Local systems over complements of hyperplanes and the Kac-Kazhdan
  conditions for singular vectors

10 pages, latex. A small error and a title in the bibliography are
  corrected

Abstract:
  In this note we strenghten a theorem by Esnault-Schechtman-Viehweg which
states that one can compute the cohomology of a complement of hyperplanes in a
complex affine space with coefficients in a local system using only logarithmic
global differential forms, provided certain "Aomoto non-resonance conditions"
for monodromies are fulfilled at some "edges" (intersections of hyperplanes).
We prove that it is enough to check these conditions on a smaller subset of
edges.
  We show that for certain known one dimensional local systems over
configuration spaces of points in a projective line defined by a root system
and a finite set of affine weights (these local systems arise in the geometric
study of Knizhnik-Zamolodchikov differential equations), the Aomoto resonance
conditions at non-diagonal edges coincide with Kac-Kazhdan conditions of
reducibility of Verma modules over affine Lie algebras.

\section{Quasiisomorphism.}

Let $
\{H_{i} \}_{i\in I}
$ be an affine arrangement
of hyperplanes, i.e.,
$\{H_{i} \}_{i\in I}
$ is a finite collection of (distinct) hyperplanes
in the affine complex space ${{\bf {\rm C}}\hspace{-.4em^{n}$.
Define $U = {{\bf {\rm C}}\hspace{-.4em^{n} - \bigcup_{i\in I} H_{i}.$
We denote by $\Omega^{p}_{U}$ the sheaves of holomorphic
forms on $U$ for $0\leq p\leq n$.
We set ${\cal O}_{U} := \Omega^{0}_{U}.   $

For any $i\in I$, choose a degree one polynomial function
$f_{i} $ on ${{\bf {\rm C}}\hspace{-.4em^{n} $ whose zero locus is equal to $H_{i}. $
Define $\omega_{i} := d\log f_{i} = df_{i}/f_{i} \in
\Gamma(U, \Omega^{1}_{U}). $
For a given $r\in {\rm I}\hspace{-.2em}{\rm N} - \{0\}$ we choose matrices
$P_{i} \in \mbox{\rm End\,} {{\bf {\rm C}}\hspace{-.4em^{s}, i\in I. $
Define
\[
\omega := \sum_{i\in I} \omega_{i} \otimes P_{i}
\in
\Gamma (U, \Omega^{1}_{U}) \otimes \mbox{\rm End\,} {{\bf {\rm C}}\hspace{-.4em^{s}.
\]

The form $\omega$ defines the connection $d + \omega$ on the trivial bundle
${\cal E} := {\cal O}_{U}^{s}.  $
We suppose that $(d + \omega)$ is {\it integrable} which is equivalent to
the condition $\omega \wedge \omega = 0$ as $d\omega = 0$.
Let $\Omega^{\bullet}_{U}({\cal E}) =
\Omega^{\bullet}_{U} \otimes_{{\cal O}_{U}} {\cal E}     $
be the de Rham complex with the differential $d+\omega$.

Define finite dimensional subspaces $$A^{p} \subset
\Gamma(U, \Omega^{p}_{U}({\cal E})) = \Gamma(U, \Omega^{p}_{U})
\otimes_{\bf C} {{\bf {\rm C}}\hspace{-.4em^{s}$$ as the ${{\bf {\rm C}}\hspace{-.4em$-linear subspaces generated by all
forms $\omega_{i_{1} }\wedge\cdots\wedge\omega_{i_{p} }\otimes v,
$
$v\in {{\bf {\rm C}}\hspace{-.4em^{s}. $
Then the exterior product by $\omega$ defines
\[
A^{\bullet} : 0 \longrightarrow A^{0} \stR{\omega} A^{1}
\stR{\omega} \cdots \stR{\omega} A^{n} \stR{} 0
\]
as a subcomplex of $\Gamma(U, \Omega^{\bullet}_{U}({\cal E}))$.

Let $
\overline{\cn}^{n}
$
be any smooth compactification of ${{\bf {\rm C}}\hspace{-.4em^{n} $
such that $H_{\infty}$ is a divisor.  Write
$H = H_{\infty} \cup \left(\bigcup_{\in I} H_{i} \right).$
Then $U = \overline{\cn}^{n} - H.$
(Typical examples for $\overline{\cn}^{n}$ include the complex projective space
${\bf P}^{n}, $ $({\bf P}^{1})^{n} $  and any toric manifold.)
Note that $\omega\in \Gamma(U, \Omega^{1}_{U}) \otimes \mbox{\rm End\,}
{{\bf {\rm C}}\hspace{-.4em^{s} $ can be uniquely extended to be an $\mbox{\rm End\,} {{\bf {\rm C}}\hspace{-.4em^{s}$-
coefficient rational $1$-form $\overline{\omega}$ on
${{\bf {\rm C}}\hspace{-.4em^{n}$.

\begin{theorem}
\label{theorem1}
Suppose $\pi : X \rightarrow \overline{\cn}^{n}$ be a blowing up of
$\overline{\cn}^{n}$ with centers in $H$ such that 1) $X$ is nonsingular,
2) $\pi^{-1}H $ is a normal crossing divisor,
and 3) none of the eigenvalues of the resideu of
$\pi^{-1}\bar{\omega} $ along any component of
$\pi^{-1}H $ lies in ${\rm I}\hspace{-.2em}{\rm N} - \{0\}$.
Then the inclusion
\[
A^{\bullet} \hookrightarrow \Gamma (U, \Omega_{U}^{\bullet}({\cal E}))
\]
is quasiisomorphism.
\end{theorem}

{\it Proof.~}
Same as the proof of the first theorem in \cite{ESV}.
~~\mbox{$\Box$}

\section{Decomposable arrangements}

Let  ${\cal A}$ be a central
arrangement in $V$, i.e., a finite collection of
hyperplanes with $
\bigcap_{A\in {\cal A}} A
 \neq \emptyset$.
Then ${\cal A}$ is called {\bf decomposable}
if
there exist nonempty subarrangements ${\cal A}_{1} $ and ${\cal A}_{2} $
with ${\cal A} = {\cal A}_{1} \cup {\cal A}_{2} $ (disjoint) and,
after a certain linear coordinate change,
defining equations for ${\cal A}_1$ and ${\cal A}_2$ have
no common variables.

Let ${\cal A}$ be a nonempty central arrangement
in ${{\bf {\rm C}}\hspace{-.4em^{n} $.
Let $T =
\bigcap_{A\in {\cal A}} A
\neq \emptyset$.
Suppose $\codim T = k+1 > 0$.
Then the points of ${\bf P}_{T} := {\bf P}^{k}$
parametrize the $(\dim X + 1)$-dimensional linear
subspaces
of ${{\bf {\rm C}}\hspace{-.4em^{n}$ which contain $T$.
In particular, if $H$ is a hyperplane containing
$T$, it uniquely determines a
hyperplane $H'$ in
${\bf P}^{k}$.
Define $P({\cal A})
:=
{\bf P}^{k}  - \bigcup_{H\in{\cal A}} H'$.

\begin{definition}
\label{definition1}
Define the {\bf beta invariant} of a central arrangement
${\cal A}$ by
\[
\beta({\cal A}) = (-1)^{r}\chi(P({\cal A}))
\]
where $\chi$ denotes the Euler characteristic.
\end{definition}

Let $L({\cal A})$ be
the set of all edges of
${\cal A}$.  We regard $L({\cal A})$  as a lattice with the reverse
inclusion as its partial order.
Then ${{\bf {\rm C}}\hspace{-.4em^{n}$ itself is the minimum element
of $L({\cal A})$.  Let $\mu$ be the M\"obius function of $L({\cal A})$.

\begin{definition}
\label{defchar}
(\cite[Def.2.52]{OrT})
Define the characteristic polynomial of ${\cal A}$ by
\[
\chi({\cal A}, t) = \sum_{X\in L({\cal A})}\mu(V, X) t^{\dim X}.
\]
\end{definition}

\begin{proposition}
\label{proposition1}
\[
\beta({\cal A})
=
(-1)^{k} \frac{d}{dt} \chi({\cal A}, 1).
\]
\end{proposition}

{\it Proof.~}
Since $P({\cal A})$ is homotopy equivalent to
the complement of
of the decone
$d{\cal A}$ \cite[p.15]{OrT} of ${\cal A}$
by \cite[Prop. 2.51, Thm.5.93]{OrT}, one has
\[
(1+t) \mbox{\rm Poin}(P({\cal A}), t)= \mbox{\rm Poin}(U, t),
\]
where $U$ is the complement of ${\cal A}$ and Poin stands for
the Poincar\'e polynomial.
Thus, by \cite[Def. 2.52]{OrT},
\begin{eqnarray*}
(t-1)^{-1} \chi({\cal A}, t)
&=&
(t-1)^{-1} t^{\ell}\mbox{\rm Poin}(U, -t^{-1})\\
&=&
(t-1)^{-1} t^{\ell} (1-t^{-1})
\mbox{\rm Poin}(P({\cal A}), -t^{-1}) \\
&=&
t^{\ell-1}\mbox{\rm Poin}(P({\cal A}), -t^{-1}).
\end{eqnarray*}
Take the limit as $t$ approaches $1$.
(Note $\chi({\cal A}, 1) = 0.$)
~~\mbox{$\Box$}

\medskip
Proposition \ref{proposition1} shows that
the beta invariant for the matroid
determined by ${\cal A}$.
The beta invariant for a  matroid
was introduced by
Crapo \cite{Cra}.

\begin{theorem}
\label{theorem2}
(\cite[Theorem 2]{Cra})

(1) If ${\cal A}$ is not empty, then $\beta({\cal A}) \geq 0$.

(2) $\beta({\cal A}) = 0$ if and only if ${\cal A}$ is decomposable.
~~\mbox{$\Box$}
\end{theorem}

Let ${\cal A}$ be an affine arrangement of hyperplanes in
${{\bf {\rm C}}\hspace{-.4em^{n} $.
Let $L$ be an edge of ${\cal A}$.

\begin{definition}
\label{}
An edge $L$
is called {\bf dense}
in ${\cal A}$ if and only if
the central arrangement
\[
{\cal A}_{L} := \{A\in {\cal A} ~|~ L \subseteq A\}
\]
is not decomposable.
\end{definition}

By Theorem \ref{theorem2}, we have

\begin{proposition}
\label{ESVproposition}
Let $L\in L({\cal A})$ with
$\codim L = r + 1.$
Then the following conditions are equivalent:

(1) $L$ is dense,

(2) ${\cal A}_{L} $ is not decomposable,

(3) $\chi(P({\cal A}_{L})) \neq 0$,

(4) $\beta({\cal A}_{L}) := (-1)^{r} \chi(P({\cal A}_{L})) > 0.$
~~\mbox{$\Box$}
\end{proposition}

\section{Resolution of a hyperplanelike divisor}
Let $Y $ be a smooth complex compact manifold
of dimension $n$, ${\cal D}$ a divisor.
${\cal D}$ is {\bf hyperplanelike} if $Y$ can be covered by
coordinate charts such that in each chart ${\cal D}$
is a union of hyperplanes.   Such charts will be called
{\bf linearizing}.

Let ${\cal D}$ be a hyperplanelike divisor, $U$ a linearizing
chart.  A {\bf local edge} of ${\cal D}$ in $U$ is any nonempty irreducible
intersection in $U$ of hyperplanes of ${\cal D}$ in $U$.
An {\bf edge} of ${\cal D}$ is the maximal analytic continuation
in $Y$ of a local edge.
Any edge is an immersed submanifold in $Y$.
An edge is called {\bf dense} if it is locally dense.

