlik-Solomon algebras

Following Orlik-Solomon’s work [7] we define:

######  Definition 2.5.1 (OS-complex).

The OS-complex $\mathcal{M}$ of a local graded lattice $(L,\leq)$ is a
$(L,\leq)$-chains complex such that:

  1. 1.

$\mathcal{M}_{0}=\mathbb{Q}$

  2. 2.

$\mathcal{M}_{s}$ sits in degree $\ -r(s)$.

  3. 3.

$Tot(\mathcal{M}|_{[0,s]})$ is acyclic for all $s>0\in L$. In other words, the
negative-graded vector space

$\bigoplus\limits_{t\leq s}\mathcal{M}_{t}$

with $\partial$ defined by:

$\partial v_{y}=\sum\limits_{x:y:>x}\partial_{xy}v_{y}$

is an acyclic complex for $s>0\in L$.

######  Remark 2.5.1.

The OS-complex $\mathcal{M}$ is the categorification of the Mobious function
$\mu$ of $L$, i.e. $\mu[0,s]=(-1)^{r(s)}dim\ \mathcal{M}_{s}$.

######  Proposition 2.5.1.

For any local graded lattice $(L,\leq)$ there is a natural OS-complex
$\mathcal{M}$. It is unique up to a canonical isomorphism.

###### Proof.

Since the poset is graded, it is straighforward to prove both claims by
induction. ∎

######  Definition 2.5.2.

OS-algebra of a local lattice $(L,\leq)$ is an OS-complex $\mathcal{M}$ which
is $(L,\leq)$-chains algebra.

Thus the multiplication in OS-algebra is non-zero only on independend
components.

######  Theorem 2.5.1 (Orlik-Solomon).

If $(L,\leq)$ is a locally geometric lattice, then there is a natural OS-
algebra $OS(L)$ underlying $L$.

###### Proof.

We sketch the construction, the details can be found in the original work [7]
by Orlik-Solomon. First will reduce to global geometric lattice by putting
$OS(L)_{x}=OS(L[0,x])_{x}$. If $L$ is geometric lattice, then
$OS(L)=\wedge^{*}\langle d\nu_{i},i\in Atoms(L)\rangle/I$, where $I$ is an
ideal in the Grassman algebra we want to descibe. The Kozsul differential
$\partial$ acts on the Grassman algebra and is equal to
$\sum\limits_{i}\iota_{i}$. Let $\langle dep\rangle$ denotes an ideal spanned
by $\nu_{I}$ for $I\subset Atoms(L)$ is dependend, i.e. $r(sup(I))<|I|$. Then
$I$ as an ideal is generated by $\partial\langle dep\rangle$. One can check
that $I=\langle dep\rangle+\partial\langle dep\rangle$ as vector spaces and
that $I$ is $L$-graded. The Koszul differential acts on $OS(L)$ and there is a
contraction $\partial^{*}=\sum\limits_{i\in Atoms(L)}d\nu_{i}\wedge-\ $ such
that $[\partial,\partial^{*}]$ is invertible, hence $(OS(L[0,x]),\partial)$ is
acyclic unless $x=0\in L$. ∎

######  Example 2.5.1.

(i) If $L=\mathcal{C}(X,Z_{*})$ is the cubical poset, then the corresponding
OS-algebra $\mathcal{M}=OS(L)$ is the Grassman algebra generated by free
variables $d\nu_{*},deg(d\nu_{*})=-1$ corresponding to $Z_{*}$. The
differential $\partial=\sum\limits_{i}\iota_{i}$ is the whole Koszul
differential. The algebra is clearly $L$-graded and $L$-filtred according to
the differential: the vertice $I\in\mathcal{C}(X,Z_{*})$ of the cube
corresponds to the element $d\nu_{I}=d\nu_{I_{1}}\wedge\ldots\wedge
d\nu_{I_{k}}\in\mathcal{M}_{I}$, for $I_{1}<\ldots<I_{k}$ forming $I$.

(ii) The intersection poset $L=(X^{n},\Delta_{*})$ of all diagonals is a
geometric lattice with atoms $\Delta_{ab}$ given by big diagonals
$x_{a}=x_{b}$. The minimal element is $\infty\in L$ corresponds to
$x_{1}=\ldots=x_{n}$. Then $\mathcal{M}[0,\infty]$ for $\mathcal{M}=OS(L)$ is
$\Lambda\langle\Delta_{ab},a\neq b\in 1\ldots n\rangle$,
$deg(\Delta_{ab})=-1,d\Delta_{ab}=1$ modulo Arnold’s relation:

$\Delta_{ab}\Delta_{bc}+\Delta_{bc}\Delta_{ca}+\Delta_{ca}\Delta_{ab}=0$

Algebra $\mathcal{M}[0,\infty]$ admits a natural $L$-grading with $L$-pieces
$\mathcal{M}[0,\infty]_{s},s\in L$ spanned by grassman monomials which define
the given diagonal $L_{s}$. This defines an $(L,\leq)$-chains algebra
$\mathcal{M}$ with $\mathcal{M}_{s}=\mathcal{M}[0,\infty]_{s}$.

Let $(L,\leq)$ be a locally geometric lattice.

######  Definition 2.5.3.

The quasi-cubical lattice of $(L,\leq)$ is a poset $(\mathcal{Q}^{L},\leq)$
formed by pairs $(x,I)$ for $I\subset Atoms(L)$ such that $sup(I)=x\in
L[0,x]$. Then $(x,I)\leq(y,J)$ iff $x\leq y$ and $I\subset J$.

Note that if $(L,\leq)$ is geometric, then
$\mathcal{Q}^{L}=(2^{Atoms(L)})^{op}$ is the cubical lattice. There is a
natural map $p:(\mathcal{Q}^{L},\leq)\to(L,\leq)$ with $p((x,I))=x$.

######  Proposition 2.5.2.

If $L$ is locally geometric, then $\mathcal{Q}^{L}$ is a locally geometric
lattice and the map $p$ is contractible homomorphism.

###### Proof.

One can check that $(x,I)<:(y,J)$ iff $|J-I|=1$. Thus $r(x,I)=|I|$ defines
grading and $p$ is contractible. Then $(x,I)\vee(y,J)=(x\vee y\in L[0,t],I\cup
J)\in[0,(t,K)]$ for any $(t,K)$ with $(x,I),(y,J)\leq(t,K)$. ∎

Let $OS(\mathcal{Q}^{L})$ denote the natural OS-algebra of $\mathcal{Q}^{L}$.
If $L$ is geometric it is the usual Grassman algebra $\Lambda\langle
d\nu_{i},i\in Atoms(L)\rangle,deg(d\nu_{i})=-1$ with the Koszul differential
$\sum\limits_{i\in Atoms(L)}\iota_{i}$ and the obvious well-grading by $L$.

The following is an extension to the local case of the notion of relative
atomic complex used by Feichter-Yuzvinsky in [11].

######  Definition 2.5.4 (Atomic complex).

The atomic complex $\mathcal{D}^{L}$ of $(L,\leq)$ is $(L,\leq)$-chains
algebra equal to the direct image $p_{*}OS(\mathcal{Q}^{L})$ under natural map
$p:\mathcal{Q}^{L}\to L$.

Explicitely $(\mathcal{D}^{L}_{x},\partial^{\prime})$ is a complex spanned by
$d\nu^{(x)}_{I},I\subset Atoms(L)$ with $sup(I)=x\in L[0,x]$. The differential
$\partial^{\prime}$ acts as follows:

$\partial^{\prime}d\nu^{(x)}_{I}=\sum\limits_{i\in I:sup(I-\\{i\\})=x\in
L[0,x]}(\iota_{i}d\nu_{I})^{(x)}$

The structure morphisms
$\partial^{\mathcal{D}^{L}}_{yx}:\mathcal{D}^{L}_{x}\to\mathcal{D}^{L}_{y}[1],x:>y$
are equal to the resting part of the Koszul differential. The multiplication
is straighforward.

We have the following homological interpretation of $OS(L)$. Consider $OS(L)$
as $(L,\leq)$-chains algebra with trivial inner differential. There is a map
$\mathcal{D}(L)\to OS(L)$ of $(L,\leq)$-chains algebras induced by the
identity. In other words sending $d\nu_{I}$ to $d\nu_{I}\in
OS(L)=\Lambda^{*}\langle d\nu_{i}\rangle/\sim$ is well-defined, i.e.
$\partial^{\prime}d\nu_{I}=0\in OS(L)$. The following result is proved in
[11]:

######  Theorem 2.5.2 (Formality of $\mathcal{D}(L)$).

The morphism $\mathcal{D}(L)\to OS(L)$ is a quasi-isomorphism of
$(L,\leq)$-chains algebras. Thus the atomic algebra is formal, one has a
natural isomorphism $H^{*}(\mathcal{D}(L)_{x})\simeq OS(L)_{x}$ commuting with
$\partial_{yx}$.