For $0\leq j\leq n-2$, let ${\cal L}_{j} $ be the collection of
all dense edges of ${\cal D}$ of dimension $j$.
The following theorem is essentially in
\cite[10.8]{Var}.

\begin{theorem}
\label{theorem3}
Let $W_{0} = Y$.
 Let
$\pi_1 : W_1 \rightarrow W_0$ be
the blow up along points in ${\cal L}_0$.
In general, for $1\leq s\leq \ell-1$, let  $\pi_s : W_s  \rightarrow
W_{s-1}$ be the
blow up along the proper transforms of the $(s-1)$-dimensional
dense edges
in ${\cal L}_{s-1}$ under $\pi_1\circ \cdots \circ \pi_{s-1}$.
Let $\pi = \pi_1\circ \cdots \circ \pi_{n-1}$.
Then $W := W_{n-1}$ is nonsingular and
$\pi^{-1} ({\cal D})$ normal crossing.
\end{theorem}

\section{Arrangements in ${\bf P}^{n} $ }

Let $
\{H_{i} \}_{i\in I}
$ be an affine arrangement
of hyperplanes in
 ${{\bf {\rm C}}\hspace{-.4em^{n}$.
Recall $U, f_{i}, \omega_{i}, P_{i}, \omega, {\cal E},$ and
$A^{\bullet} $ from Section 1.
Choose ${\bf P}^{n}$        as the compactification of
${{\bf {\rm C}}\hspace{-.4em^{n}. $
Let $H_{\infty} = {\bf P}^{n} - {\cn}^{n}$
and ${\cal A} =
\{\overline{H}_{i} \}_{i\in I}  \cup \{ H_{\infty} \}.
$
($\overline{H}_{i} $ is the closure of $H_{i} $ in ${\bf P}^{n}$.)
Obviously
$
\left(\bigcup_{i\in I} \overline{H}_{i} \right)  \cup H_{\infty}
$
is a hyperplanelike divisor.
Suppose $(z_{0}:\cdots :z_{n})$ be a homogeneous coordinate
system with $H_{\infty} : z_{0}=0. $
Then each $\omega_{i} $ is uniquely extended to be a rational form
$\overline{\omega}_{i} $ on ${\bf P}^{n}$;
$\overline{\omega}_{i} = \omega_{i} - (dz_{0}/z_{0}).$
Thus the form $\omega =
\sum_{i\in I} \omega_{i} \otimes P_{i}
\in
\Gamma (U, \Omega^{1}_{U}) \otimes \mbox{\rm End\,} {{\bf {\rm C}}\hspace{-.4em^{s}.
$
can be uniquely extended to $\overline{\omega}$:
\[
\overline{\omega}
=
\sum_{i\in I} \overline{\omega}_{i} \otimes P_{i}
=
\sum_{i\in I} \omega_{i} \otimes P_{i}  - (dz_{0}/z_{0}) \otimes
\left(
\sum_{i\in I}
P_{i}
\right).
\]
Define
$P_{\infty} = - \sum_{i\in I} P_{i}. $
For any edge $L$ of ${\cal A}$, let $I_{L} =
\{
i\in I\cup\{\infty\} | L\subseteq H_{i}
\}.$
Let
$P_{L} := \sum_{i\in I_{L}} P_{i}. $
By Theorems \ref{theorem1} and \ref{theorem3}, we get

\begin{theorem}
\label{theorem4}
We set ${\cal L}$ be
the set of all dense edges of ${\cal A}$.
Suppose that

\smallskip
\noindent
{\rm {\bf (Mon)* :}}  for all $L\in {\cal L}$, none of the eigenvalues of
$P_{L}$ lies in ${\rm I}\hspace{-.2em}{\rm N} - \{0\}$.
\smallskip

\noindent
Then the inclusion
\[
A^{\bullet} \hookrightarrow \Gamma (U, \Omega_{U}^{\bullet}({\cal E}))
\]
is quasiisomorphism. ~~\mbox{$\Box$}
\end{theorem}

\noindent
{\bf Remark.}  Since ``dense'' implies ``bad''
\cite{ESV}, Theorem \ref{theorem4} improves the main theorem of
\cite{ESV}.

\begin{corollary}
\label{corollary1}
Under the assumption of Theorem \ref{theorem4}, one has
\[
H^{p}(U, {\cal S}) \cong H^{p}(A^{\bullet})~~~
\mbox{\rm for~~}0\leq p\leq n
\]
where ${\cal S}$ is the local system of flat
sections of $({\cal E}, d + \omega)$
on $U$.   ~~\mbox{$\Box$}
\end{corollary}

\begin{corollary}
\label{corollary2}
Suppose
that

\smallskip
\noindent
{\rm {\bf (Mon)** :}}  for all $L\in {\cal L}$, none of the eigenvalues of
$P_{L} $ lies in ${\rm I}\hspace{-.2em}{\rm N} \cup \{0\}$.
\smallskip

\noindent
Also suppose that $P_{i} P_{j} = P_{j} P_{i}$ for all $i, j$.
Then
\[
H^{p}(U, {\cal S}) = 0~~~\mbox{\rm for~~} p \neq n.
\]
\end{corollary}

{\it Proof.~}
By Theorem \ref{theorem4} and
\cite[4.1]{Yuz}.
~~\mbox{$\Box$}

\section{Discriminantal arrangements in $({\bf P}^{1})^{n}$}

See \cite{ScV} for discriminantal arrangments.

Let $\Gamma$ be a graph without loops with vertices
$v_{1}, \ldots, v_{p}.  $
Let $
n_{1}$, $\ldots, n_{r}
$ be nonnegative integers,
$n =
n_{1}+\cdots + n_{r},
$
$
X = \{(i, \ell) | \ell = 1,\ldots, r, i=1,\ldots, n_{\ell} \},$
$
Y = ({\bf P}^{1})^{n}. $
Label the factors of $Y$ by elements of $X$
and for every $(i, \ell)\in X$ fix an affine coordinate
$t_{i}(\ell)$ on the $(i, \ell)$-th factor.

For pairwise distinct $z_{1},\ldots, z_{k} \in {{\bf {\rm C}}\hspace{-.4em, $
$
z_{k+1} = \infty$,
introduce in $Y$ a {\bf discriminantal arrangement}
${\cal A}$ of ``hyperplanes''
\[
H_{(i,\ell), j}
 :
t_{i}(\ell) = z_{j}
\mbox{\rm ~for~} (i, \ell)\in X, j=1,\ldots, k+1,
\]
\[
H_{(i,\ell), (j, \ell)}
:
t_{i}(\ell)=t_{j}(\ell)
\mbox{\rm ~for~} 1\leq i < j \leq n_{\ell},
\]
and
\[
H_{(i,\ell), (j, m)}
:
t_{i}(\ell)=t_{j}(m)
\]
for $\ell, m$ such that $v_{\ell} $ and $v_{m} $ are joined by an edge
in the graph and
$i=1,\cdots, n_{\ell},$
$j=1,\cdots, n_{m}.$
The union of these ``hyperplanes'' is a hyperplanelike divisor.
Let $\Delta \subseteq \Gamma$ be a connected subgraph with vertices
labelled by $V \subseteq \{1,\ldots, r\}.$
For every $\ell \in V$ fix a nonempty subset
$I_{\ell} \subseteq \{1,\ldots, n_{\ell} \}.$
Fix $j\in \{1,\ldots, k+1\}$.
Introduce edges
\[
L(\{I_{\ell}\}, j)
:=
\{t\in Y ~|~ t_{i}(\ell) = z_{j} \mbox{\rm ~for~} \ell\in V, i\in
I_{\ell}  \}.
\]
Next assume that the graph $\Delta$ remains connected after any vertex
$\ell\in V$ with $|I_{\ell}| = 1$ is removed.
Under these assumptions, define edges
\[
L(\{I_{\ell}\})
:=
\{t\in Y ~|~ t_{i}(\ell) = t_{h}(\ell), t_{i}(\ell) = t_{g}(m)
\mbox{\rm ~for~} \ell, m\in V;
i, h\in
I_{\ell}; g\in I_{m}\}.
\]

\begin{proposition}
\label{proposition3}

(1)
$
L(\{I_{\ell}\}, j)
$,
$
L(\{I_{\ell}\})
$
are dense.

(2)
Every dense edge has the form above.

\end{proposition}

{\it Proof.~}
For any graph $G$ with vertices $\{1,\ldots, m\}$
and edges $E$, associate a central arrangement
${\cal A}_{G} $ in ${{\bf {\rm C}}\hspace{-.4em^{m} $ consisting of
$\{x_{i} = 0 (1\leq i\leq m)\}$
and
$\{x_{i} = x_{j} | \{i, j\} \in E\}$.
Define a central arrangement ${\cal B}_{G} $ consisting of
$\{x_{i} = x_{j} | \{i, j\} \in E\}$.
(The arrangement ${\cal B}_{G} $ is called a graphic arrangement
\cite[2.4]{OrT}.)
In order to prove (1) and (2), it is enough to show
 the following lemma;

\begin{lemma}
\label{lemma1}

(a)
${\cal A}_{G} $ is not decomposable iff $G$ is connected,

(b)
${\cal B}_{G} $ is not decomposable iff $G$ is $2$--connected,
that is, $G$ remains connected after any vertex is removed.
\end{lemma}

{\it Proof.~}
(a):  If $G$ is disconnected, ${\cal A}_{G} $ is obviously decomposable.
If $G$ is connected, let $T$ be a maximal tree inside $G$.
Choose an edge $\{i, j\}$ such that $j$ is a terminal point of $T$.
Let ${\cal A}'$ and ${\cal A}''$ be the deletion and the restriction
of ${\cal A}_{T}$ with respect to the hyperplane $\{x_{i} = x_{j} \}.$
Since $\beta({\cal A}') + \beta({\cal A}'') = \beta({\cal A}_{T})$
\cite[Theorem1]{Cra}, we can prove $\beta({\cal A}_{T} ) = 1$
for any tree by induction on the number of edges.
This shows $\beta({\cal A}_{G} ) \geq \beta({\cal A}_{T} ) = 1.$

(b): Note that the matroid associated with the arrangement ${\cal B}_{G} $
is the same as the matroid associated with the graph $G$.
The matroid is connected if and only if $G$ is $2$--connected
\cite{Tut}.
~~\mbox{$\Box$}

Let ${\cn}^{n} = Y - \bigcup_{(i, \ell)\in X} H_{(i, \ell), k+1}. $
Let $U$ be the complement in $Y$ to the union of ``hyperplanes''
of ${\cal A}$.
Recall $f_{i}, \omega_{i}, P_{i}, \omega, {\cal E},$
and $A^{\bullet} $ from Section 1.
$\omega$ can be uniquely extended to be an $\mbox{\rm End\,} {{\bf {\rm C}}\hspace{-.4em^{s}$--coefficient
rational $1$-form
$
\overline{\omega}
$ on $Y$.
For $(i, \ell)\in X$ the residue of
$
\overline{\omega}
$ at $
H_{(i, \ell), k+1} $ is
\[
P_{(i, \ell), k+1}
=
-
\sum_{j=1}^{k}
P_{(i, \ell), j}
-
\sum_{\scriptstyle j=1 \atop
\scriptstyle j \neq i}
^{n_{\ell} }
P_{(j, \ell), (i, \ell)}
-
\sum
P_{(i, \ell), (j, m)}
\]
where the last sum is over all $m$ such that
$v_{\ell} $ and
$v_{m} $ are joined by an edge in $\Gamma$ and $j=1,\ldots,n_{m}. $

For any edge $L$ in ${\cal A}$, let $P_{L} $ be the sum of residues
of
$
\overline{\omega}
$
at all ``hyperplanes'' of ${\cal A}$ contaning $L.$

\begin{theorem}
\label{theorem5}
Let ${\cal L}$ be the set
of dense
edges of ${\cal A}$.
Suppose that

\smallskip
\noindent
{\rm {\bf (Mon)* :}}  for all $L\in {\cal L}$, none of the eigenvalues of
$P_{L}$ lies in ${\rm I}\hspace{-.2em}{\rm N} - \{0\}$.
\smallskip

\noindent
Then the inclusion
\[
A^{\bullet} \hookrightarrow \Gamma (U, \Omega_{U}^{\bullet}({\cal E}))
\]
is quasiisomorphism. ~~\mbox{$\Box$}
\end{theorem}

\begin{corollary}
\label{corollary3}
Suppose
that

\smallskip
\noindent
{\rm {\bf (Mon)** :}}  for all $L\in {\cal L}$, none of the eigenvalues of
$P_{L} $ lies in ${\rm I}\hspace{-.2em}{\rm N} \cup \{0\}$.
\smallskip

\noindent
Also suppose that $P_{i} P_{j} = P_{j} P_{i}$ for all $i, j$.
Then
\[
H^{p}(U, {\cal S}) = 0~~~\mbox{\rm for~~} p \neq n.
~~\mbox{$\Box$}
\]
\end{corollary}

\section{Kac-Kazhdan conditions}
Let ${\cal G}$
be a finite dimensional simple complex Lie algebra with Chevalley
generators $e_{i}, f_{i}, h_{i},\ i=1,\ldots, r.   $
Let $\cal G = \cal N_{-} \oplus \cal H \oplus \cal N_{+} $ be the corresponding
Cartan decomposition;
$\alpha_{1}, \ldots, \alpha_{r}\in \cal H^{*}   $ the simple roots,
$\theta$ the highest root.  Let $(~,~)$ be the symmetric
non-degenerate bilinear  form on $\cal G$ such that $(\theta, \theta) = 2$.

Let $T$ be an independent variable,  ${{\bf {\rm C}}\hspace{-.4em[T]$ the ring of polynomials,
${{\bf {\rm C}}\hspace{-.4em[T,T^{-1}]$ the ring of Laurent polynomials.
For $f(T), g(T) \in{{\bf {\rm C}}\hspace{-.4em[T,T^{-1}]$, set
\[
\mbox{\rm res}_{0}(f(T)dg(T))  = \mbox{\rm coefficient~at~} T^{-1} \mbox{\rm ~in~}
f(T)g'(T).
\]
The space
$\cal G\otimes_{\bf C} {{\bf {\rm C}}\hspace{-.4em[\it{T,T^{-1}}]$
is a Lie algebra with bracket
\[
[b\otimes f(T), c\otimes g(T)]
=
[b, c] \otimes f(T)g(T)
\]
for $b, c\in\cal G$.
Define $\hat{\cal G}$ as a central extension of
$
\cal G\otimes_{\bf C} {{\bf {\rm C}}\hspace{-.4em[\it{T,T^{-1}}],$
\[
\hat{\cal G} = \cal G \otimes {{\bf {\rm C}}\hspace{-.4em[\it{T,T^{-1}}] \oplus {{\bf {\rm C}}\hspace{-.4em {\it K},
\]
where $K$ is a central element of $\hat{\cal G},$
and
\[
[b\otimes f(T), c\otimes g(T)]
=
[b, c]\otimes f(T)g(T)
+
(b, c) \mbox{\rm res}_{0} (f(T)dg(T))K.
\]
Set $\hat{\DG}^{+} =\cal G \otimes {{\bf {\rm C}}\hspace{-.4em[\it T]\oplus {{\bf {\rm C}}\hspace{-.4em {\it K}$; it is a Lie
subalgebra of $\hat{\DG}$.

Fix a complex number $k$. Set $\kappa=k+g$ where $g$ is the dual Coxeter
number of $\cal G$, cf. \cite{K}, 6.1.

For $\Lambda \in
\cal H^{*}, $ let $M(\Lambda)$ be the Verma module over $\cal G$
with highest weight $\Lambda$.
Consider $M(\Lambda)$ as a $\hat{\DG}^{+}$-module by setting
$
\cal G \otimes \it{T}{{\bf {\rm C}}\hspace{-.4em[\it T]
$
to act as zero and $K$ as multiplication by $k$.
Set
\[
\hat{M}(\Lambda) := U(\hat{\DG}) \otimes_{U(\hat{\DG}^{+})} M(\Lambda).
\]
It is a Verma module over $\hat{\DG}$.

\begin{proposition}
\label{proposition4}
(Kac-Kazhdan conditions)
$\hat{M}(\Lambda)
$
is reducible if and only if
at least one of the following
three conditions is satisfied.

(1) $\kappa = 0$.

(2) There exist a positive root $\alpha$ of $\cal G$
and natural numbers $p, s \in {\rm I}\hspace{-.2em}{\rm N} - \{0\}$
such that
\[
(\Lambda, \alpha) + (\rho, \alpha)
=
p \frac{(\alpha, \alpha)}{2} - (s-1) \kappa,
\]
where $\rho$ is half-sum of positive roots
of $\cal G$.

(3) There exist a positive root $\alpha$ of $\cal G$
and natural numbers $p, s \in {\rm I}\hspace{-.2em}{\rm N} - \{0\}$
such that
\[
(\Lambda, \alpha) + (\rho, \alpha)
=
- p \frac{(\alpha, \alpha)}{2} + s \kappa.
\]
\end{proposition}

{\it Proof.~}
We use notations of \cite{K}, Ch. 6,7. In these notations the Kac-Kazhdan
reducibility condition, \cite{KK}, Thm 1, reads as
$$
\left<\Lambda,\nu^{-1}(\beta)\right>+ \left<\hat{\rho},\nu^{-1}(\beta)\right>
-
p\frac{(\beta,\beta)}{2}=0
$$
for some positive root $\beta$ of $\hat{\DG}$ and a positive integer $p$.
(Here we denoted by $\hat{\rho}$ an element denoted by $\rho$ in \cite{K},
to distinguish it from our $\rho$.)

By {\em loc. cit.}, 6.3, every
such $\beta$ has one of the following forms:
(1) $\beta=m\delta,\ m>0$; (2) $\beta=\alpha+m\delta,\ m\geq 0$;
(3) $\beta=-\alpha +m\delta,\ m>0$, where $\alpha$ is a positive root
of $\cal G$, $m$ an integer. From {\em loc. cit} it follows easily that
$\left<\Lambda,\nu^{-1}(\delta)\right>
=k$, $\left<\hat{\rho},\nu^{-1}(\delta)\right>=g$ and
$\left<\hat{\rho},\nu^{-1}(\alpha)\right>=(\rho,\alpha)$.
This implies the proposition.
~~\mbox{$\Box$}
\medskip

Let $w$ be the longest element of the
Weyl group of $\cal G$.
For $\Lambda\in\cal H^{*}, $
set $\Lambda' = -w(\Lambda)$.

\begin{proposition}
\label{proposition5}
$\hat{M}(\Lambda')$ is reducible if and only if
$\hat{M}(\Lambda) $
is reducible.  The Kac-Kazhdan conditions
for $\Lambda'$ expressed in terms of $\Lambda$
coincide with the Kac-Kazhdan conditions
for $\Lambda$.
\end{proposition}

{\it Proof.~}
For a positive root $\alpha$, $-w(\alpha)$ is a positive root.
This implies the proposition.
~~\mbox{$\Box$}

\section{Resonances of discriminantal arrangements}

Let $\Gamma$ be the Dynkin diagram of a complex simple
Lie algebra $\cal G.$
The vertices of the diagram are labelled by simple roots
$\alpha_{1},\ldots, \alpha_{r}  $ of the algebra.  Let $n_{1},\ldots, n_{r}  $
be nonnegative integers, $n = n_{1} +\cdots + n_{r}. $
For pairwise distinct $z_{1},\ldots, z_{k}\in {{\bf {\rm C}}\hspace{-.4em, z_{k+1}=\infty, $
consider in $Y = ({\bf P}^{1})^{n} $ the discriminantal arrangement
${\cal A}$ associated to these data.

Let $\Lambda_{1},\ldots, \Lambda_{k} \in \cal H^{*}. $
Set
$
\Lambda_{k+1} =
-\omega (\Lambda_{1} +\cdots +\Lambda_{k} - n_{1} \alpha_{1} -\cdots - n_{r}\alpha_{r}).  $
Fix a nonzero complex number $\kappa$.
Introduce an integrable connection $d + \omega$ on the trivial bundle
${\cal E} := {\cal O}_{U} $ with
\[
\omega = \sum_{(i,\ell)\in X}\sum_{j=1}^{k}
P_{(i, \ell), j} \omega_{(i, \ell), j}
+  \sum_{\ell=1}^{r} \sum_{1 \leq i < j \leq n_{\ell}}
P_{(i, \ell), (j, \ell)} \omega_{(i, \ell), (j, \ell)}
 +  \sum_{1\leq \ell < m \leq r} \sum_{i=1}^{n_{\ell}}
\sum_{j=1}^{n_{m}}
P_{(i, \ell), (j, m)} \omega_{(i, \ell), (j, m)},
\]
where
\[
\omega_{(i, \ell), j}
=
d(t_{i}(\ell) - z_{j})
/
(t_{i}(\ell) - z_{j}),
{}~~
\omega_{(i, \ell), (j, m)}
=
d(t_{i}(\ell) - t_{j}(m))
/
(t_{i}(\ell) - t_{j}(m)),
\]
\[
P_{(i, \ell), j} = -(\alpha_{\ell}, \Lambda_{j})/\kappa,
{}~~
P_{(i, \ell), (j, m)} = -(\alpha_{\ell}, \alpha_{m})/\kappa,
\]
see \cite{ScV} and \cite{Var}.
$\omega$ extends to be a rational $1$-form $\overline{\omega}$ on $Y$.

For any edge $L$ in ${\cal A}$, let $P_{L} $ be the sum of residues of
$\overline{\omega}$ at all ``hyperplanes''
of ${\cal A}$ containing $L$.
For $p\in{\rm I}\hspace{-.2em}{\rm N} \cup \{0\}$, we say that the connection $d+\omega$
has a {\bf resonance at} $L$ {\bf of level} $p$, if $P_{L} = p.$

The following theorem connects resonances of ${\cal A}$ with the Kac-Kazhdan
conditions for the Verma modules
$
\hat{M}(\Lambda_{1}),
\ldots,
\hat{M}(\Lambda_{k+1})
$
of the affine algebra $\hat{\DG}.$
Let $\alpha = \sum a_{\ell} \alpha_{\ell} $
be a positive root of $\cal G,$ $p$ a natural number.
Assume that $a_{\ell} p \leq n_{\ell} $ for all $\ell$.
For every $\ell$, fix a subset $I_{\ell} \subseteq
\{1,\ldots, n_{\ell} \}$ consisting of $a_{\ell} p$
elements.

\begin{theorem}
\label{theorem6}

(1) For every $j = 1,\ldots, k+1,$ the edge $L_{j}
= L(\{I_{\ell}\}, j)$ is dense.

(2) For $j = 1,\ldots, k$ and every natural number $s$,
the resonance condition at $L_{j} $ of level $ps$,
$P_{L_{j} } = ps,$ coincides with the Kac-Kazhdan
condition of type (2) for $\hat{M}(\Lambda_{j}),$
\[
(\Lambda_{j}, \alpha) + (\rho, \alpha)
=
p \frac{(\alpha, \alpha)}{2} - s\kappa.
\]
(3) For $j = k+1$ and every natural number $s$,
the resonance condition at $L_{k+1} $ of level $ps$,
$P_{L_{k+1} } = ps,$ coincides with the Kac-Kazhdan
condition of type (3) for $\hat{M}(\Lambda_{k+1}),$
\[
(\Lambda_{k+1}, \alpha) + (\rho, \alpha)
=
- p \frac{(\alpha, \alpha)}{2} + s\kappa.
\]
\end{theorem}

{\bf Remarks.}
(1) For resonance values of $\Lambda_{1},\ldots ,\Lambda_{k}, \kappa,$
nontrivial cohomological relations occur in the image of
$A^{\bullet} \subset \Gamma(U, \Omega_{U}({\cal E}))$.
The Theorem suggests that the relations correspond to singular
vectors in the Verma modules $\hat{M}(\Lambda_{1}), \ldots,
\hat{M}(\Lambda_{k+1})$.
In \cite{FSV}
this correspondence was established for the simplest singular
vector in $\hat{M}(\Lambda_{k+1}),$
the correspondence implied algebraic equations
satisfied by conformal blocks in the WZW model of conformal
field theory.

(2) For $j = 1,\ldots, k$ and natural number $p$, the Kac-Kazhdan
condition,
$
(\Lambda_{j}, \alpha)
+
(\rho, \alpha)
=
p
\frac{(\alpha, \alpha)}{2},
$
appears as a degeneration condition for a certain contravariant
form of the arrangement ${\cal A}$, see \cite[Secs. 3, 6]{ScV}.

\medskip

{\it Proof.~}
(1) For a positive root $\alpha = \sum a_{\ell} \alpha_{\ell} $
consider the subset $\{\alpha_{\ell}~|~a_{\ell} > 0\}$ of the set
of simple roots.  The subset distinguishes a subgraph of the Dynkin
diagram.  The subgraph is connected \cite[ch. 7, sec. 1]{Bou}.
Now $L_{j} $ is dense by Proposition \ref{proposition3}.

(2)
\begin{eqnarray*}
P_{L_{j}} - ps
&=& \frac{1}{\kappa}
[
(-\Lambda_{j}, \alpha) p + \sum_{r=1}^{r}
\frac{p a_{\ell} (p a_{\ell} - 1) }{2} (\alpha_{\ell}, \alpha_{\ell})
 +
\sum_{1\leq \ell < m \leq r} p a_{\ell} p a_{m} (\alpha_{\ell}, \alpha_{m})
]
- ps\\
&=&
\frac{p}{\kappa}
\left[
-(\Lambda_{j}, \alpha) + p \frac{(\alpha, \alpha)}{2} -
\sum_{\ell=1}^{r} a_{\ell} \frac{(\alpha_{\ell}, \alpha_{\ell})}{2}
-s \kappa
\right]\\
&=&
\frac{p}{\kappa}
\left[
-(\Lambda_{j}, \alpha) - (\rho, \alpha) + p
\frac{(\alpha_{\ell}, \alpha_{\ell})}{2}
-s \kappa
\right].
\end{eqnarray*}
This proves (2).  Part (3) is proved by similar direct computations
using Proposition \ref{proposition5}.
~~\mbox{$\Box$}

\medskip
{\small \bf Acknowledgement.}
{\small The authors thank H\'el\`ene Esnault
for pointing out an error in an earlier version.
The second author thanks G. Ziegler for directing
his attention to the work of H. Crapo about the beta invariant.
He is also thankful to A. Libgober and S. Yuzvinsky for useful
discussions.}


Localization of $\frak{u}$-modules. I. Intersection cohomology of real
  arrangements

26 pages, amslatex. A misprint corrected

Abstract:
  This paper is the first in a series. The main goal of the series is to
present a geometric construction of certain remarkable tensor categories
arising from quantum groups coresponding to the value of deformation parameter
$q$ equal to a root of unity.
  In the present paper we study perverse sheaves over a complex affine space
which are smooth along the stratification determined by a finite arrangement of
complex affine hyperplanes defined by real equations. In particular, we
construct explicitely (in terms of combinatorial data) complexes computing
cohomology of Goresky-MacPherson extensions of one-dimensional local systems
over the complement of hyperplanes.

\section{Introduction}

This paper is the first in a series. The main goal of the series
is to present a geometric construction of certain remarkable
tensor categories arising from quantum groups corresponding
to the value of deformation parameter $q$ equal to a root of unity,
cf. ~\cite{ajs}, these categories playing an important role in Lusztig program.

\subsection{} Let $\BA$ be a complex affine space, $\CH$ a finite collection
of hyperplanes in $\BA$. We suppose that all $H\in\CH$ are given by real
equations, and denote by $H_{\BR}\subset H$ the subspace of real points.
An arrangement $\CH$ induces naturally a stratification of
$\BA$ denoted by $\CS_{\CH}$ (for precise definitions see section 2 below).
The main goal of this paper is the study
of the category $\CM(\BA;\CS_{\CH})$ of perverse sheaves (of vector spaces
over a fixed ground field) over $\BA$ which are smooth along $\CS_{\CH}$.

In {\em section 1} we collect topological notations and known facts we will
need
in the sequel.

{\em Section 2.}
To each object $\CM\in\CM(\BA;\CS_{\CH})$ we assign "a linear algebra data".
Namely, for each {\em facet} of $\CH$ --- i.e. a connected component
of an intersection of some $H_{\BR}$'s --- we define a vector space
$\Phi_F(\CM)$; if $F$ is contained and has codimension $1$ in the closure of
another facet $E$, we define two linear operators between $\Phi_F(\CM)$ and
$\Phi_E(\CM)$ acting in opposite directions. These data contain all
information about $\CM$ we need.