$\square$

##  3 Towards the complement

In this section we will define several versions of the Mayer-Vietoris complex:
$MV,(MVHC_{\mathbb{Q}},MVHC_{\mathbb{C}})$,
$(MVHD_{\mathbb{Q}},MVHD_{\mathbb{C}})$ adapted for calculation of the
cohomology and the rational homotopy type of the complement
$U=X-\bigcup\limits_{x>0\in L}L_{x}$.

In subsection (1) we recall basic preliminaries about sheaves. Then, if
$i:(L,\leq)^{op}\to 2^{X}$ is an arragement poset with a given OS-complex
$\mathcal{M}$, we define $MV$ as the convolution $i_{*}i^{!}G*\mathcal{M}$
with a flabby $(L,\leq)$-sheaf over $X$, where $G$ is the Godement resolution
of $\underline{\mathbb{Z}}$. This provides a natural resolution $MV\sim
j_{*}j^{*}\underline{\mathbb{Z}}$. If $\mathcal{M}$ is OS-algebra and $G$ was
multiplicative, then $MV$ is dg-algebra of sheaves.

In subsection (2) we construct a multiplicative model for the cohomology of
$U$. Though the similar results appeared in [6], we decided to reprove the
corresponding theorems under weaker assumptions. Namely, we omit the
requirement on $(L,\leq)$ to be locally geometric, only assuming that the
local lattice is graded and an OS-algebra $\mathcal{M}$ is given, further we
do not assume that intersections of varieties in the arrangement are "clean".

The outline is following. The $L$-grading defines the rank filtration
(definition 2.3.15) $W^{\prime}$ on $i_{*}i^{!}G*\mathcal{M}$ and the
associated spectral sequence ${}_{W^{\prime}}E_{1}$ is easily expressed in
terms of the arrangement. Then, any OS-algebra $\mathcal{M}$ of $(L,\leq)$
allows to introduce a Hodge complex
$((MVHC_{\mathbb{Q}},W^{\prime}),(MVHC_{\mathbb{C}},F,W^{\prime}))$. By the
construction $MVHC_{\mathbb{Q}}$ is a dg-algebra, thus the corresponding
${}_{W^{\prime}}E_{2}$-term tensored by $\mathbb{C}$ is canonically isomorphic
to $H^{*}(U,\mathbb{C})$. This also provides the Hodge structure on the
associated graded pure Hodge structure of $H^{*}(U)$.

In the last subsection (3) we define a Hodge diagram
$((MVHD_{\mathbb{Q}},W^{\prime}),(MVHD_{\mathbb{C}},F,W^{\prime}))$ in case
$(L,\leq)$ is geometric. Then Morgan’s machinery provides an expression for
$\mathbb{Q}$-homotopy type of $U$ as the corresponding
${}_{W^{\prime}}E_{1}$-term.

###  3.1 Mayer-Vietoris resolution of $j_{*}j^{*}\underline{\mathbb{Q}}$

If $V_{i}$ is a collection of subvarieties in $X$, then for a multi-index $I$
we write $V_{I}=\bigcap\limits_{i\in I}V_{i},V_{\emptyset}=X$ and
$V^{I}=\bigcup\limits_{i\in I}V_{i}$. Put $U_{i}=X-Z_{i}$, then
$U_{I}=X-Z^{I}$ and $U^{I}=X-Z_{I}$. Consider $Z_{\bullet}$ and $U_{\bullet}$
as cubical varieties augmented by inclusions $i_{\bullet}:Z_{\bullet}\to X$
and $j_{\bullet}:U_{\bullet}\to X$.

######  Definition 3.1.1 (see [12], §10).

If $X$ is a smooth manifold, we denote by $S(X)_{\bullet}\in sSet$ the smooth
singular simplicial set of $X$.

######  Definition-Construction 3.1.1.

If $A_{\bullet}$ is a simplicial dg-algebra and $X$ is a smooth manifold, then
the presheaf

$X\to A(X)^{q}=Hom_{sSet}(X_{\bullet},A_{\bullet}^{q})$

is naturally a dg-algebra with point-wise multiplication.

Thus $A(pt)=A_{\bullet}$. The basic fact is that if $A_{\bullet}\to
A^{\prime}_{\bullet}$ is a quasi-isomorphism, then $A^{*}(X)\to
A^{\prime*}(X)$ is a quasi-isomorphism if $A$ and $A^{\prime}$ are Kan
simplicial objects (see [12]).

######  Example 3.1.1.

(i) Take ${A_{PL}}_{\bullet}$ to be polynomial differential forms over
$\mathbb{Q}$ on the standard simplex, i.e.
${A_{PL}}_{n}=\Omega^{*}_{PL}(\Delta^{n})$.

(ii) Taking ${CS_{sm}}_{\bullet}=C^{*}_{sm,sing}(\Delta^{n})$ gives a non-
commutative dg-algebra denoted by $CS_{sm}^{*}(X)$. Taking $C^{*}_{sing}$
defines $CS^{*}(X)$.

(iii) Similarly we can define $A_{sm}(X)$ over $\mathbb{R}$.

Note that $A(pt)=A_{\bullet}$. The basic fact is that if $A_{\bullet}\to
A^{\prime}_{\bullet}$ is a quasi-isomorphism, then $A^{*}(X)\to
A^{\prime*}(X)$ is a quasi-isomorphism if $A$ and $A^{\prime}$ are Kan
simplicial objects, thus

######  Proposition 3.1.1 (see [12], §10).

The following natural maps are quasi-isomorphism:

${A_{PL}}_{\bullet}\to{CS_{sm}}_{\bullet}\longleftarrow CS_{\bullet}$

The corresponding presheaves are naturally quasi-isomorphic to the singular
cochains $C^{*}$ functor.

######  Definition 3.1.2.

The sheaffication of presheaf $X\to A^{*}(X)$ is denoted by
$\underline{A}^{*}$.

Since all this presheaves are quasi-isomorphic to singular cochains $C^{*}$,
the corresponding sheaffications are also quasi-isomorphic as presheaves.

######  Definition 3.1.3.

We call a sheaf $G\in Sh(X)$ adapted, if $G|_{U}$ is $\Gamma$-acyclic for all
open $U\subset X$.

######  Proposition 3.1.2.

Over smooth manifolds there are natural sheaves of commutative dg-algebras
$\underline{A}_{PL},\underline{A}_{Sm}$ over $\mathbb{Q}$ and $\mathbb{R}$
respectively. There are natural quasi-isomorphisms of sheaves of dg-algebras:

$\underline{A}_{PL}\otimes_{\mathbb{Q}}\mathbb{C}\overset{\sim}{\rightarrow}\underline{A}_{Sm}\otimes_{\mathbb{R}}\mathbb{C}\overset{\sim}{\leftarrow}\Lambda^{*}\otimes_{\mathbb{R}}\mathbb{C}$

where $\Lambda^{*}$ is the de Rham sheaf. All quasi-isomorphisms holds on the
level of presheaves, i.e. all introduced sheaves are adapted. The
sheaffication $\underline{C}^{*}_{sing}\otimes\mathbb{Q}$ of singular cochains
is naturally quasi-isomorphic to $\underline{A}_{PL}$.

######  Definition 3.1.4.

Call $F$ a Godement sheaf, if it has the form $F(U)=\prod_{x\in
U}\tilde{F}_{x}$, for some sheaf $\tilde{F}$.

Fix $(L,\leq)^{op}\to 2^{X}$ to be an arrangement poset enhanced with an OS-
complex $\mathcal{M}$. Denote by $i_{ts}:L_{s}\to L_{t}\subset X,s\geq t$ and
$j:X-\bigcup\limits_{s\in L}L_{s}\to X$ the natural inclusions. Put
$i_{s}:=i_{0s}$.

Any sheaf $F$ over $X$ defines a $(L,\leq)$-sheaf $(i_{*}i^{!}F)_{\bullet}$
given by $(i_{*}i^{!}F)_{s}:={i_{s}}_{*}{i_{s}}^{!}F$.

######  Definition 3.1.5 (Mayer-Vietoris complex).

For any $(L,\leq)$-sheaf $\mathcal{F}_{\bullet}$, the Mayer-Vietoris complex
$MV(\mathcal{F}_{\bullet})$ is the convolution
$\mathcal{F}_{\bullet}*\mathcal{M}_{\bullet}$.

Recall that $MV(\mathcal{F}_{\bullet})$ is $L$-filtred and thus admits the
rank filtration $W^{\prime}$:

$W^{\prime}_{k}MV_{\bullet}(\mathcal{F})=\bigoplus\limits_{x\in L:r(x)\leq
k}MV_{x}(\mathcal{F})$

If $\mathcal{F}_{\bullet}$ is a $(L,\leq)$-algebra of (complexes of) sheaves
and $\mathcal{M}$ is an OS-algebra, then $MV(\mathcal{F}_{\bullet})$ becomes a
dg-algebra of sheaves.