In fact, it is natural to expect that the sheaf $\CM$ may be {\em uniquely
reconstructed} from the linear algebra data above (one can check this in
the simplest cases).

The spaces $\Phi_F(\CM)$ are defined using certain relative cohomology.
They are similar to a construction by R. MacPherson (see ~\cite{b2}).
The spaces $\Phi_F(\CM)$ are analogous to functors of vanishing cycles.

The main technical properties of functors $\CM\mapsto\Phi_F(\CM)$ are
(a) {\em commutation with Verdier duality}, and (b) {\em exactness}.
In fact, (b) follows from (a) without difficulty; the proof of (a)
is the principal geometrical result of this paper (see Theorem ~\ref{dual}).
Actually, we prove a more general statement concerning all complexes
smooth along $\CS_{\CH}$. This is a result of primary importance for us.

Using these linear algebra data, we define an exact functor from
$\CM(\BA;\CS_{\CH})$ to complexes of finite dimensional vector spaces
computing cohomology $R\Gamma(\BA;\bullet)$, see Theorem ~\ref{rgamma}.
This is the main result we will need below. The idea of constructing such
functorial complexes on categories of perverse sheaves
(quite analogous to the usual computation of singular cohomology using
cell decompositions) is due to A. Beilinson,
{}~\cite{b1}. It was an important source of inspiration for us.

A similar problem was considered in ~\cite{c}.

In {\em section 3} we present explicit computations for standard extensions
of an arbitrary one-dimensional local system over an open stratum.
The main result is the computation of {\em intersection cohomology}, see
Theorem ~\ref{inters}. In our computations we use some simple geometric
ideas from M. Salvetti's work, ~\cite{sa}. However, we do not use a more
difficult  main theorem of this paper. On the contrary, maybe our
considerations shed some new light on it.

\subsection{} The idea of this work appeared several years ago,
in some discussions with H\'{e}l\`{e}ne Esnault and Eckart Viehweg.
We use this occasion to express to them our sincere gratitude. We are also
very grateful to Paul Bressler for a stimulating discussion.
We thank Kari Vilonen for pointing out an error in an earlier version of the
manuscript.
We are greately
indebted to D. Kazhdan for his constant encouragement and attention to the
work.

\section{Topological preliminaries}

In this section we introduce our notations and recall some basic
facts from topology. Main references are ~\cite{ks}, \cite{bbd}.

\subsection{}
\label{derived} Throughout the paper, all our topological
spaces will be {\em locally compact}, in particular {\em Hausdorff}.
$\{ pt\}$ will denote a one-point space. For a topological space $X$,
$a_X:X\lra \{pt\}$ will be the unique map. If $Y\subset X$ is a subspace,
$\bar{Y}$ will denote the closure of $Y$.

Throughout this chapter we fix an arbitrary ground field $B$. {\em A vector
space} will mean a vector space over $B$. For a finite dimensional
vector space $V$, $V^*$ will denote a dual space.
$\Vect$ will denote a category
of vector spaces.

{\em A sheaf} will mean a sheaf of vector spaces. For a topological
space $X$, $\Sh(X)$ will denote a category of sheaves over $X$,
$\CD^*(X)$ will denote the derived category of $\Sh(X)$,
$*=+,-,b$ or $\emp$ will have the usual meaning.

For $p\in\Bbb Z$, $\CD^{\leq p}(X)\subset \CD(X)$ will denote
a full subcategory of complexes $\CK$ with $H^i(\CK)=0$ for all
$i>p$.

If $\CA,\CB$ are abelian categories, we will say that a left exact functor
$F:\CD^*(\CA)\lra\CD(\CB)$ ($*=+,-,b$ or $\emp$)
has {\em cohomological dimension $\leq r$}, and write
$\cd (F)\leq r$, if $H^i(F(A))=0$ for any $A\in\CA$ and $i>r$. (Left exactness
here means that $H^i(F(A))=0$ for $i<0$).

\subsection{} Let $f:X\lra Y$ be a continuous map. We will use the following
notations for standard functors.

$f^*:\CD(Y)\lra\CD(X)$ --- the inverse image;
$f_*:\CD^+(X)\lra\CD^+(Y)$ --- (the right derived of) the direct image;
$f_!:\CD^+(X)\lra\CD^+(Y)$ --- (the right derived of) the direct image
with proper supports;
$f^!:\CD^+(Y)\lra\CD^+(X)$ --- the right adjoint to $f_!$ (defined
when $f_!$ has finite cohomological dimension), see ~\cite{ks}, 3.1.

We will denote the corresponding functors on sheaves as follows:
$f^*:\Sh(Y)\lra\Sh(X);\ R^0f_*:\Sh(X)\lra\Sh(Y);\
R^0f_!:\Sh(X)\lra\Sh(Y)$.

We will denote $R\Gamma(X;\cdot):=p_{X*};\  R\Gamma_c(X;\cdot):=p_{X!}$.
For $\CK\in\CD^+(X),\ i\in\BZ$, we set
$H^i(X;\CK):=H^i(R\Gamma(X;\CK));\  H^i_c(X;\CK):=H^i(R\Gamma_c(X;\CK))$.

For $V\in\Vect=\Sh(\{ pt\})$ we denote by $V_X$ the constant sheaf $p^*_XV$.

If $Y$ is a subspace of $X$, we will use a notation $i_{Y,X}$,
or simply $i_Y$ if $X$ is understood, for the
embedding $Y\hra X$. If $Y$ is open in $X$, we will also write $j_Y$
instead of $i_Y$.

For $\CK\in\CD(X)$, we will use notations $\CK|_Y:=i^*_{Y,X}\CK$;
$R\Gamma(Y;\CK):=R\Gamma(Y;\CK|_Y),\ H^i(Y;\CK):=H^i(Y;\CK_Y)$,
etc.

\subsection{} We have functors
\begin{equation}
R\CHom :\CD^-(X)^{opp}\times\CD^+(X)\lra\CD^+(X),
\end{equation}
\begin{equation}
\otimes:\CD^-(X)\times\CD^*(X)\lra\CD^*(X)
\end{equation}
where $*=,-,b$ or $\emp$;
\begin{equation}
R\Hom=R\Gamma\circ R\CHom:\CD^-(X)^{opp}\times\CD^+(X)\lra\CD^+(\Vect)
\end{equation}
For $\CK,\CL\in\CD^b(X)$ we have
\begin{equation}
\Hom_{\CD(X)}(\CK,\CL)=H^0(R\Hom(\CK,\CL))
\end{equation}

We denote $\CK^*:=R\CHom(\CK,B_X)$.

\subsection{} Let $X$ be a topological space, $j:U\lra X$ an embedding
of an open subspace, $i: Y\lra X$ an embedding of the complement.
In this case $i_*=i_!$ and $j^!=j^*$. $R^0i_*$ and $R^0j_!$ are
exact, so we omit $R^0$ from their notation.
$j_!$ is the functor of extension by zero.
$i^!$ is the right derived of the functor $R^0i^!:\Sh(X)\lra \Sh(Y)$
of the subsheaf of sections with supports in $Y$ (in notations of ~\cite{ks},
$R^0i^!(F)=\Gamma_Y(F)$).

Let $\CK\in\CD^+(X)$. We will use the following notations for relative
cohomology: $R\Gamma(X,Y;\CK):=R\Gamma(X;j_!\CK|_U);\
H^i(X,Y;\CK):=H^i(R\Gamma(X,Y;\CK))$.

If $Z\hra Y$ is a closed subspace,
we have a canonical exact triangle
\begin{equation}
i_{X-Y!}\CK|_{X-Y}\lra i_{X-Z!}\CK|_{X-Z}
\lra i_{Y-Z!}\CK|_{Y-Z} \lra i_{X-Y!}\CK|_{X-Y}[1]
\end{equation}
(of course, $i_{X-Y}=j$)
inducing a long cohomology sequence
\begin{equation}
\label{cohom}
\ldots\lra H^i(X,Y;\CK)\lra H^i(X,Z;\CK)\lra H^i(Y,Z;\CK)
\overset{\dpar}{\lra} H^{i+1}(X,Y;\CK)\lra\ldots
\end{equation}

\subsection{} Let $X$ be a topological space such that the functor
$R\Gamma_c(X;\cdot)$ has a finite cohomological dimension.
We define {\em a dualizing complex}:
\begin{equation}
\omega_X:=a_X^!B\in\CD^b(X)
\end{equation}
For $\CK\in\CD^b(X)$ we set $D_X\CK=R\CHom(\CK,\omega_X)\in\CD^b(X)$.
If there is no risk of confusion we denote $D\CK$ instead of $D_X\CK$.
We get a functor
\begin{equation}
D:\CD^b(X)^{opp}\lra\CD^b(X)
\end{equation}
which comes together with a natural transformation
\begin{equation}
\Id_{\CD^b(X)}\lra D\circ D
\end{equation}

\subsection{Orientations} (Cf. ~\cite{ks}, 3.3.) Let $X$ be an $n$-dimensional
$C^0$-manifold.
We define an {\em orientation sheaf} $\Or_X$ as the sheaf associated to
a presheaf $U\mapsto \Hom_{\BZ}(H^n_c(U;\BZ),\BZ)$. It is a locally constant
sheaf of abelian groups of rank $1$. Its tensor square is constant.
By definition, {\em orientation of $X$} is an isomorphism
$\Or_X\iso\BZ_X$.

We have a canonical isomorphism
\begin{equation}
\omega_X\iso\Or_X\otimes_{\BZ}B[n]
\end{equation}

\subsection{Homology} Sometimes it is quite convenient to use
homological notations. Let $Y\subset X$ be a closed subspace of
a topological space $X$ ($Y$ may be empty), $\CK\in\CD^b(X)$.
We define homology groups as
$$
H_i(X,Y;\CF):=H^i(X,Y;\CF^*)^*.
$$
These groups behave covariantly with respect to continuous mappings.

Let $\sigma$ be {\em a relative singular $n$-cell}, i.e. a continuous
mapping
$$
\sigma: (D^n,\dpar D^n)\lra (X,Y)
$$
where, $D^n$ denotes a standard closed unit ball in $\BR^n$.
We supply $D^n$ with the standard orientation. Let $\DO^n$ denote the interior
of $D^n$.

Suppose that $\sigma^*\CK|_{\DO^n}$ is constant. Then by Poincar\'{e} duality
we have isomorphisms
$$
H_n(D^n,\dpar D^n;\sigma^*\CK)=H^n_c(\DO^n;\sigma^*\CK^*)^*\iso
H^0(\DO^n;\sigma^*\CK).
$$
(recall that $\DO^n$ is oriented).
Thus, given an element $s\in H^0(\DO^n;\CK)$, we can define a homology class
$$
cl(\sigma;s):=\sigma_*(s)\in H_n(X,Y;\CK).
$$
We will call the couple $(\sigma;s)$
{\em a relative singular $n$-cell for $\CK$}.

These classes enjoy the following properties.

\subsubsection{Homotopy} Let us call to cells $(\sigma_0;s_0)$
and $(\sigma_1,s_1)$ {\em homotopic} if there exists a map
$$
\sigma: (D^n\times I,\dpar D^n\times I)\lra (X,Y)
$$
(where $I$ denotes a unit interval) such that $\sigma^*\CK|_{\DO^n\times I}$
is constant,  and a section
$$
s\in H^0(\DO^n\times I;\sigma^*\CK)
$$
such that $(\sigma,s)$ restricted to $D^n\times\{ i\}$
is equal to $(\sigma_i;s_i),\ i=0,1$.

We have

(H) {\em if $(\sigma_0;s_0)$ is homotopic to $(\sigma_1;s_1)$ then
$cl(\sigma_0;s_0)=cl(\sigma_1;s_1)$.}

\subsubsection{Additivity} Suppose $D^n$ is represnted as a union of its
upper and lower half-balls $D^n=D^n_1\cup D^n_2$ where
$D^n_1=\{ (x_1,\ldots, x_n)\in D^n|\ x_n\geq 0\}$ and
$D^n_2=\{ (x_1,\ldots, x_n)\in D^n|\ x_n\leq 0\}$. Let us supply $D^n_i$
with the induced orientation.

Suppose we are given a relative $n$-cell $(\sigma;s)$ such that
$\sigma(D^n_1\cap D^n_2)\subset Y$.
Then its restriction to $D^n_i$ gives us two relative $n$-cells
$(\sigma_i;s_i)$,
$i=1,2$. We have

(A) $cl(\sigma;s)=cl(\sigma_1;s_1)+cl(\sigma_2;s_2)$.

\subsection{} We will call {\em a local system} a locally constant sheaf
of finite rank.

Let $X$ be a topological space, $i:Y\hra X$ a subspace. We will say that
$\CK\in\CD(X)$ is {\em smooth} along $Y$ if all cohomology sheaves
$H^p(i^*\CK),\ p\in\BZ,$ are local systems.

We will call a {\em stratification} of $X$ a partition $X=\bigcup_{S\in\CS}S$
into a finite disjoint of locally closed subspaces ---
{\em strata} --- such that the closure of each stratum is the union
of strata.