Note that, if $F$ is a sheaf of dg-algebras, then, since $L_{x}\cap
L_{y}=\bigsqcup\limits_{t\in x\overset{\circ}{\vee}y}L_{t}$,
$(i_{*}i^{!}F)_{\bullet}$ naturally a $(L,\leq)$-algebra of sheaves. For
similicity we force $(i_{*}i^{!}F)_{\bullet}$ to be well-graded. Namely, we
put the component
$m^{t}_{xy}:(i_{*}i^{!}F)_{x}\otimes(i_{*}i^{!}F)_{y}\to(i_{*}i^{!}F)_{t}$ for
$t\in x\overset{\circ}{\vee}y$ to be zero if $r(t)<r(x)+r(y)$ and equal to the
product otherwise.

Let $F$ be a sheaf over $X$.

######  Proposition 3.1.3.

There is a natural augmentation $MV_{\bullet}:=MV((i_{*}i^{!})_{\bullet}F)\to
j_{*}j^{*}F$ of complexes mapping $MV_{s}$ to zero if $s>0\in L$. Assume $F$
is a Godement sheaf then

  1. 1.

There is a natural inclusion $j_{*}j^{*}F\to F=MV_{0}\subset MV_{\bullet}$ of
complexes. It is inverse to the augmentation.

  2. 2.

The augmentation makes $MV_{\bullet}$ a flabby resolution of $F$.

###### Proof.

Clearly the augmentation is a map of complexes, since $\partial MV_{t},t>0$ is
formed by sections supported on the complement $X-\bigcup\limits_{s>0\in
L}L_{s}$.

We want to show it is a quasi-isomorphism. Take an open $U\subset X$.

Since $F(U)=\prod_{x\in U}G_{x}$ for some sheaf $G$, we have natural
identification:

$MV_{\bullet}(U)=\prod_{x\in U}\bigoplus\limits_{t\in L:L_{t}\ni
x}\mathcal{M}(t)\otimes G_{x}$

and

$j_{*}j^{*}F(U)=\prod_{x\in U:x\not\in\bigcup\limits_{s>0\in L}L_{s}}G_{x}$

Note that $S_{x}:=\\{t\in L:L_{t}\ni x\\}\subset L$, by definition of $L$, is
a sublattice with the maximal element $s_{x}\in L$, thus $S_{x}=L[0,s_{x}]$.
Moreover, $s_{x}>0$ iff $x\not\in X-\bigcup\limits_{s>0}L_{s}$. Hence

$(\bigoplus\limits_{L_{t}\ni x}\mathcal{M}(t)\otimes
G_{x},\partial)=(\mathcal{M}[0,s_{x}],\partial)\otimes G_{x}$

is naturally quasi-isomorphic to $G_{x}$ if $x\in X-\bigcup\limits_{s>0}L_{s}$
and is acyclic otherwise.

Recall that $\mathcal{M}[0,0]=\mathbb{Z}$. Thus $MV_{\bullet}(F)(U)$ is a
product over $x\in U$ of complexes equal to $G_{x}$ for $x\in
U-\bigcup\limits_{s>0\in L}L_{s}$ and of acyclic complexes for
$x\in\bigcup\limits_{s>0\in L}L_{s}$. This defines the inclusion in the
obvious manner. Clearly both maps are quasi-isomorphism. ∎

If $F$ is a sheaf of dg-algebras, then by the construction the augmentation is
a morphism of dg-algebras.

Recall that the Godement cosimplicial construction applied to
$\underline{\mathbb{Q}}$ provides a canonical multiplicative (non-commutative)
resolution of $\underline{\mathbb{Q}}\to G$ by Godement sheaves.

Consider the sheaffication of singular cochains $\underline{C}_{sing}$. Then
there is a natural chain of quasi-isomorphisms of presheaves of dg-algebras
$\underline{C}_{sing}\to\underline{C}_{sing}\otimes G\leftarrow G$. It is
well-known that $(i_{*}i^{!})_{\bullet}\underline{C}_{sing}$ and $i_{*}i^{!}G$
both compute cohomology with support, thus are naturally quasi-isomorphic.

If $j_{x}:X-L_{x}\to X$ is an open inclusion, then we denote by
$(j_{*}j^{*})_{\bullet}\underline{C}_{sing}$ a $(L,\leq)$-sheaf with value at
$x\in L$ equal to $(j_{x})_{*}j^{*}_{x}\underline{C}_{sing}$. If
$\underline{C}_{sing}|_{X}$ denotes the constant $(L,\leq)$-sheaf on $X$, then
$(i_{*}i^{!})_{\bullet}\underline{C}_{sing}$ is quasi-isomorphic to the
shifted cone (definition 1.2.1)
$[\underline{C}_{sing}|_{X}\to(j_{*}j^{*})_{\bullet}\underline{C}_{sing}]$.

Then, by naturality of $MV$ along $(L,\leq)$-sheaves we obtain

######  Corollary 3.1.1.

Let $\mathcal{M}$ be a OS-complex (-algebra) of arrangement poset $(L,\leq)$
on $X$. There is a natural quasi-isomorphism

$\underline{\mathbb{Q}}_{U}\sim
MV((i_{*}i^{!})_{\bullet}\underline{C}_{sing})$

of complexes (algebras). Similarly one has a natural quasi-isomorphisms of
complexes:

$\underline{\mathbb{Q}}_{U}\sim
MV([{\underline{C}_{sing}}|_{X}\to(j_{*}j^{*})_{\bullet}\underline{C}_{sing}])\sim
MV([{\underline{A}_{PL}}|_{X}\to(j_{*}j^{*})_{\bullet}\underline{A}_{PL}])$

###  3.2 Čech model for $i_{*}i^{!}\underline{C}^{*}_{sing}$

In course of the proof of the main theorem we will need a multiplicative Čech
resolution model for $i_{*}i^{!}G$, let us recall the construction. Here $G$
stands for the Godement multiplicative resolution of $\underline{\mathbb{Q}}$
or $\underline{C}^{*}_{sing}$.

Let $U_{i}\subset X,i\in N$ be a collections of open subsets and
$\underline{A}$ denotes one of the natural sheaves
$\Omega^{*},\underline{C}^{*}_{sing}$ etc.

Let $i:X-\cup_{i}U_{i}\to X$ and $j_{I}:U_{I}\to X$ be the natural inclusions
and $g_{JI}:(j_{*}j^{*})_{I}\underline{A}\to(j_{*}j^{*})_{J}\underline{A}$
denotes the restriction map for $J\supset I$. We define the corresponding Čech
model for $i_{*}i^{!}G$ as follows.

Let

$\check{C}(\underline{A},U_{*},N)=\bigoplus\limits_{I\subset
N}(j_{*}j^{*})^{I}\underline{A}\cdot d\tau^{I}$

with an additional Čech differential

$\delta_{Ch}=\sum\limits_{I,k}g_{I+\\{k\\},I}\cdot d\tau^{k}\wedge-$

Let $U_{*}+U^{\prime}_{*}$ denotes the union of collections $U_{*}$ and
$U^{\prime}_{*}$ parametrized by ordered set $N\sqcup N^{\prime}$.

With our notation consider the obvious multiplication map:

$m_{U,V}:\check{C}(\underline{A},U_{*},N)\otimes\check{C}(\underline{A},U^{\prime}_{*},N)\to\check{C}(\underline{A},V_{*},M)$

Let $U^{\prime}_{j}\subset X,j\in N^{\prime}$ be other collection of opens.

######  Proposition 3.2.1.

With notation above

  1. 1.

the multiplication is associative, i.e. if
$U_{*},U^{\prime}_{*},U^{\prime\prime}_{*}$ are three collections of open
subsets, then

$m_{U+U^{\prime},U^{\prime\prime}}\circ m_{U,U^{\prime}}\otimes
id=m_{U,U^{\prime}+U^{\prime\prime}}\circ id\otimes
m_{U^{\prime},U^{\prime\prime}}$

  2. 2.

there is a natural quasi-isomorphism of sheaves:

$\check{C}(\underline{A},U_{*},N)\sim(i_{*}i^{!})_{X-U}G$

Such that the following diagram is commutative:

${\check{C}(\underline{A},U_{*},N)\otimes\check{C}(\underline{A},U^{\prime}_{*},N^{\prime})}$${\check{C}(\underline{A},U_{*}+U^{\prime}_{*},M)}$${(i_{*}i^{!})_{X-U}G\otimes(i^{\prime}_{*}i^{\prime!})_{X-U^{\prime}}G}$${(i_{*}i^{!})_{X-V}G}$$\scriptstyle{m}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{m}$

###### Proof.

Associativity is obvious. Then, using the natural quasi-isomorphism:
$\check{C}(G,U_{*},N)\sim\check{C}(\underline{A},U_{*},N)$ we reduce to case
$\underline{A}=G$. Since the sheaf $i_{*}i^{!}G$ is naturally a multiplicative
subcomplex in $G$, which is the first term of $\check{C}(G,U_{*},N)$. ∎

###  3.3 Hodge complex $(MVHC_{\mathbb{Q}},MVHC_{\mathbb{C}})$

Here $(L,\leq)$ is an arrangement poset over $X$ with an OS-complex
$\mathcal{M}$. Recall that $U=X-\bigcup\limits_{s>0\in L}L_{s}$. We shall
construct a functorial replacement for $i_{*}i^{!}G$ to define a Hodge complex
of sheaves.