We say that $\CK\in\CD(X)$ is {\em smooth along $\CS$} if it is smooth
along each stratum. We will denote by $\CD^*(X;\CS)$ the full subcategory
of $\CD^*(X)$ ($*=+,-,b$ or $\emp$) consisting of complexes smooth along $\CS$.

\subsection{} Let us call a stratification $\CS$ of a topological space
$X$ {\em good} if the following conditions from ~\cite{bbd}, 2.1.13
hold.

(b) All strata are equidimensional topological varieties. If a stratum
$S$ lies in the closure of a stratum $T$, $\dim S<\dim T$.

(c) If $i:S\hra X$ is a stratum, the functor $i_*:\CD^+(S)\lra \CD^+(X)$
has a finite cohomological dimension. If $\CF\in\Sh(S)$ is a local system,
$i_*(\CF)$ is smooth along $\CS$.

Let $\CS$ be a good stratification such that all strata have even dimension.
We will denote by $\CM(X;\CS)\subset\CD^b(X;\CS)$ the category of
smooth along $\CS$
perverse sheaves corresponding to the middle perversity $p(S)=-\dim\ S/2$, cf.
{\em loc. cit.}

\subsection{} Let $X$ be a complex algebraic variety. Let us call
its stratification $\CS$ {\em algebraic} if all strata are algebraic
subvarieties. Following ~\cite{bbd}, 2.2.1, let us call a sheaf
$\CF\in\Sh(X)$ {\em constructible} if it is smooth along some algebraic
stratification. (According to Verdier, every algebraic stratification
admits a good refinement.) We denote by $\CD^b_c(X)$ a full subcategory
of $\CD^b(X)$ consisting of complexes with constructible cohomology.

We denote by $\CM(X)\subset\CD^b_c(X)$ the category of perverse sheaves
corresponding to the middle perversity. It is a filtered union of categories
$\CM(X;\CS)$ over all good algebraic stratifications $\CS$, cf. {\em loc.cit.}

\subsection{} Let $X$ be a complex manifold, $f:X\lra \BC$ a holomorphic map.
Set $Y=f^{-1}(0),\ U=X-Y$, let $i:Y\hra X,\ j:U\hra X$ denote the
embeddings.

We define a functor of {\em nearby cycles}
\begin{equation}
\label{near}
\Psi_f:\CD^b(U)\lra\CD^b(Y)
\end{equation}
as $\Psi_f(\CK)=\psi_f j_*(\CK)[-1]$ where $\psi_f$ is defined in ~\cite{ks},
8.6.1.

We define a functor of {\em vanishing cycles}
\begin{equation}
\label{vanish}
\Phi_f:\CD^b(X)\lra\CD^b(Y)
\end{equation}
as $\phi_f$ from {\em loc. cit.}

\subsection{}
\label{limit}
\begin{lem}{} Let $X$ be topological space, $Y\hra X$ a closed subspace,
$\CF\in\Sh(X)$. Then natural maps
$$
\dirlim\ H^i(U;\CF)\lra H^i(Y;\CF),\ i\in\BZ,
$$
where $U$ ranges through all the open neighbourhoods of $U$, are
isomorphisms.
\end{lem}

{\bf Proof.} See ~\cite{ks}, 2.5.1, 2.6.9. $\Box$

\subsection{Conic sheaves}
\label{conic} (Cf. ~\cite{ks}, 3.7.) Let $\BR^{*+}$ denote
the multiplicative group of positive real numbers. Let $X$ be a topological
space endowed with an $\BR^{*+}$-action.

Following {\em loc.cit.}, we will call a sheaf $\CF$ over $X$ {\em conic}
(with respect to the given $\BR^{*+}$-action)
if its restriction to every $\BR^{*+}$-orbit is constant.
We will call a complex $\CK\in\CD(X)$ {\em conic} if all its cohomology
sheaves are conic.

We will denote by $\Sh_{\BR^{*+}}(X)\subset \Sh(X)$,
$\CD^*_{\BR^{*+}}(X)\subset\CD^*(X),\ *=b,+,-$ or $\emp$,
the full subcategories of conic objects.

\subsubsection{}
\label{conicl}
\begin{lem}{} Let $U\hra X$ be an open subset. Suppose that for every
$\BR^{*+}$-orbit $O\subset X$, $O\cap U$ is contractible (hence, non-empty).
Then for every conic $\CK\in\CD^+(X)$ the restriction morphism
$$
R\Gamma(X;\CK)\lra R\Gamma(U;\CK)
$$
is an isomorphism.
\end{lem}

{\bf Proof.} See {\em loc. cit.}, 3.7.3. $\Box$

\section{Vanishing cycles functors}
\label{vanishcycl}

\subsection{Arrangements} Below we use some terminology from ~\cite{sa}.
Let $\BA_{\BR}$ be a
real affine space $\BA_{\BR}$ of dimension $N$, and $\CH_{\BR}=\{ H_{\BR}\}$
a finite set of distinct real affine hyperplanes in $\BA_{\BR}$. Such a set
is called a {\em real arrangement}. We pick once and for all a square root
$i=\sqrt{-1}\in\BC$.

Let $\BA=\BA_{\BR}\otimes_{\BR}\BC$ denote the complexification of
$\BA_{\BR}$, and $\CH=\{ H\}$ where $H:=H_{\BR}\otimes_{\BR}\BC$.
(A finite set of complex hyperplanes in a complex affine space
will be called a {\em complex arrangement}).

We will say that $\CH$ is {\em central} if $\bigcap_{H\in\CH}H$ consists
of one point.

For a subset $\CK\subset \CH$ denote
$$
H_{\CK}=\bigcap_{H\in \CK}H;\ _{\CK}H=\bigcup _{H\in \CK}H,
$$
and
$$
H_{\BR,\CK}=\bigcap_{H\in \CK}H_{\BR},
\ _{\CK}H_{\BR}=\bigcup_{H\in \CK}H_{\BR}.
$$
The nonempty subspaces $H_{\CK}$ and $H_{\CK,\BR}$ are called complex
and real {\em edges} respectively. Set
$$
\HO_{\CK}=H_{\CK}-\cup L,
$$
the union over all the complex edges $L\subset H_{\CK},\ L\neq H_{\CK}$. We set
$\HO_{\BR,\CK}=H_{\BR,\CK}\cap \BA_{\BR}$.
Connected components of $\HO_{\BR,\CK}$ are called {\em facets} of $\CH_{\BR}$.
Facets of codimension $0$ (resp., $1$) are called {\em chambers}
(resp., {\em faces}).

Let us denote by $\CS_{\CH}$ a stratification of $\BA$ whose strata are
all non-empty $\HO_{\CK}$. We will denote by $\BAO_{\CH}$ a unique open
stratum
$$
\BAO_{\CH}=\HO_{\emp}=\BA- _{\CH}H.
$$

In this Section we will study categories of sheaves $\CD(\BA;\CS_{\CH})$
and $\CM(\BA;\CS_{\CH})$.

We will denote by $\CS_{\CH_{\BR}}$ a stratification of $\BA_{\BR}$ whose
strata are all facets. We set
$$
\BAO_{\CH,\BR}=\BA_{\BR}- _{\CH}H_{\BR}.
$$
It is a union of all chambers.

\subsection{Dual cells}
\label{salv}(cf. ~\cite{sa}).
Let us fix a point $^Fw$ on each facet $F$. We will call this set of points
$\bw =\{^Fw\}$ {\em marking} of our arrangement.

For two facets $F,\ E$ let us write $F<E$ if $F\subset\BE$ and
$\dim\ F<\dim\ E$.
We will say that $E$ is {\em adjacent} to $F$. We will denote by $\Ch (F)$
the set of all chambers adjacent to $F$.

Let us call a {\em flag} a sequence
of $q-p+1$ facets $\bF=(F_p<F_{p+1}<\ldots <F_q)$ with $\dim\ F_i=i$.
We say that $F_p$ is {\em the beginning} and $F_q$ {\em the end} of $\bF$.

Let us denote by $^{\bF}\Delta$ a closed $(q-p)$-symplex with vertices
$^{F_p}w,\ldots,^{F_q}w$. Evidently, $^{\bF}\Delta\subset\BF_q$.

Suppose we are given two facets $F_p<F_q$, $\dim F_i=i$. We will denote
$$
D_{F_p<F_q}=\bigcup\ ^{\bF}\Delta,
$$
the sum over all flags beginning at $F_p$ and ending at $F_q$. This is a
$(q-p)$-dimensional cell contained in $\BF_q$.

For a facet $F$ let us denote
$$
D_F=\bigcup_{C\in \Ch (F)}D_{F<C}
$$
We set
$$
S_F:=\bigcup_{E,C: F<E<C;}D_{E<C},
$$
the union over all facets $E$ and chambers $C$. The space $S_F$ is contained
in $D_F$ (in fact, in the defintion of $S_F$ it is enough to take the union
over all facets $E$ such that
$\dim\ E=\dim\ F+1$). If $q=\codim\ F$, then
$D_F$ is homeomorphic to a $q$-dimensional disc and $S_F$ --- to a $(q-1)$-
dimensional sphere, cf. ~\cite{sa}, Lemma 6. We denote $\DO_F:=D_F-S_F$.
We will call $D_F$ {\em a dual cell corresponding to $F$}.

We set $\DO_{F<C}:=D_{F<C}\cap \DO_F$.

\subsection{Generalized vanishing cycles} Let $\CK\in\CD^b(\BA;\CS_{\CH})$,
$F$ a $p$-dimensional facet. Let us introduce a complex
\begin{equation}
\label{vc}
\Phi_F(\CK):=R\Gamma(D_F,S_F;\CK)[-p]\in\CD^b(pt)
\end{equation}
This complex will be called {\em a complex of generalized vanishing
cycles of $\CK$ at a facet $F$}.

Formally, the definition of functor $\Phi_F$ depends
upon the choice of a marking $\bw$. However, functors defined using
two different markings are canonically isomorphic. This is evident.
Because of this, we omit markings from the notations.

\subsection{Transversal slices}
\label{slice} Let $F$ be a facet of dimension $p$ which
is a connected component of a real edge $M_{\BR}$ with the complexification
$M$.
Let us choose a real affine subspace $L_{\BR}$ of codimension $p$ transversal
to $F$ and passing through $^Fw$. Let $L$ be its complexification.

Let us consider a small disk $L_{\epsilon}\subset L$ with the centrum
at $^Fw=L\cap M$. We identify $L_{\epsilon}$ with an affine space
by dilatation. Our arrangement induces a central arrangement $\CH_L$
in $L_{\epsilon}$. Given  $\CK\in\CD^b(\BA;\CS_{\CH})$,
define $\CK_L:=i_{L_{\epsilon}}^*\CK[-p]$.

\subsubsection{} {\bf Lemma.} {\em We have a natural isomorphism
\begin{equation}
\label{slicedu}
D(\CK_L)\iso (D\CK)_L
\end{equation}}

{\bf Proof.} Consider an embedding of smooth complex manifolds
$i_L: L\lra \BA$. Let us consider the following complexes:
$\omega_{L/\BA}:=i^!_L B_{\BA}$ (cf. ~\cite{ks}, 3.1.16 (i)) and
$\Or_{L/\BA}:=\Or_{L}\otimes_{\BZ}\Or_{\BA}$.
We have canonical isomorphism
$$
\omega_{L/\BA}\iso\Or_{L/\BA}[-2p]\otimes B.
$$
The chosen orientation of $\BC$ enables us to identify
$\Or_{\BA}$ and $\Or_L$, and hence $\Or_{L/\BA}$ with constant sheaves;
consequently, we get an isomorphism
$$
\omega_{L/\BA}\iso B[-2p].
$$
The canonical map
\begin{equation}
\label{closed}
i^!_L\CK\lra i^*\CK\otimes\omega_{L/\BA}
\end{equation}
(cf. ~\cite{ks}, (3.1.6)) is an isomorphism since singularities of $\CK$
are transversal to $L$ (at least in the neighbourhood of $^Fw$).
Consequently we get an isomorphism $i^!_L\CK\iso i^*\CK_L[-2p]$.

Now we can compute:
$$
D(\CK_L)=D(i_L^*\CK[-p])\iso D(i^*_L\CK)[2p]\iso i^!_LD\CK[p]
\iso i^*_LD\CK[-p]=(D\CK)_L,
$$
QED. $\Box$

\subsubsection{} {\bf Lemma.} {\em We have a natural
isomorphism
\begin{equation}
\label{sliceiso}
\Phi_{F}(\CK)\cong\Phi_{\{^Fw\} }(\CK_L)
\end{equation}}

This follows directly from the definition of functors $\Phi$.

This remark is often useful for reducing the study of functors $\Phi_F$ to
the case of a central arrangement.

\subsubsection{} {\bf Lemma.} {If $\CM\in\CM(\BA;\CS_{\CH})$
then $\CM_L\in\CM(L_{\epsilon};\CS_{\CH_{L_{\epsilon}}})$.}

{\bf Proof} follows from transversality of $L_{\epsilon}$ to singularities
of $\CM$. $\Box$

\subsection{Duality}
\label{dual}
\begin{thm}{} Functor $\Phi_F$ commutes with Verdier duality.
More precisely, for every $\CK\in\CD^b(\BA,\CS_{\CH})$ there exists a natural
isomorphism
\begin{equation}
\label{duala}
\Phi_F(D\CK)\iso D\Phi_F(\CK)
\end{equation}
\end{thm}

This is the basic property of our functors.