####  3.3.1 The construction

Recall that, for a subset $I\subset Atoms(L)$ we write
$L^{I}=\bigcup\limits_{x\in I}L_{x}$ and $L_{I}=\bigcap\limits_{x\in i}L_{x}$.

Let $\mathcal{C}$ be the cubical poset with vertices corresponding to all
subsets of $Atoms(L)$. Consider the open $\mathcal{C}$-subspace $U_{\bullet}$
of $X$, defined by $U_{I}=X-L^{I},I\subset Atoms(L)$. Thus $U_{0}=X$ and
$U_{a}=X-L^{a}\to U_{0},r(a)=1$ etc. For each $x\in L$ let $I_{x}:=\\{t\in
Atoms(L):t\leq x\\}$. Since $I_{x}\subset I_{y}$ for $x\leq y$ we may consider
$I:(L,\leq)\to(\mathcal{C},\leq)$ as an order preserving map. By the
definition of the arrangement poset $L_{I_{x}}=L_{x}$.

######  Proposition 3.3.1.

There is a smooth compact algebraic $\mathcal{C}$-variety
$\tilde{U}_{\bullet}$ with a morphism $U_{\bullet}\to\tilde{U}_{\bullet}$
which is NCD-embedding and $\tilde{U}_{0}=X$.

###### Proof.

The diagram of spaces $U_{\bullet}$ is ranked. So that $U_{0}=X$, $U_{a}\to
X,r(a)=1$ etc. Put $\tilde{U}_{0}=X$, we construct $\tilde{U}_{I}$ by
induction on $r(I)=|I|$. If $\tilde{U}_{I}$ for $r(I)<h$ are given, then for
$J,r(J)=h$ take the diagonal embedding $j:U_{J}\to\prod_{I<:J}\tilde{U}_{I}$.
By Hironaka’s theorem there is an extension of $j$ to a smooth projective
$\tilde{U}_{J}$ such that $U_{J}\to\tilde{U}_{J}$ is an NCD-embedding. ∎

Let $g_{I}:\tilde{U}_{I}\to\tilde{U}_{0}=X$ be the natural map. We can
consider a $\mathcal{C}^{op}$-sheaf on $X$

$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{\bullet}):\mathcal{C}\to Sh(X)$

defined as

$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{I})=(g_{I})_{*}\Omega^{*}_{\tilde{U}_{I}}(\ln\tilde{U}_{I}-U_{I})$

The RHS can be viewed as a subsheaf of $j_{*}j^{*}\Omega^{*}_{X}$ and is
formed by forms $\omega$ on $X$ with appropriate singularites near $L^{I}$,
meaning that the pull-back of $g^{*}_{I}\omega$ has logarithmic singularites
along $\tilde{U}_{I}-U_{I}$. Since $\tilde{U}_{\bullet}$ is a diagram of
varieties, the restriction induced by $X-L^{I}\to X-L^{J},I\subset J$ of a
form on $X$ with appropriate singularity near $L^{I}$ has an appropriate
singularity near $L^{J}$.

Define $A^{\mathcal{C}^{op}}_{PL}(X-L^{\bullet}):\mathcal{C}\to Sh(X)$ by
$A^{\mathcal{C}^{op}}_{PL}(X-L^{I})={\underline{A}_{PL}}_{(X-L^{\bullet})}$
and similarly $\Omega^{\mathcal{C}^{op}}(X-L^{\bullet})$. Clearly
$A^{\mathcal{C}^{op}}_{PL}(X-L^{\bullet})\otimes\mathbb{C}$,
$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{\bullet})$ and
$\Omega^{\mathcal{C}^{op}}(X-L^{\bullet})$ are naturally quasi-isomorphic
$\mathcal{C}^{op}$-sheaves. Denote by
$\underline{C}^{\mathcal{C}}_{sing}(X,X-L_{\bullet})$ the $\mathcal{C}$-sheaf
formed by terms $\underline{C}^{*}_{sing}(X,X-L_{I})$.

Let $\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})$ denotes a
$\mathcal{C}$-algebra of sheaves on $X$ defined as the Čech resolution
(proposition 3.2.1)i:

$\Omega^{\mathcal{C}}_{log}(X,X-L_{I})=\bigoplus\limits_{K\subset
I}\Omega^{\mathcal{C}^{op}}_{log}(X-L^{K})\cdot
d\tau^{K}=[\Omega^{*}(X)\to\bigoplus\limits_{a\leq
x,r(a)=1}\Omega^{\mathcal{C}^{op}}_{log}(X-L^{a})\to\ldots]$

Here $d\tau^{K}$ are usual fictive Grassmann monomials with
$deg(d\tau^{K})=|K|$ such that the differential is equal to $\delta_{Ch}=\sum
g^{\mathcal{C}^{op}}_{K+\\{k\\},K}d\tau^{k}\wedge-$, where
$g^{\mathcal{C}^{op}}_{K+\\{k\\},K}$ is the restriction map which is the
structure map of $\mathcal{C}^{op}$-sheaf
$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{\bullet})$.

The structure maps
$g^{\mathcal{C}}_{II^{\prime}}:\Omega^{\mathcal{C}}_{log}(X,X-L_{I}^{\prime})\to\Omega^{\mathcal{C}}_{log}(X,X-L_{I}),I\subset
I^{\prime}$ are defined by mapping the component
$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{K^{\prime}}),K^{\prime}\subset
I^{\prime}$ identically if $K^{\prime}\subset I$ and to zero otherwise. On the
level of Čech resolutions there is a natural map

$\Omega^{\mathcal{C}}_{log}(X,X-L_{I})\otimes\Omega^{\mathcal{C}}_{log}(X,X-L_{I^{\prime}})\to\Omega^{\mathcal{C}}_{log}(X,X-L_{II^{\prime}}),I\cap
J=\emptyset$

which acts on components
$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{K})\cdot\tau^{K},K\subset I$ and
$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{K^{\prime}})\cdot\tau^{K^{\prime}},K^{\prime}\subset
I^{\prime}$ as the inclusion to
$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{KK^{\prime}})$ followed by the usual
multiplication of forms. Here we used the fact that the log-complex is dg-
algebra.

The natural quasi-isomorphisms of the previous section provides the natural
quasi-isomorphism
$\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})\sim\underline{C}^{\mathcal{C}}_{sing}(X,X-L_{\bullet})\otimes\mathbb{C}$
of $(\mathcal{C},\leq)$-sheaf algebras. This is equivalent to Čech model
described in the previous section.

Denote by $\underline{C}^{L}_{sing}(X,X-L_{\bullet})$ an $(L,\leq)$-sheaf with
values
$\underline{C}^{L}_{sing}(X,X-L_{t}):=\underline{C}^{*}_{sing}(X,X-L_{t}),t\in
L$. Similarly for $\underline{A}^{L}_{PL}(X,X-L_{\bullet})$,
$\underline{C}^{L}_{sing}(X,X-L_{\bullet})$ and $\Omega^{L}(X,X-L_{\bullet})$.

Contrary, we define

$\Omega^{L}_{log}(X,X-L_{\bullet}):=I^{*}\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})$

as an $(L,\leq)$-sheaf given by the pulling back along
$I:(L,\leq)\to(\mathcal{C},\leq)$.

Clearly
$\underline{C}^{L}_{sing}(X,X-L_{\bullet})\otimes_{\mathbb{Q}}\mathbb{C}$ and
$\Omega^{L}_{log}(X,X-L_{\bullet})$ are quasi-isomorphic.

Thus the data we needed so far is an arrangement poset $(L,\leq)^{op}\to X$,
the OS-complex $\mathcal{M}$ of $(L,\leq)$ and the Hironaka’s compactification
$U_{\bullet}\to\tilde{U}_{\bullet}$.

Recall that $W^{\prime}$ denotes the rank filtration (definition 2.3.15).