\subsection{$1$-dimensional case}
\label{one-dim} To start with the proof, let us treat first the simplest case.
Consider an arrangement consisting of one point --- origin --- in
a one-dimensional
space $\BA$. It has one $0$-face $F$ and two $1$-faces $E_{\pm}$ --- real rays
$\BR_{>0}$ and $\BR_{<0}$. A marking consists of two points
$w_{\pm}\in E_{\pm}$ and $F$. We set $w_{\pm}=\pm 1$ (we pick a coordinate on
$\BA$).

For a positive
$r$ denote $\BA_{\leq r}:=\{ z\in\BA|\ |z|\leq r\},\ S_r:=\dpar\BA_{\leq r}$;
$\BA_{< r},\ \BA_{\geq r}$, etc. have an evident meaning. We also set
$\BA_{(r',r'')}:=\BA_{>r'}\cap \BA_{<r''}.$ A subscript $\BR$
will denote an intersection of these subsets with $\BA_{\BR}$.
Evidently, $D_F=\BA_{\leq 1,\BR},\ S_F=S_{1,\BR}$. Define $D_F^{opp}:=
\BA_{\geq 1,\BR},\ Y:=i\cdot D_F^{opp}$.

One sees easily (cf. {\em infra}, Lemma ~\ref{isom}) that one has isomorphisms
\begin{equation}
\label{isom-one}
\Phi_F(\CK)\cong R\Gamma(\BA,S_F;\CK)\iso R\Gamma(\BA,D_F^{opp};\CK).
\end{equation}

Let us choose real numbers $\epsilon, r', r''$ such that $0<\epsilon<r'<1<r''$.
Set $Y:=\epsilon i\cdot D_F^{opp}$. Denote $j:=j_{\BA-S_F}$.
We have natural isomorphisms
$$
D\Phi_{F}(\CK)\cong DR\Gamma(\BA,S_{F};\CK)
\cong R\Gamma_c(\BA; j_{*}j^*D\CK)
$$
(Poincar\'{e} duality)
$$
\cong R\Gamma(\BA,\BA_{\geq r''};j_{*}j^*D\CK)\cong
R\Gamma(\BA,Y\cup \BA_{\geq r''};j_{*}j^*D\CK)
$$
(homotopy).
Consider the restriction map
\begin{equation}
res: R\Gamma(\BA,Y\cup \BA_{\geq r''};j_*j^*D\CK)\lra
R\Gamma(\BA_{\leq r'},Y\cap \BA_{\leq r'};D\CK)
\end{equation}
We claim that $res$ is an isomorphism.
In fact, $\Cone (res)$ is isomorphic to
\begin{eqnarray}
R\Gamma(\BA,\BA_{\leq r'}\cup\BA_{\geq r''}\cup Y;j_*j^*D\CK)=
R\Gamma_c(\BA_{< r''},\BA_{\leq r'}\cup Y;j_*j^*D\CK)\cong\nonumber\\
\cong DR\Gamma(\BA_{< r''}-(\BA_{\leq r'}\cup Y);j_!j^*\CK)\nonumber
\end{eqnarray}
We have by definition
$$
R\Gamma(\BA_{< r''}-(\BA_{\leq r'}\cup Y);j_!j^*\CK)=
R\Gamma(\BA_{(r',r'')}-Y,S_F;\CK)
$$
On the other hand, evidently $\CK$ is smooth over $\BA_{(r',r'')}$, and we
have an evident retraction of $\BA_{(r',r'')}$ on $S_F$ (see Fig. 1).
Therefore, $R\Gamma(\BA_{(r',r'')}-Y,S_F;\CK)=0$ which proves the claim.

\begin{picture}(20,8)(-10,-4)

\put(0,0){\circle{0.2}}
\put(0,0){\oval(6,6)}
\put(-1.5,3.3){$S_{r''}$}

\put(-4,0){\line(1,0){8}}

\put(0,3){\line(0,-1){2}}
\put(0,-3){\line(0,1){2}}
\put(-0.4,1.8){$Y$}

\put(0,1){\circle*{0.2}}
\put(0,0.5){$i\epsilon w_+$}
\put(0,-1){\circle*{0.2}}

\put(2,0){\circle*{0.2}}
\put(2,-0.5){$w_+$}
\put(-2,0){\circle*{0.2}}
\put(-2,-0.5){$w_-$}

\put(0,0){\oval(3,3)}
\put(-1.5,1.6){$S_{r'}$}

\put(0,0){\oval(5,5)[tr]}
\put(0.5,2.5){\circle*{0.2}}
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\put(0.5,2.5){\vector(1,0){1}}

\put(-0.5,-4){Fig. 1}

\end{picture}

A clockwise rotation by $\pi/2$ induces an isomorphism
$$
R\Gamma(\BA_{\leq r'},Y\cap \BA_{\leq r'};D\CK)\cong
R\Gamma(\BA_{\leq r'},\epsilon\cdot D_F^{opp};D\CK),
$$
and the last complex is isomorphic to $\Phi_F(D\CK)$ by dilatation and
{}~(\ref{isom-one}).
This proves the theorem for $\Phi_F$. The statement for  functors
$\Phi_{E_{\pm}}$ is evident.

Let us return to the case of an arbitrary arrangement.

\subsection{}
\label{isom}
\begin{lem}{} Suppose that $\CH$ is central. Let $F$ be the unique $0$-
dimensional facet. The evident restriction maps induce canonical isomorphisms
$$
\Phi_{F}(\CK)\overset{(1)}{\cong} R\Gamma(\BA_{\BR},S_{F};\CK)
\overset{(2)}{\cong} R\Gamma(\BA,S_{F};\CK)
\overset{(3)}{\cong} R\Gamma(\BA,D_{F}^{opp};\CK)
\overset{(4)}{\cong} R\Gamma(\BA_{\BR},D_{F}^{opp};\CK)
$$
where $D_{F}^{opp}=\BA_{\BR}-\DO_{F}$.
\end{lem}

{\bf Proof.} Let us fix a coordinate system with the origin at $F$, and
hence a metric on $\BA$.
For $\epsilon>0$ let $U_{\epsilon}\subset \BA_{\BR}$ denote
the set of points $x\in \BA_{\BR}$ having distance $<\epsilon$ from
$D_{F}$.

It follows from ~\ref{limit} that
$$
\Phi_{F}(\CK)=R\Gamma(D_{F},S_{F};\CK)\cong
\dirlim_{\epsilon}\ R\Gamma(U_{\epsilon},S_{F};\CK).
$$
On the other hand, from ~\ref{conicl} it follows that restriction maps
$$
R\Gamma(\BA_{\BR},S_{F};\CK)\lra R\Gamma(U_{\epsilon},S_{F};\CK)
$$
are isomorphisms. This establishes an isomorphism (1). To prove (3), one
remarks that its cone is acyclic.

The other isomorphisms are proven by the similar arguments. We leave
the proof to the reader. $\Box$

\subsection{Proof of ~\ref{dual}}
\label{mainconst} First let us suppose that $\CH$ is central
and $F$ is its $0$-dimensional facet. We fix a coordinate system in $\BA$ as
in the proof of the lemma above.
We have a decomposition $\BA=\BA_{\BR}\oplus i\cdot \BA_{\BR}$. We will denote
by $\Re,\Im:\BA\lra\BA_{\BR}$ the evident projections.

Let
$\BA_{\leq r}$, etc. have the meaning similar to the one-dimensional case
above.
Let us denote $j:=j_{\BA-S_F}$. We proceed as in one-dimensional case.

Let us choose positive numbers $r',r'',\ \epsilon,$ such that
$$
\epsilon D_F\subset \BA_{<r'}\subset \DO_F\subset D_F\subset
\BA_{< r''}
$$
Let us introduce a subspace
$$
Y=\epsilon i\cdot D_F^{opp}
$$
We have isomorphisms
$$
D\Phi_{F}(\CK)\cong DR\Gamma(\BA,S_{F};\CK)
$$
(by Lemma ~\ref{isom})
$$
\cong R\Gamma_c(\BA; j_{*}j^*D\CK)
\cong R\Gamma(\BA,\BA_{\geq r''};j_{*}j^*D\CK)\cong
R\Gamma(\BA,Y\cup \BA_{\geq r''};j_{*}j^*D\CK)
$$
(homotopy).
Consider the restriction map
\begin{equation}
res: R\Gamma(\BA,Y\cup \BA_{\geq r''};j_*j^*D\CK)\lra
R\Gamma(\BA_{\leq r'},Y\cap \BA_{\leq r'};D\CK)
\end{equation}
$\Cone (res)$ is isomorphic to
\begin{eqnarray}
R\Gamma(\BA,\BA_{\leq r'}\cup\BA_{\geq r''}\cup Y;j_*j^*D\CK)=
R\Gamma_c(\BA_{< r''},\BA_{\leq r'}\cup Y;j_*j^*D\CK)\cong\nonumber\\
\cong DR\Gamma(\BA_{< r''}-(\BA_{\leq r'}\cup Y);j_!j^*\CK)\nonumber
\end{eqnarray}
We have by definition
$$
R\Gamma(\BA_{< r''}-(\BA_{\leq r'}\cup Y);j_!j^*\CK)=
R\Gamma(\BA_{(r',r'')}-Y,S_F;\CK)
$$

\subsubsection{}
\label{acycl} {\bf Lemma.\ }{\em Set $\BA'_{\BR}:=\BA_{\BR}-\{ 0\}.$ We have
$R\Gamma(\BA-i\cdot \BA_{\BR},\BA'_{\BR};\CK)=0$.}

{\bf Proof.}
We have to prove that the restriction
map
\begin{equation}
\label{puncture}
R\Gamma(\BA-i\cdot \BA_{\BR};\CK)\lra R\Gamma(\BA'_{\BR};\CK)
\end{equation}
is an isomorphism. Consider projection to the real part
$$
\pi: \BA-i\cdot \BA_{\BR}\lra\BA'_{\BR}.
$$
Evidently,
$$
R\Gamma(\BA-i\cdot \BA_{\BR};\CK)=R\Gamma(\BA'_{\BR},\pi_*\CK).
$$
On the other hand, since $\CK$ is conic along fibers of $\pi$, we have
$$
\pi_*\CK\iso e^*\CK
$$
where $e:\BA'_{\BR}\lra \BA-i\cdot \BA_{\BR}$ denotes the embedding
(cf. \ref{conic}). This implies our lemma. $\Box$

It follows easily that $\Cone(res)$ is acyclic, therefore the map $res$ is an
isomorphism. In other words, we have constructed an isomorphism
$$
D\Phi_F(\CK)\iso R\Gamma(\BA_{\leq r'},Y\cap \BA_{\leq r'};D\CK).
$$
A clockwise rotation by $\pi/2$ induces an isomorphism
$$
R\Gamma(\BA_{\leq r'},Y\cap \BA_{\leq r'};D\CK)\cong
R\Gamma(\BA_{\leq r'},\epsilon\cdot D_F^{opp};D\CK),
$$
and the last complex is isomorphic to $\Phi_F(D\CK)$ by dilatation and
{}~\ref{isom} (3).

This proves the theorem for $\Phi_F$. Note that we have constructed an
explicit isomorphism.

\subsubsection{}
\label{transv} If $F$ is an arbitrary facet, we consider a transversal
slice $L$ as in ~\ref{slice}. By the results of {\em loc.cit.}, and
the above proven case, we have natural isomorphisms
$$
D\Phi_F(\CK)\iso D\Phi_{\{^Fw\} }(\CK_L)\iso \Phi_{\{^Fw\} }(D\CK_L)\iso
\Phi_{\{^Fw\}}((D\CK)_L)\iso \Phi_F(D\CK).
$$
This proves the theorem. $\Box$

\subsection{}
\label{exact}
\begin{thm}{} For every $\CM\in\CM(\BA;\CS_{\CH})$ and every facet $F$
we have $H^i(\Phi_{F}(\CM))=0$ for all $i\neq 0$.
\end{thm}

In other words, functors $\Phi_F$ are $t$-exact with respect to the middle
perversity.

{\bf Proof.} First let us suppose that $\CH$ is central and $F$ is its
$0$-dimensional facet. Let us prove that $\Phi_F$ is right exact,
that is, $H^i(\Phi_F(\CM))=0$ for $i>0$ and every $\CM$ as above.

In fact, we know that

\subsubsection{}
\label{estim} {\em if $S$ is any stratum of $\CS_{\CH}$ of dimension $p$
then $\CM|_S\in\CD^{\leq -p}(S)$}

by the condition of perversity. In particular, $R\Gamma(\BA;\CM)=
i_F^*\CM\in\CD^{\leq 0}(\{ pt\})$.

On the other hand, one deduces from ~\ref{estim} that
$R\Gamma(S_F;\CM)\in\CD^{\leq -1}(\{ pt\})$. In fact, by definition
$S_F$ is a union of certain simplices $\Delta$, all of whose edges
lie in strata of positive dimension. This implies that
$R\Gamma(\Delta;\CM|_{\Delta})\in \CD^{\leq -1}(\{ pt\})$, and one concludes
by Mayer-Vietoris argument, using similar estimates for intersections
of simplices.

Consequently we have
$$
\Phi_{F}(\CM)\cong R\Gamma(\BA,S_F;\CM)\in\CD^{\leq 0}(\{ pt\}),
$$
as was claimed.

On the other hand by Duality theorem ~\ref{dual} we have an opposite
inequality, which proves that $\Phi_F$ is exact in our case.