######  Definition 3.3.1 (Mayer-Vietoris Hodge complex).

Using convolutions define

$MVHC_{\mathbb{Q}}(L,\mathcal{M})=(\underline{C}^{L}_{sing}(X,X-L_{\bullet})*\mathcal{M},W^{\prime})$
$MVHC_{\mathbb{C}}(L,\mathcal{M})=(\Omega^{L}_{log}(X,X-L_{\bullet})*\mathcal{M},F,W^{\prime})$

then the rank Hodge complex

$MVHC(L,\mathcal{M}):=((MVHC_{\mathbb{Q}}(L,\mathcal{M}),W^{\prime}),(MVHC_{\mathbb{C}}(L,\mathcal{M}),F,W^{\prime}))$

is called the rank Mayer-Vietoris Hodge complex.

All filtrations are provided by the lemma 2.4.1 on convolutions of Hodge
complexes. Similarly, the lemma implies the existence of the natural Hodge
complex structure $((MVHC_{\mathbb{Q}},W),(MVHC_{\mathbb{C}},F,W))$.

Below we will often omit $(L,\leq)$ and $\mathcal{M}$ from the notation. Thus
$MVHC_{\mathbb{Q}}$ is equal to $MV(i_{*}i^{!}\underline{C}_{sing})$
introduced in previous section.

######  Remark 3.3.1.

If $\mathcal{M}$ is an OS-algebra, then $MVHC_{\mathbb{Q}}(\mathcal{M})$ is
dg-algebra. However, since $I:L\to\mathcal{C}$ is not a homomorphism in
general, the pull-back
$\Omega^{L}(X,X-L_{\bullet})=I^{*}\Omega^{\mathcal{C}}(X,X-L_{\bullet})$ is
not $(L,\leq)$-algebra. Thus $MVHC_{\mathbb{C}}$ in general does not has a
natural algebra structure.

####  3.3.2 An explicit expression for $MVHC(\mathcal{M})$

For clarity let us describe $MVHC$ explicitely. Since $L_{I_{x}}=L_{x}$ one
can write:

$MVHC_{\mathbb{Q}}=\bigoplus\limits_{x\in
L}\underline{C}_{sing}(X,X-L_{x})\otimes\mathcal{M}_{x}$

with the differential $d+\partial$, where $d$ is the inner differential on
cochains and $\partial=\sum\limits_{x^{\prime}<:x}\partial_{x^{\prime}x}$ is
the sum of degree $1$. Then $d,\partial$ clearly commute if we impose the
usual sign-rules, e.g.:

$\partial(c_{x}\otimes
v_{x})=\sum\limits_{x^{\prime}<:x}(-1)^{deg(c_{x})}c_{x^{\prime}}\otimes\partial_{x^{\prime}x}v_{x}$

Similarly

$MVHC_{\mathbb{C}}=\bigoplus\limits_{x\in L}\bigoplus\limits_{K\subset
I_{x}}\Omega^{\mathcal{C}^{op}}_{log}(X-I^{K})\cdot
d\tau^{K}\otimes\mathcal{M}_{x}$

where $d\tau^{K}$ are usual fictive monomials of the corresponding Grassmann
algebra, here $deg(d\tau^{K})=|K|$.

The whole differential is $d_{dR}+\partial+\delta_{Ch}$. Here $d_{dR}$ is de
Rham differential,
$\partial=\sum\limits_{x^{\prime}<:x}g^{\mathcal{C}}_{I_{x^{\prime}},I_{x}}\partial_{x^{\prime}x}$
and
$\delta_{Ch}=\sum\limits_{k}g^{\mathcal{C}^{op}}_{K+\\{k\\},K}d\tau^{k}\wedge-$
, we consider structure maps $g$ as having degree $0$. All maps between
different vertices in cubical diagrams of sheaves are geometric and induced by
maps of compactified algebraic varieties with NCD-complement.

Thus $\mathcal{C}^{op}$-diagram
$(\underline{C}_{sing}(X-L^{\bullet}),\Omega_{log}(X-L^{\bullet}))$ form a
$\mathcal{C}^{op}$-diagram of Hodge complexes with the Hodge and weight
filtrations equal to $F,W$. Then
$(\underline{C}_{sing}(X,X-L_{\bullet}),\Omega_{log}(X,X-L_{\bullet}))$ is
$(L,\leq)$-diagram of Hodge complexes.

Since diagram $(L,\leq)$ is graded, one can consider the convolution as an
iterated cone, then the mixed cone construction (definition 1.2.1) produces
the natural Hodge complex $(MVHC,F,W)$ and the rank Hodge complex
$(MVHC,F,W^{\prime})$ by the lemma 2.4.1.

####  3.3.3 The homotopy type of $U$

######  Proposition 3.3.2.

The hypercohomology of the Hodge complex
$((MVHC_{\mathbb{Q}},W),(MVHC_{\mathbb{C}},F,W))$ gives a mixed Hodge
structure naturally isomorphic to the cohomology of the complement $H^{n}(U)$,
for $U=X-\bigcup\limits_{x>0\in L}L_{x}$.

###### Proof.

Recall that $\tilde{U}_{N},N=Atoms(L)$ denotes the chosen Hironaki
compactification of the complement $X-L^{N}=U_{N}=U$. Since the morphisms
between mixed Hodge structures are strict, it is enough to provide a quasi-
isomorphism of $(MVHC_{\mathbb{C}},F,W)$ with the log-complex
$(\Omega^{*}_{\tilde{U}}(ln\ \tilde{U}_{N}-U),F,W)$ compatible with both-
filtrations. To do this let $\Omega^{\prime\mathcal{C}^{op}}_{\bullet}$ be the
constant $\mathcal{C}^{op}$-sheaf with all values equal to
$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{N})$. We have a natural restriction map
$\Omega^{\mathcal{C}^{op}}_{log}(X-L^{\bullet})\to\Omega^{\prime\mathcal{C}^{op}}_{\bullet}$.
Applying the construction of $\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})$ to
$\Omega^{\prime\mathcal{C}^{op}}_{\bullet}$ instead of
$\Omega^{\mathcal{C}^{op}}_{log}(X,X-L_{\bullet})$ we get $\mathcal{C}$-sheaf
$\Omega^{\prime\mathcal{C}}_{\bullet}$. Then we have a natural restriction
morphism:

$\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})\to\Omega^{\prime\mathcal{C}}_{\bullet}$

Note that Čech resolutions $\Omega^{\prime\mathcal{C}}_{I}$ are acyclic except
for $I=\emptyset$ with
$\Omega^{\prime\mathcal{C}}_{\emptyset}=\Omega^{*}_{\tilde{U}}(ln\
\tilde{U}-U)$. Then we have a chain of quasi-isomorphisms compatible with
filtrations $F,W$:

$\Omega^{*}_{\tilde{U}}(ln\
\tilde{U}-U)\overset{\sim}{\to}I^{*}\Omega^{\prime\mathcal{C}}_{\bullet}*\mathcal{M}\overset{\sim}{\leftarrow}I^{*}\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})*\mathcal{M}$

which follows since all three complexes are natural resolutions of
$j_{*}j^{*}\mathbb{C}$. ∎

The following is clear.

######  Proposition 3.3.3.

The spectral sequence corresponding to the rank Hodge complex structure
$((MVHC_{\mathbb{Q}}(\mathcal{M}),W^{\prime}),(MVHC_{\mathbb{C}}(\mathcal{M}),F,W^{\prime}))$
satisfy the following:

  1. 1.

${}_{W^{\prime}}E^{pq}_{1}(MVHC_{\mathbb{Q}},W^{\prime})=\bigoplus\limits_{x\in
L:r(x)=-p}H^{q}(X,X-L_{x})\otimes\mathcal{M}_{x}$

  2. 2.

The differential is equal to $d_{1}=\sum
g_{x^{\prime}x}\otimes\partial_{x^{\prime}x}$, where
$g_{x^{\prime}x}:H^{*}_{L_{x}}(X)\to H^{*}_{L_{x^{\prime}}}(X)$ are natural
homomorphisms for $x^{\prime}<:x\in L$ and $\partial_{x^{\prime}x}$ are the
structure maps of the OS-algebra $M$.

  3. 3.

${}_{W^{\prime}}E_{2}={}_{W^{\prime}}E_{\infty}$

Thus $E_{1}$ term is simply the convolution of $(L,\leq)$-algebra
$H^{*}(X,X-L_{\bullet})$ with $\mathcal{M}$.

Note that, thanks to lemma 2.4.2 ${}_{W}Gr\
\mathbb{H}^{*}(MVHC_{\mathbb{Q}})={}_{W^{\prime}}Gr\
\mathbb{H}^{*}(MVHC_{\mathbb{Q}})$.

Assume $\mathcal{M}$ is an OS-algebra. Then, according to the remark 3.3.1
$MVHC_{\mathbb{Q}}(\mathcal{M})$ is a dg-algebra. Since the spectral sequence
is multiplicative, it converges to the algebra ${}_{W}Gr\ H^{*}(U)$.

Moreover, since the weight filtrations $W,W^{\prime}$ canonicaly split over
$\mathbb{C}$, one can use the spectral sequence to describe the algebra
$H^{*}(U,\mathbb{C})$. As a direct corollary of what been done, we conclude
with the following theorem.

######  Theorem 3.3.1 (The cohomology of the complement).

Let $\mathcal{M}$ be an OS-complex (algebra) of an arrangment poset $(L,\leq)$
on compact algebraic $X$. Consider the corresponding Mayer-Vietoris Hodge
complex $(MVHC_{\mathbb{Q}},MVHC_{\mathbb{C}})$. There is a canonical
isomorphism of vector spaces (algebras)

$E^{**}_{2}(MVHC_{\mathbb{C}},W^{\prime})\simeq H^{*}(U,\mathbb{C})$

It induces (remark 1.2.1) an isomorphism of pure Hodge structures of weight
$q$:

$E^{pq}_{2}(MVHC_{\mathbb{C}},W^{\prime})\simeq Gr^{Dec\
W}_{q}H^{p+q}(U,\mathbb{C})$

###### Proof.

Let $H^{n}=H^{n}(U)\simeq H^{n}(MVHC_{\mathbb{Q}})$. Recall that
$(H^{n},F,Dec\ W)$ is a mixed Hodge structure with $(Dec\
W)_{q}H^{n}=W^{n-q}H^{n}$ and thus the weight of ${}_{W}Gr^{p}H^{p+q}$ is $q$.
The Deligne splitting (proposition 1.2.2) implies the natural isomorphsim

$H^{*}(U,\mathbb{C})\simeq{}_{W}Gr\ H^{*}(U,\mathbb{C})$

of algebras. Since
$E^{pq}_{2}(MVHC_{\mathbb{C}},W)=E^{pq}_{2}(MVHC_{\mathbb{C}},W^{\prime})={}_{W}Gr^{p}H^{p+q}(U)$
the claim follows. ∎

######  Remark 3.3.2.

This result is also proved in [6] by a similar method. We want to note, that
the splitting does not holds in general over $\mathbb{Q}$. Even over
$\mathbb{R}$ the Deligne splitting of the weight filtration is not conjugate-
invariant.

Since a non canonical descent of the quasi-isomorphism still exists (theorem
1.2.3), we have:

######  Corollary 3.3.1.

If $\mathcal{M}$ is OS-algebra, then there is a non-canonical isomorphism
$E_{2}(MVHC_{\mathbb{Q}},W^{\prime})\simeq H^{*}(U,\mathbb{Q})$ of algebras.

###### Proof.

Morgan’s theorem 1.2.3 applied to a NCD-compactification of $U$ implies the
existence of a non canonical isomorphism of algebras
$Gr(H^{*}(U,\mathbb{Q}),W)\simeq H^{*}(U,\mathbb{Q})$. The result follows,
since

$E_{2}(MVHC_{\mathbb{Q}},W^{\prime})\simeq E_{2}(MVHC_{\mathbb{Q}},W)\simeq
Gr(H^{n}(U,\mathbb{Q}),W)$

∎

###  3.4 Hodge diagram $(MVHD_{\mathbb{Q}},MVHD_{\mathbb{C}})$

According to remark 3.3.1, the map $I:L\to\mathcal{C}$ is not a homomorphism
of lattices, thus in order to obtain an algebra structure on a complex part of
Mayer-Vietoris complex we have to impose an additional requirements on the
arrangement poset $(L,\leq)$.

Assume $(L,\leq)$ is good arrangement poset (definition 2.2.2), i.e.
$(L,\leq)$ is geometric lattice and hence $L_{I}=L_{x}$ for $x=sup(I)\in L$.

In this case the quasi-cubical lattice $\mathcal{Q}^{L}$ coincides with the
cubical lattice $\mathcal{C}$ introduced above and $OS(\mathcal{C})$ is just
the Grassman algebra $\Lambda^{*}\langle d\nu_{i},i\in Atoms(L)\rangle$ viewed
as $(L,\leq)$-chains algebra.

Let $p:(\mathcal{C},\leq)\to(L,\leq)$ be the homomorphism of lattices defined
by $p(I)=sup(I)\in L$.

######  Definition 3.4.1.

Let

$MVHD_{\mathbb{Q}}(L):=Tot(p_{*}(A^{\mathcal{C}}_{PL}(X,X-L_{\bullet})\otimes
OS(\mathcal{C})))$
$MVHD_{\mathbb{C}}(L):=Tot(p_{*}(\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})\otimes
OS(\mathcal{C})))$

The pair

$((MVHD_{\mathbb{Q}},W^{\prime}),(MVHD_{\mathbb{C}},F,W^{\prime}))$

is called Mayer-Vietoris Hodge Diagram.

Thus $MVHD$ is a convolution over cubical lattice with the rank filtration
induced by that of $L$. Note that the lemma 2.4.1 on convolutions implies that
$((MVHD_{\mathbb{Q}},W),(MVHD_{\mathbb{C}},F,W))$ is a Hodge complex. Despite
the definition given above we need an additional argument to prove that the
rank filtration $W^{\prime}$ indeed induces the Hodge complex structure on
$MVHD$.

Explicitely

$MVHD_{\mathbb{C}}=\bigoplus\limits_{I\subset
Atoms(L)}\bigoplus\limits_{K\subset
I}\Omega^{\mathcal{C}^{op}}_{log}(X-L^{K})\cdot d\tau^{K}\cdot d\nu_{I}$

All terms with $r(sup(I))\leq k$ contributes to
$W^{\prime}_{k}MVHD_{\mathbb{C}}$. The whole differential has four components
$d_{dR},\delta_{Ch},\partial^{\prime},\partial^{\prime\prime}$, where $d_{dR}$
and $\delta_{Ch}$ are de Rham and Čech differentials as for $MVHC$
respectively, while $\partial=\partial^{\prime}+\partial^{\prime\prime}$
corresponds to the Koszul differential on $OS(\mathcal{C})$ and decomposes
according to $L$-filtration:
$\partial^{\prime}=\sum\limits_{i:sup(I)=sup(I-\\{i\\})}g^{\mathcal{C}}_{I-\\{i\\},I}\iota_{i}$.

Recall that $j:X-\bigcup\limits_{x>0\in L}L_{x}\to X$ is inclusion of the
complement.

######  Proposition 3.4.1.

  1. 1.

The pair
$((MVHD_{\mathbb{Q}}(L),W^{\prime}),(MVHD_{\mathbb{C}}(L),F,W^{\prime}))$ form
a Hodge diagram.

  2. 2.

There is a natural quasi-isomorphism of Hodge complexes:

$(MVHD(L),F,W^{\prime})\sim(MVHC(L,OS(L)),F,W^{\prime})$

Thus $MVHD_{\mathbb{iQ}}(L)$ is a cdga resolution
$j_{*}j^{*}\underline{\mathbb{Q}}$.

###### Proof.

Let $MVHC=MVHC^{\prime}(L,OS(L))$ denotes a Hodge complex obtained by
$MVHC^{\prime}_{\mathbb{Q}}=(\underline{A}^{L}_{PL}(X,X-L_{\bullet})*\mathcal{M},W^{\prime})$
and
$MVHC^{\prime}_{\mathbb{C}}=\Omega^{L}_{log}(X,X-L_{\bullet}*\mathcal{M},F,W^{\prime})$.
Thus $MVHC^{\prime}_{\mathbb{C}}=MVHC_{\mathbb{C}}$.

Since $MVHD$ is already a filtered algebra all claims will follow if we show
an existence of a natural quasi-isomorphisms $MVHD(L)\sim
MVHC^{\prime}(L,OS(L))$ compatible with the rank filtration $W^{\prime}$ and
giving an isomorphism of ${}_{W^{\prime}}E_{1}\otimes\mathbb{C}$-terms
compatible with the Hodge filtration.

We focus on the complex part, since the rational case is similar. Consider
$I\circ p:\mathcal{C}\to\mathcal{C}$. This is a order-preserving selfmap of
the lattice such that $(I\circ p)(J)=I(sup(J))\supset J$, applying lemma 2.3.1
on endomorphism we get a morphism of $(\mathcal{C},\leq)$-sheaves:

$p^{*}I^{*}\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})\to\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})$

Since in geometric lattice case $L_{I(sup(J))}=L_{J}$ this map is a quasi-
isomorphism.