The case of an arbitrary facet is reduced immediately to the central one by
noting that an operation of the restriction to a transversal slice
composed with a shift by its codimension is $t$-exact and using
{}~(\ref{sliceiso}). The theorem is proved. $\Box$

\subsection{} By the above theorem, the restriction of functors
$\Phi_F$ to the abelian subcategory $\CM(\BA,\CS_{\CH})$ lands
in subcategory $\Vect\subset\CD(\{ pt\})$.

In other words, we get exact functors
\begin{equation}
\label{phi}
\Phi_F:\CM(\BA,\CS_{\CH})\lra\Vect
\end{equation}
These functors commute with Verdier duality.

We will also use the notation $\CM_F$ for $\Phi_F(\CM)$.

\subsection{Canonical and variation maps} Suppose we have a facet $E$.
Let us denote by $\Fac^1(E)$ the set of all facets $F$ such that
$E<F$, $\dim\ F=\dim\ E+1$. We have
\begin{equation}
\label{unican}
S_E=\bigcup_{F\in\Fac^1(E)}D_F
\end{equation}

Suppose we have $\CK\in\CD(\BA,\CS_{\CH})$.

\subsubsection{}
\label{addcan} {\bf Lemma.} {\em We have a natural isomorphism
$$
R\Gamma(S_E,\bigcup_{F\in\Fac^1(E)}S_F;\CK)\cong
\oplus_{F\in\Fac^1{E}}R\Gamma(D_F,S_F;\CK)
$$}

{\bf Proof.} Note that $S_E-\bigcup_{F\in\Fac^1(E)}S_F=
\bigcup_{F\in\Fac^1(E)}\DO_F$ (disjoint union). The claim follows now from the
Poincar\'{e} duality. $\Box$

Therefore, for any $F\in\Fac^1(E)$ we get a natural inclusion map
\begin{equation}
\label{projcan}
i^F_E: R\Gamma(D_F,S_F;\CK)\hra R\Gamma(S_E,\bigcup_{F'\in\Fac^1(E)}S_{F'};\CK)
\end{equation}

Let us define a map
$$
u^F_E(\CK):\Phi_F(\CK)\lra\Phi_E(\CK)
$$
as a composition
$$
R\Gamma(D_F,S_F;\CK)[-p]\overset{i^F_E}{\lra}
R\Gamma(S_E,\bigcup_{F'\in\Fac^1(E)}S_{F'};\CK)[-p]\lra
R\Gamma(S_E;\CK)[-p]\lra
R\Gamma(D_E,S_E)[-p+1]
$$
where the last arrow is the coboundary map for the couple $(S_E,D_E)$,
and the second one is evident.

This way we get a natural transormation
\begin{equation}
\label{can}
u^F_E:\Phi_F\lra \Phi_E
\end{equation}
which will be called a {\em canonical map}.

We define a {\em variation map}
\begin{equation}
\label{var}
v^E_F:\Phi_E\lra\Phi_F
\end{equation}
as follows. By definition, $v^E_F(\CK)$ is the map dual to
the composition
$$
D\Phi_F(\CK)\iso\Phi_F(D\CK)\overset{u^F_E(D\CK)}{\lra}\Phi_E(D\CK)
\iso D\Phi_E(\CK).
$$

\subsection{Lemma}
\label{trans} {\em Suppose we have $4$ facets $A,B_1,B_2,C$ such that
$A<B_1<C,\ A<B_2<C$ and $\dim\ A=\dim\ B_i-1=\dim\ C-2$ (see Fig. 2). Then
$$
u^{B_1}_A\circ u^C_{B_1}=-u^{B_2}_A\circ u^C_{B_2}
$$
and
$$
v^{B_1}_C\circ v^A_{B_1}=-v^{B_2}_C\circ v^A_{B_2}.
$$}

For a proof, see below, ~\ref{prooftr}.

\begin{picture}(20,8)(-10,-4)

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\put(2.5,0){$B_2$}
\put(0,-2){\line(-1,1){4}}
\put(-3,0){$B_1$}

\put(0,-2){\circle*{0.2}}
\put(0,-2.7){$A$}

\put(0,3){$C$}

\put(-0.5,-4){Fig. 2}

\end{picture}

\subsection{Cochain complexes}
For each integer $p$, $0\leq p\leq N$, and $\CM\in\CM(\BA,\CS_{\CH})$
introduce vector spaces
\begin{equation}
C^{-p}_{\CH}(\BA;\CM)=\oplus_{F:\dim F=p}\ \CM_F
\end{equation}
For $i>0$ or $i<-N$ set $C^i_{\CH}(\BA;\CM)=0$.

Define operators
$$
d: C^{-p}_{\CH}(\BA;\CM)\lra C^{-p+1}_{\CH}(\BA;\CM)
$$
having components $u^F_E$.

\subsubsection{}
\label{nilp} {\bf Lemma.} {\em $d^2=0$}.

{\bf Proof.} Let us denote $X_p:=\bigcup_{F:\dim F=p}D_F$. We have
$X_p\supset X_{p+1}$. Evident embeddings
of couples $(D_F,S_F)\hra (X_p,X_{p+1})$ induce maps
$$
R\Gamma(X_p,X_{p+1}; \CM)\lra \oplus_{F:\dim F=p}\ R\Gamma(D_F,S_F;\CM)
$$
which are easily seen to be isomorphisms. Thus, we can identify
$C^{-p}(\BA;\CM)$ with $R\Gamma(X_p,X_{p+1};\CM)$. In these terms,
$d$ is a boundary homomorphism for the triple $X_p\supset X_{p+1}
\supset X_{p+2}$. After this description, the equality $d^2=0$ is a general
fact from homological algebra. $\Box$

\subsubsection{}
\label{prooftr} {\bf Proof of ~\ref{trans}.} The above lemma is equivalent
to the statement of ~\ref{trans} about maps $u$, which is thus proven.
The claim for variation maps follows by duality. $\Box$

This way we get a complex $C^{\bullet}_{\CH}(\BA;\CM)$ lying in
degrees from $-N$ to $0$. It will be called the {\em cochain
complex} of our arrangement $\CH_{\BR}$ with coefficients in $\CM$.

\subsection{Theorem}
\label{rgamma} {\em (i) A functor
$$
\CM\mapsto C^{\bullet}_{\CH}(\BA;\CM)
$$
is an exact functor from $\CM(\BA;\CS_{\CH})$ to the category of complexes
of vector spaces.

(ii) We have a canonical natural isomorphism in
$\CD(\{ pt\})$
$$
C^{\bullet}_{\CH}(\BA;\CM)\iso R\Gamma(\BA;\CM)
$$}

{\bf Proof.} (i) is obvious from the exactness of functors $\Phi_F$,
cf. Thm. ~\ref{exact}.
To prove (ii), let us consider the filtration
$$
\BA\supset X_0\supset X_1\supset\ldots X_N\supset 0.
$$
It follows easily from homotopy argument (cf. ~\ref{limit}, ~\ref{conic})
that the restriction
$$
R\Gamma(\BA;\CM)\lra R\Gamma(X_0;\CM)
$$
is an isomorphism. On the other hand, a "Cousin" interpretation of
$C^{\bullet}_{\CH}(\BA;\CM)$ given in the proof of Lemma ~\ref{nilp},
shows that one has a canonical isomorphism
$R\Gamma(X_0;\CM)\iso C^{\bullet}_{\CH}(\BA;\CM)$. $\Box$

\section{Computations for standard sheaves}

\subsection{}
\label{poinc} Suppose we have a connected locally simply connected
topological space $X$ and a subspace $Y\subset X$ such that each connected
component of $Y$ is simply connected.
Recall that a {\em groupoid} is a category all of whose morphisms are
isomorphisms. Let us define a {\em Poincar\'{e} groupoid}
 $\pi_1(X;Y)$ as follows.

We set $\Ob \pi_1(X;Y)=\pi_0(Y)$. To define morphisms, let us choose
a point $y_i$ on each connected component $Y_i\subset Y$. By definition,
for two connected components $Y_i$ and $Y_j$, the set of homomorphisms
$\Hom_{\pi_1(X,Y)}(Y_i,Y_j)$ is the set of all homotopy classes of paths
in $X$ starting at $y_i$ and ending at $y_j$.

A different choice of points $y_i$ gives a canonically isomorphic
groupoid. If $Y$ is reduced to one point we come back to a usual definition of
the fundamental group.

Given a local system $\CL$ on $X$, we may assign to it a "fiber" functor
$$
F_{\CL}:\pi_1(X;Y)\lra\Vect,
$$
carrying $Y_i$ to the fiber $\CL_{y_i}$. This way we get an equivalence
of the category of local systems on $X$ and the category of functors
$\pi_1(X,Y)\lra\Vect$.

\subsection{} Return to the situation of the previous section. It is known
(cf. ~\cite{br}) that the homology group $H_1(\BAO_{\CH};\BZ)$ is a free
abelian group with a basis consisting of classes of small loops around
hyperplanes $H\in\CH$. Consequently, for each map
\begin{equation}
\label{monodr}
\bq: \CH\lra\BC^*
\end{equation}
there exists a one-dimensional local system $\CL(\bq)$ whose monodromy
around $H\in \CH$ is equal to $\bq(H)$. Such a
local system is unique up to a non-unique isomorphism.

Let us construct such a local system explicitely, using a language of the
previous subsection.

\subsection{} From now on we fix
a real equation for each $H\in\CH$, i.e. a linear function
$f_H:\BA_{\BR}\lra\BR$
such that $H_{\BR}=f^{-1}(0)$. We will denote also by $f_H$ the induced
function $\BA\lra\BC$.

The hyperplane $H_{\BR}$ divides $\BA_{\BR}$ into two halfspaces:
$\BA^+_{\BR,H}=\{ x\in\BA_{\BR}|f_H(x)>0\}$ and
{}~{$\BA^-_{\BR,H}=\{ x\in\BA_{\BR}|f_H(x)<0\}$}.

Let $F\subset H_{\BR}$ be a facet of dimension $N-1$.
We have two chambers $F_{\pm}$ adjacent to $F$, where
$F_{\pm}\subset\BA_{\BR,H}^{\pm}$. Pick a point $w\in F$.
Let us choose a real affine line $l_{\BR}\subset\BA_{\BR}$ transversal
to $H_{\BR}$ and passing through $w$. Let $l$ denote its complexification.

The function $f_H$ induces isomorphism $l\iso\BC$, and
$f_H^{-1}(\BR)\cap l=l_{\BR}$. Let us pick a real $\epsilon>0$ such that two
points
$f_H^{-1}(\pm\epsilon)\cap l_{\BR}$ lie in $F_{\pm}$ respectively. Denote
these points by $w_{\pm}$.

\begin{picture}(20,8)(-10,-4)

\put(-1,-3){\line(1,1){5}}
\put(-1.5,-3.5){$H_{\BR}$}
\put(0.3,-1.2){$F$}

\put(1.5,-0.5){\circle*{0.2}}
\put(1.2,-1){$w$}

\put(1.5,-0.5){\line(1,-1){1}}
\put(2,-1){\circle*{0.2}}
\put(2.3,-1){$w_+$}

\put(1.5,-0.5){\line(-1,1){1}}
\put(1,0){\circle*{0.2}}
\put(0.2,0){$w_-$}

\put(1,-3){\line(-1,1){5}}

\put(-1,0){$F_-$}
\put(3,0){$F_+$}

\put(-4,1){\line(1,0){8}}
\put(-3,1){\circle*{0.2}}
\put(3,1){\circle*{0.2}}

\put(0,-2){\circle*{0.2}}

\put(-0.5,-4){Fig. 3}

\end{picture}

Let us denote by  $\tau^+$ (resp., $\tau^-$) a counterclockwise
(resp., clockwise) path in the upper (resp., lower) halfplane connecting
$\epsilon$ with $-\epsilon$. Let us denote
$$
\tau^{\pm}_{F}=f_H^{-1}(\tau^{\pm})
$$
This way we get two well-defined homotopy classes of paths
connecting chambers $F_+$ and $F_-$. The argument $\arg\ f_H$
increases by $\mp\frac{\pi}{2}$ along $\tau^{\pm}_F$.

Note that if $H'$ is any other hyperplane of our arrangement then
$\arg f_{H'}$ gets no increase along $\tau^{\pm}_F$.

\subsection{} Now suppose we have $\bq$ as in ~(\ref{monodr}).
Note that all connected components of $\BAO_{\CH,\BR}$ --- chambers
of our arrangement --- are contractible.
Let us define a functor
$$
F(\bq^2):\pi_1(\BAO_{\CH},\BAO_{\CH,\BR})\lra\Vect_{\BC}
$$
as follows. For each chamber $C$ we set $F(\bq^2)(C)=\BC$.
For each facet $F$ of codimension $1$ which lies in a hyperplane $H$
we set
$$
F(\bq^2)(\tau^{\pm}_F)=\bq(H)^{\pm 1}
$$
It follows from the above remark on the structure of $H_1(\BAO_{\CH})$ that
we get a correctly defined functor.

The corresponding abelian local system over $\BA_{\CH}$ will be denoted
$\CL(\bq^2)$; it has a monodromy $\bq (H)^2$ around $H\in\CH$.
If all numbers $\bq(H)$ belong to some subfield $B\subset\BC$ then
the same construction gives a local system of $B$-vector spaces.