Then we have a quasi-isomorphism of $(L,\leq)$-sheaves:

$p_{*}(p^{*}I^{*}\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})\otimes
OS(\mathcal{C}))\to p_{*}(\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})\otimes
OS(\mathcal{C}))$

By the projection formula of lemma 2.3.2 this is equivalent to a quasi-
isomorphism

$I^{*}\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})\otimes\mathcal{D}(L)\to
p_{*}(\Omega^{\mathcal{C}}_{log}(X,X-L_{\bullet})\otimes OS(\mathcal{C}))$

Since the atomic complex $\mathcal{D}(L)=p_{*}OS(\mathcal{C})$ is formal
(theorem 2.5.2), i.e. quasi-isomorphic to $OS(L)$, applying $Tot(-)$ to both
sides we get a filtered quasi-isomorphism of complexes compatible with the
Hodge filtration:

$(MVHC_{\mathbb{C}},W)\sim(MVHD_{\mathbb{C}},W^{\prime})$

∎

######  Remark 3.4.1.

Note that our resolution is actually the cubical resolution from the previous
section, the role of the lattice $L$ is to control the rank filtration making
$E_{1}$ term more accessible.

Thus ${}_{W^{\prime}}E_{1}$-term of $MVHD_{\mathbb{Q}}(L)$ have the same
description as $MVHC_{\mathbb{Q}}(L,OS(L))$ (proposition 3.3.3).

Recall that $OS(L)_{x}$ sits in degree $-r(x)$. The natural multiplication is
non-zero only on components $x,y\in L$ with $r(x\vee y)=r(x)+r(y)$ and define
a cdga structure on $\bigoplus\limits_{x}OS_{x}(L)$ by $OS_{x}(L)\otimes
OS_{y}(L)\to OS_{x\vee y}(L)$ with differential
$\partial=\sum\limits_{y<:x}\partial_{yx}$.

As a corollary we get

######  Theorem 3.4.1 (The main theorem for lattices).

Assume $(L,\leq)$ is an arrangment poset on $X$ with the complement
$X-\bigcup\limits_{x>0\in L}L_{x}$ and $(L,\leq)$ is a geometric lattice. Then
there is a multiplicative spectral sequence $E^{pq}_{r}$ such that:

  1. 1.

$E^{pq}_{1}=\bigoplus\limits_{x\in L:r(x)=-p}H^{q}(X,X-L_{x})\otimes
OS(L)_{x}$

is a cdga with the differential
$d_{1}=\sum\limits_{y<:x}g_{yx}\otimes\partial_{yx}$, where
$g_{yx}:H^{*}_{L_{x}}(X)\to H^{*}_{L_{y}}(X)$ are are the natural
homomorphisms.

  2. 2.

$E_{2}=E_{\infty}$.

  3. 3.

There is a natural quasi-isomorphism of dg-algebras

$E^{**}_{1}\otimes\mathbb{C}\sim C^{*}_{sing}(X-\bigcup\limits_{x>0\in
L}L_{x};\mathbb{C})$

thus $E^{**}_{1}\otimes\mathbb{C}$ provides a natural model for
$\mathbb{C}$-homotopy type of the complement.

  4. 4.

There is a non canonical quasi-isomorphism of $E^{**}_{1}$ with the rational
homotopy type of the complement.

###### Proof.

Here $E_{r}=E_{r}(MVHD_{\mathbb{Q}},W^{\prime})$ and since $MVHD$ is a Hodge
diagram which is quasi-isomorphic to $MVHC$ as a Hodge complex, the above
descripton of $E_{1}(MVHC_{\mathbb{Q}},W^{\prime})$ and Morgan’s theorem 1.2.3
immediatly yields the result. The minor issue is to pass from the Hodge
diagram $MVHD$ of sheaves to a Hodge diagram of vector spaces. To do this one
can consider, for example, a pair of dg-algebras $((\Gamma\
MVHD_{\mathbb{Q}},W^{\prime}),(\Gamma(MVHD_{\mathbb{C}}\otimes_{\mathcal{O}_{X}}\Omega^{0,*}_{X}),F,W^{\prime}))$.
Here the functor of global sections of the complex part is followed by the
Dolbeault resolution (see [1] part 2, 3.2.3). This provides a Hodge diagram to
which the theorem 1.2.3 is applied. ∎

Since the cubical lattice is always geometric, the following result is most
simple and general result of this type.

######  Corollary 3.4.1 (The main theorem).

For a smooth arrangement $Z_{I}\subset X$ of projective varieties there is a
multiplicative spectral sequence $E^{pq}_{r}$:

  1. 1.

$E^{pq}_{1}=\bigoplus_{|I|=-p}H^{q}_{Z_{I}}(X)\cdot d\nu_{I}$

  2. 2.

The differential $d_{1}=\sum g_{I-\\{i\\},I}\otimes\iota_{i}$, where
$g_{I^{\prime}I},I^{\prime}\subset I$ are natural homorphisms:

$H^{*}_{Z_{I}}(X)\to H^{*}_{Z_{I^{\prime}}}(X)$

induced by inclusion $Z_{I}\to Z_{I^{\prime}}$.

  3. 3.

The multiplication

$H^{q}_{Z_{I}}(X)\cdot d\nu_{I}\otimes H^{q^{\prime}}_{Z_{J}}(X)\cdot
d\nu_{J}\to H^{q+q^{\prime}}_{Z_{IJ}}(X)\cdot d\nu_{I}\wedge d\nu_{J}$

makes $E^{pq}_{1}$ into cdga.

  4. 4.

There is a natural quasi-isomorphism of cdga

$E^{**}_{1}\otimes\mathbb{C}\sim
C^{*}_{sing}(X-\bigcup\limits_{i}Z_{i};\mathbb{C})$

  5. 5.

There is a non canonical quasi-isomorphism of rational hompotopy type of
$X-\bigcup\limits_{i}Z_{i}$ with $E^{**}_{1}$.

##  4 Applications

###  4.1 Formality of linear subspace arrangement with geometric intersection
poset

Consider an intersection poset of linear subspaces in $\mathbb{C}^{n}$. Let
$(L,\leq)$ be the corresponding intersection poset. Assume that $(L,\leq)$ is
geometric. The following result is proved in [11]:

######  Theorem 4.1.1 (Feichtner, Yuzvinsky).

The complement $\mathbb{C}^{n}-\bigcup\limits_{x>0}L_{x}$ is formal.

To prove this theorem using our methods let
$X=\mathbb{P}^{n}\supset\mathbb{C}^{n}$ and $\tilde{L}$ be the intersection
poset of the compactification of $L$ together with a hyperplane at infinity
$\tilde{L}_{\infty}:=\mathbb{P}^{n-1}$. Clearly
$U:=\mathbb{P}^{n}-\bigcup\limits_{x\in\tilde{L}}\tilde{L}_{x}=\mathbb{C}^{n}-\bigcup\limits_{x\in
L}L_{x}$.

Let us first describe a simple model for the homotopy type of $U$ without any
assumptions on the intersection poset. By our results on the cubical model the
following cdga controls the rational homotopy type of $U$:

$(\bigoplus\limits_{I\subset
Atoms(\tilde{L})}H^{*}_{\tilde{L}_{I}}(\mathbb{P}^{n})\cdot
d\nu_{I},\partial)$

Recall that $d\nu_{I}$ has degree $-|I|$ and the differential
$\partial=\sum\limits_{i}\iota_{i}$ is the whole Koszul differential.

Since $H^{*}(\mathbb{P}^{n})=\mathbb{Q}[t]/t^{n+1},deg(t)=2$ and
$\tilde{L}_{I}$ are linear one can identify
$H^{*}_{\tilde{L}_{I}}(\mathbb{P}^{n})\subset H^{*}(\mathbb{P}^{n})$ with
$t^{codim\ L_{I}}H^{*}(\mathbb{P}^{n})$ for $\tilde{L}_{I}\neq\emptyset$ and
with $0$ otherwise. This description is compatible with the algebra structure,
the element $t^{codim\ L_{I}}$ corresponds to the Thom class $\tau_{I}$ of
$L_{I}$.

Further, since taking the direct product with an affine space does not change
the homotopy type of the affine complement and the codimension of subspaces in
the arrangement, one can assume that for any $I\subset Atoms(\tilde{L})$,
$\tilde{L}_{I}\subset\mathbb{P}^{n}\neq\emptyset$.

Consider the following complexes over $\mathbb{Q}[t]$:

  1. 1.

$K=\bigoplus\limits_{I\subset Atoms(\tilde{L})}t^{codim\
L_{I}}\mathbb{Q}[t]/t^{n+1}\cdot d\nu_{I}$

  2. 2.

$K^{\prime}=\bigoplus\limits_{I\subset Atoms(\tilde{L})}t^{codim\
L_{I}}\mathbb{Q}[t]\cdot d\nu_{I}$

  3. 3.

$K^{\prime\prime}=\bigoplus\limits_{I\subset Atoms(\tilde{L})}d\mu_{I}$

The Koszul differential naturally acts on each complex. We consider $K$ and
$K^{\prime}$ as cdga’s with the multplication $d\nu_{I}\cdot
d\nu_{J}=d\nu_{I}\wedge d\nu_{J}$. By our assumption $K$ controls the rational
homotopy type of $U$.

Define $i:K^{\prime\prime}{\to}K^{\prime}$ by $i(d\mu_{I})=t^{n+1}d\nu_{I}$
and $p:K^{\prime}\to K$ to be the natural projection. Both maps commute with
the Koszul differential and $p$ is a morphism of cdga’s. Clearly $0\to
K^{\prime\prime}\overset{i}{\to}K^{\prime}\overset{p}{\to}K\to 0$ is exact.
The Koszul complex $K^{\prime\prime}$ is acyclic and one can pass from $K$ to
$K^{\prime}$. It is natural to think of $K^{\prime}$ as a stable version of
the model $K$ as $n\to\infty$.

Let $s:K^{\prime}\to K^{\prime}[1]$ be the multipication by $t\cdot
d\nu_{\infty}\in K^{\prime}$.

######  Lemma 4.1.1.

  1. 1.

The commutator $[s,\partial]:K^{\prime}\to K^{\prime}$ is the multiplication
by $t$.

  2. 2.

The subcomplex $sK^{\prime}+tK^{\prime}\subset K^{\prime}$ is acyclic.

###### Proof.

(1) It is straighforward.

(2) By the previous $sK^{\prime}+tK^{\prime}$ is a subcomplex. Let
$sc_{1}+tc_{2}$ be a cocycle, i.e. $0=\partial(sc_{1}+tc_{2})=-s\partial
c_{1}+tc_{1}+t\partial c_{2}$. Then, since $s^{2}=0$, $t(sc_{1}+s\partial
c_{2})=0$, thus $sc_{1}+s\partial c_{2}=0$. Hence $\partial(sc_{2})=-s\partial
c_{2}+tc_{2}=sc_{1}+tc_{2}$ is a coboundary. ∎

As a corollary the rational homotopy type of $U$ is quasi-isomorphic to
$M:=K^{\prime}/(sK^{\prime}+tK^{\prime})$. Factorizing $sK^{\prime}$ out of
$K^{\prime}$ we reduce the lattice $\tilde{L}$ to $L$. The following
factorization by $tK^{\prime}$ leaves elements spanned by Thom classes. Thus:

$M=\bigoplus\limits_{I\subset Atoms(L)}\tau^{codim\ \tilde{L}_{I}}\cdot
d\nu_{I}$

where $\tau$ is a formal variable of degree $2$. The differential $\partial$
acts by

$\partial d\nu_{I}=\sum\limits_{\begin{subarray}{c}i\in I:\\\
\tilde{L}_{I-i}=\tilde{L}_{I}\end{subarray}}\iota_{i}d\nu_{I}$

making $K^{\prime}/(sK^{\prime}+tK^{\prime})$ into cdga.

Now we may back to the theorem.

###### Proof.

Assume $(L,\leq)$ is a geometric lattice Then $\tilde{L}_{I}$ is a
compactification of $L_{I}$ and thus $M=\bigoplus\limits_{x\in L}M_{x}$, where

$M_{x}=\bigoplus\limits_{\begin{subarray}{c}I\subset Atoms(L)\\\
\tilde{L}_{I}=\tilde{L}_{x}\end{subarray}}\tau^{codim\ \tilde{L}_{I}}\cdot
d\nu_{I}=\bigoplus\limits_{\begin{subarray}{c}I\subset Atoms(L)\\\
L_{I}=L_{x}\end{subarray}}\tau^{codim\ L_{x}}\cdot d\nu_{I}$

By formality of the atomic complex (theorem 2.5.2) $M_{x}$ is naturally quasi-
isomorphic to $H^{*}(M_{x})\simeq\tau^{codim\ L_{x}}\cdot OS(L)_{x}$.

This implies formality of $U$:

$M\sim\bigoplus\limits_{x\in L}\tau^{codim\ L_{x}}\cdot OS(L)_{x}$

where the algebra structure on the right hand side is such, that non zero
multiplication occurs only for components $x,y\in L$ intersecting
transversally, i.e. $codim\ L_{x}\cap L_{y}=codim\ L_{x}+codim\ L_{y}$. ∎

###  4.2 Rational model of configuration spaces

Assume $M$ is a smooth proper algebraic variety over $\mathbb{C}$. Consider
the intersection poset $(L,\leq)$ of all diagonals in $X=M^{n}$ (example
2.2.1). Let $F(M,n)=X-\bigcup\limits_{x>0\in L}L_{x}$ be the configuration
space of $n$-tuples of ordered points in $M$. More generally, for any finite
graph $G$ with vertices $V(G)=\\{1,\ldots,n\\}$ one can consider the chromatic
configuration space $F(M,G)=\\{(x_{1},\ldots,x_{n})\in M^{n}:x_{i}\neq
x_{j},(ij)\in E(G)\\}$. The variety $F(M,n)$ corresponds to the complete graph
on $n$ vertices.

It is well-known that $(L,\leq)$ forms a geometric lattice which has an
explicit desciption in terms of $G$, see [6]. Namely, an element $x\in L$ is a
unordered collection of disjoint subsets $S_{k}\subset V(G)$ such that $S_{k}$
is connected in $G$. The diagonal $L_{x}$ in $M^{n}$ is determined by the
condition $x_{i}=x_{j}$ whenever $i,j\in S_{k}$ for some $k$. Its codimension
is equal to $dim\ M\cdot r(x)$. The atoms $\Delta_{ab}\in L$ correspond to big
diagonals $L_{\Delta_{ab}}\subset M^{n}$. An independend set of atoms
corresponds to a set of edges in $G$ containing no cycles.

Recall that the corresponding OS-algebra $OS(L)$ is a quotient of the Grassman
algebra $\Lambda\langle\Delta_{ab},(ab)\in E(G)\rangle$ with
$deg(\Delta_{ab})=-1$, modulo an ideal generated by expressions
$\partial(\Delta_{i_{1}i_{2}}\wedge\ldots\wedge\Delta_{i_{k-1}i_{k}})$, where
$(i_{1},\ldots,i_{k}=i_{1})$ form a cycle in $G$. As usual
$\partial=\sum\limits_{i\in Atoms(L)}\iota_{i}$ denotes the whole Koszul
differential in the Grassman algebra. If $G$ is complete, the corresponding
ideal is generated by famous Arnold’s relations

$\Delta_{ab}\Delta_{bc}+\Delta_{bc}\Delta_{ca}+\Delta_{ca}\Delta_{ab}$

Thus, the Mayer-Vietoris model controlling the homotopy type of $F(M,G)$ is a
cdga

$(\bigoplus\limits_{x\in L}H^{*}_{L_{x}}(M^{n})\otimes OS(G)_{x},d)$

provided by our main theorem for lattices.

Let $\tau_{x}\in H^{2r(x)\cdot dim\ M}_{L_{x}}(M^{n})$ be the Thom class
corresponding to $L_{x}\subset M^{n}$. Denote by $p_{i}:M^{n}\to M$ the
projection onto $i$-th coordinate. If $x=\Delta_{ab}$, then

$H^{*}_{L_{x}}(M^{n})=\tau_{x}\cdot H^{*}(M^{n})\simeq H^{*}(M^{n})/\langle
p_{a}^{*}u=p_{b}^{*}u,u\in H^{*}(M)\rangle[-2dim\ M]$

If $x,y\in L$ are idependend, then $L_{x}$ intersects $L_{y}$ transversally
and hence $\tau_{x\vee y}=\tau_{x}\cup\tau_{y}$. Thus the model is generated
over $H^{*}(L_{0})=H^{*}(M^{n})$ by elements
$\tilde{\Delta}_{ab}:=\tau_{\Delta_{ab}}\otimes\Delta_{ab}$ of degree $2dim\
M-1$, subject to relation determing $OS(L)$. Note that
$d\tilde{\Delta}_{ab}=[L_{\Delta_{ab}}]\in H^{2dim\ M}(M^{n})$.

As a direct corollary we obtain the chromatic version of the Kriz-Totaro
model:

######  Theorem 4.2.1.

The rational homotopy type of $F(M,G)$ is a cdga equal to the quotient

$H^{*}(M^{n})\otimes\Lambda^{*}\langle\tilde{\Delta}_{ab},(ab)\in
E(G)\rangle/I$

with $deg(\tilde{\Delta}_{ab})=2dim\ M-1$, where $I$ is an ideal generated by

  1. 1.

$\partial(\Delta_{i_{1}i_{2}}\wedge\ldots\wedge\Delta_{i_{k-1}i_{k}})$, for
each cycle $(i_{1},\ldots,i_{k}=i_{1})$ in $G$.

  2. 2.

$p^{*}_{a}u\cdot\tilde{\Delta}_{ab}=p^{*}_{b}u\cdot\tilde{\Delta}_{ab},u\in
H^{*}(M)$

The differential $d$ is determined by
$d\tilde{\Delta}_{ab}=[L_{\Delta_{ab}}]\in H^{2dim\ M}(M^{n})$.

$\square$

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