\subsection{} From now on untill the end of this section we fix a map
$$
\bq:\CH\lra B^*,
$$
$B$ being a subfield of $\BC$, and denote by $\CL$ the local system
of $B$-vector spaces $\CL(\bq^2)$ constructed above. We denote by
$\CL^{-1}$ the dual local system $\CL(\bq^{-2})$.

Let $j:\BAO_{\CH}\lra\BA$ denote an open embedding. For $?$ equal to
one of the symbols $!,*$ or $!*$,
let us consider perverse sheaves $\CL_{?}:=j_?\CL[N]$. They belong to
$\CM(\BA;\CS_{\CH})$; these sheaves will be called {\em standard extensions}
of $\CL$, or simply {\em standard sheaves}. Note that
\begin{equation}
\label{dualstand}
D\CL_!\cong\CL^{-1}_*;\ D\CL_*\cong\CL^{-1}_!
\end{equation}
We have a canonical map
\begin{equation}
\label{map!*}
m: \CL_!\lra\CL_*,
\end{equation}
and by definition $\CL_{!*}$ coincides with its image.

Our aim in this section will be to compute explicitely the cochain
complexes of standard sheaves.

\subsection{Orientations}
\label{coor} Let $F$ be a facet which is a
connected component of a real edge $L_F$. Consider a linear space
$L^{\bot}_F=\BA_{\BR}/L_F$. Let us define
$$
\lambda_F:=H^0(L^{\bot}_F;\Or_{L^{\bot}_F}),
$$
it is a free abelian group of rank $1$. To choose an orientation
of $L^{\bot}_F$ (as a real vector space) is the same as to choose
a basis vector in $\lambda_F$.
We will call an orientation of $L^{\bot}_F$ a {\em coorientation}
of $F$.

We have an evident piecewise linear homeomorphism of
$D_F=\bigcup D_{F<C}$ onto a closed disk in $L^{\bot}_F$; thus, a
coorientation of $F$ is the same as an orientation of $D_F$ (as a $C^0$-
manifold); it defines orientations of all cells $D_{F<C}$.

\subsubsection{}
\label{sign} From now on untill the end of the section, let us fix
coorientations of all facets.
Suppose we have a pair $E<F$, $\dim\ E=\dim\ F-1$. The cell $D_E$ is a part
of the boundary of $D_F$. Let us define the sign $\sgn(F,E)=\pm 1$ as follows.
Complete an orienting basis of $D_E$ by a vector directed outside
$D_F$; if we get the given orientation of $D_F$, set $\sgn(F,E)=1$,
otherwise set $\sgn(F,E)=-1$.

\subsection{Basis in $\Phi_F(\CL_!)^*$}

Let $F$ be a facet of dimension $p$. We have by definition
$$
\Phi_F(\CL_!)=H^{-p}(D_F,S_F;\CL_!)=H^{N-p}(D_F,S_F;j_!\CL)\cong
H^{N-p}(D_F,S_F\cup (_{\CH}H_{\BR}\cap D_F);j_!\CL).
$$
By Poincar\'{e} duality,
$$
H^{N-p}(D_F,S_F\cup (_{\CH}H_{\BR}\cap D_F);j_!\CL)^*\cong
H^0(D_F-(S_F\cup _{\CH}H_{\BR});\CL^{-1})
$$
(recall that we have fixed an orientation of $D_F$).
The space $D_F-(S_F\cup _{\CH}H_{\BR})$ is a disjoint union
$$
D_F-(S_F\cup _{\CH}H_{\BR})=\bigcup_{C\in \Ch (F)}\DO_{F<C}.
$$
Consequently,
$$
H^0(D_F-(S_F\cup _{\CH}H_{\BR});\CL^{-1})\cong
\oplus_{C\in \Ch (F)}H^0(\DO_{F<C};\CL^{-1}).
$$
By definition of $\CL$, we have canonical identifications
$H^0(\DO_{F<C};\CL^{-1})=B$. We will denote by $c(\CL_!)_{F<C}\in\Phi_F(\CL)^*$
the
image of $1\in H^0(\DO_{F<C};\CL^{-1})$ with respect to the embedding
$$
H^0(\DO_{F<C};\CL^{-1})\hra \Phi_F(\CL_!)^*
$$
following from the above.

Thus, classes $c(\CL_!)_{F<C},\ C\in \Ch (F)$, form a basis of
$\Phi_F(\CL_!)^*$.

\subsection{}
\label{can!} Let us describe canonical maps for
$\CL_!$. If $F<E,\ \dim\ E=\dim\ F+1$, let
$u^*:\Phi_F(\CL_!)^*\lra\Phi_E(\CL_!)^*$ denote the map dual to $u^E_F(\CL_!)$.
Let $C$ be a chamber adjacent to $F$.
Then
\begin{equation}
u^*(c(\CL_!)_{F<C})=\left\{ \begin{array}{ll}
                        \sgn(F,E)c(\CL_!)_{E<C}&\mbox{if $E<C$}\\
                          0&\mbox{otherwise}
                     \end{array}
             \right.
\end{equation}

\subsection{Basis in $\Phi_F(\CL_*)^*$} We have isomorphisms
\begin{equation}
\label{dualchains}
\Phi_F(\CL_*)\cong\Phi_F(D\CL_!^{-1})\cong\Phi_F(\CL_!^{-1})^*
\end{equation}
Hence, the defined above basis $\{ c(\CL^{-1}_!)_{F<C}\}_{C\in\Ch (F)}$
of $\Phi_F(\CL_!^{-1})^*$, gives a basis in  $\Phi_F(\CL_*)$.
We will denote by $\{ c(\CL_*)_{F<C}\}_{C\in\Ch (F)}$ the dual basis
of $\Phi_F(\CL_*)$.

\subsection{Example} Let us describe our chains explicitely in the simplest
one-dimensional case, in the setup ~\ref{one-dim}. We choose a natural
orientation on $\BA_{\BR}$.
A local system $\CL=\CL(q^2)$ is uniquely determined by one nonzero
complex number $q$. By definition, the upper (resp., lower) halfplane
halfmonodromy from  $w_+$ to $w_-$ is equal to $q$ (resp., $q^{-1}$).

\subsubsection{Basis in $\Phi_F(\CL_!)^*$} The space $\Phi_F(\CL_!)^*$
admits a basis consisting of two chains
$c_{\pm}=c(\CL_!)_{F<E_{\pm}}$ shown below, see Fig. 3(a).
By definition, a homology class is represented by a cell together with
a section of a local system $\CL^{-1}$ over it. The section of $\CL^{-1}$
over $c_+$ (resp., $c_-$) takes value $1$ over $w_+$ (resp., $w_-$).

\begin{picture}(20,8)(-10,-4)

\put(-5,0){\circle{0.2}}
\put(-5,0){\oval(6,6)}
\put(-6.5,3.3){$S_{r''}$}

\put(-3,0){\circle*{0.2}}
\put(-3,-0.5){$w_+$}
\put(-7,0){\circle*{0.2}}
\put(-7,-0.5){$w_-$}

\put(-7,0){\line(1,0){4}}
\put(-7,0){\vector(1,0){1.5}}
\put(-5,0){\vector(1,0){1.5}}

\put(-6.7,0.3){$c_-$}
\put(-3.5,0.3){$c_+$}

\put(-6,-3){\line(0,1){6}}
\put(-6,1){\vector(0,1){1}}
\put(-6,-3){\circle*{0.2}}
\put(-6,3){\circle*{0.2}}
\put(-5.8,2){$Dc_-$}

\put(-4,-3){\line(0,1){6}}
\put(-4,1){\vector(0,1){1}}
\put(-4,-3){\circle*{0.2}}
\put(-4,3){\circle*{0.2}}
\put(-3.8,2){$Dc_+$}

\put(-5.5,-3.5){(a)}

\put(5,0){\circle{0.2}}
\put(5,0){\oval(6,6)}
\put(3.5,3.3){$S_{r''}$}

\put(5,3){\line(0,-1){2}}
\put(5,-3){\line(0,1){2}}
\put(4.6,1.8){$Y$}

\put(5,1){\circle*{0.2}}
\put(5,-1){\circle*{0.2}}

\put(7,0){\circle*{0.2}}
\put(7,-0.5){$w_+$}
\put(3,0){\circle*{0.2}}
\put(3,-0.5){$w_-$}

\put(3,0){\line(1,0){4}}
\put(3,0){\vector(1,0){1.5}}
\put(5,0){\vector(1,0){1.5}}

\put(-6.7,0.3){$c_-$}
\put(-3.5,0.3){$c_+$}

\put(5,0){\oval(2,3)}
\put(6,0){\vector(0,1){0.5}}
\put(6.2,0.5){$\tilde{D}c_+$}
\put(4,0){\vector(0,1){0.5}}
\put(3,0.5){$\tilde{D}c_-$}
\put(5,1.5){\circle*{0.2}}
\put(5,-1.5){\circle*{0.2}}

\put(4.5,-3.5){(b)}

\put(-0.5,-4){Fig. 3}

\end{picture}

\subsubsection{Basis in $\Phi_F(\CL_*)^*$}
Let us adopt notations of ~\ref{mainconst}, with $\CK=\CL^{-1}_!$.
It is easy to find the basis $\{ Dc_+,Dc_-\}$ of the dual space
$$
\Phi_F(\CL_!^{-1})\cong H^0(\BA,\BA_{\geq r''};j^*j_*D\CL_!^{-1})^*=
H^1(\BA-\{ w_+,w_-\},\BA_{\geq r''};\CL)^*
$$
dual to $\{ c(\CL^{-1})_{F<E_+},c(\CL^{-1})_{F<E_-}\}$.
Namely, $Dc_{\pm}$ is represented by the relative $1$-chain
$$
\{ \pm\frac{1}{2}+y\cdot i|\ -\sqrt{(r'')^2-\frac{1}{4}}\leq y
\leq \sqrt{(r'')^2-\frac{1}{4}}\},
$$
with evident sections of $\CL^{-1}$ over them, see Fig. 3(a).

Next, one has to deform these chains to chains $\tilde{D}c_{\pm}$ with
their ends on $Y$, as in Fig. 3(b). Finally, one has to make a clockwise
rotation of the picture by $\pi/2$. As a result, we arrive at the
following two chains $c^*_{\pm}$ forming a basis of $\Phi_F(\CL_*)^*$:

\begin{picture}(20,8)(-10,-4)

\put(0,0){\circle{0.2}}

\put(2,0){\circle*{0.2}}
\put(2.2,-0.5){$w_+$}
\put(-2,0){\circle*{0.2}}
\put(-1.8,-0.5){$w_-$}

\put(0,0){\oval(4,4)}
\put(0,2){\vector(1,0){0.5}}
\put(0,2.3){$c^*_-$}
\put(0,-2){\vector(1,0){0.5}}
\put(0,-2.5){$c^*_+$}

\put(-2,0){\line(1,0){4}}
\put(-2,0){\vector(1,0){1.5}}
\put(0,0){\vector(1,0){1.5}}

\put(-1,0.3){$c_-$}
\put(0.8,0.3){$c_+$}

\put(-0.5,-4){Fig. 4}

\end{picture}

The section of $\CL^{-1}$ over $c^*_+$ (resp., $c_-$) has value $1$ at
$w_+$ (resp., $w_-$).

It follows from this description that the natural map
\begin{equation}
\label{m!*}
m:\Phi_F(\CL_*)^*\lra\Phi_F(\CL_!)^*
\end{equation}
is given by the formulas
\begin{equation}
\label{f!*}
m(c^*_+)=c_++qc_-;\ m(c^*_-)=qc_++c_-
\end{equation}

By definition, spaces $\Phi_{E_{\pm}}(\CL_?)^*$ may be identified
with fibers $\CL_{w_{\pm}}$ respectively, for both $?=!$ and $?=*$,
and hence with $B$. Let us denote by $c_{w_{\pm}}$ and
$c^*_{w_{\pm}}$ the generators corresponding to $1\in B$.

It follows from the above description that the canonical maps $u^*$ are given
by the formulas
\begin{equation}
\label{fu!}
u^*(c_+)=c_{w_+};\ u^*(c_-)=-c_{w_-};
\end{equation}
and
\begin{equation}
\label{fu*}
u^*(c_+^*)=c_{w_+}^*-qc_{w_-}^*;\
u^*(c_-^*)=qc_{w_+}^*-c_{w_-}^*;\
\end{equation}
Let us compute variation maps. To get them for the sheaf $\CL_!$, we should
by definition replace $q$ by $q^{-1}$ in ~(\ref{fu*}) and take the conjugate
map:
\begin{equation}
\label{fv!}
v^*(c_{w_+})=c_+ + q^{-1}c_-;\
v^*(c_{w_-})=-q^{-1}c_+-c_-
\end{equation}
To compute $v^*$ for $\CL_*$, note that the basis in
$$
H^0(\BA,\{ w_+,w_-\};\CL_!^{-1})^*=H_1(\BA,\{w_+,w_-,0\};\CL)
$$
dual to $\{ c^*_+,c^*_-\}$, is $\{ q^{-1}\tc_-,q^{-1}\tc_+\}$ {\em (sic!)}
where $\tc_{\pm}$ denote the chains defined in the same way as $c_{\pm}$,
with $\CL$ replaced by $\CL^{-1}$. From this remark it follows that
\begin{equation}
\label{fv*}
v^*(c_{w_+}^*)=q^{-1}c_-^*;\
v^*(c_{w_-}^*)=-q^{-1}c_+^*
\end{equation}

\subsection{} Let us return to the case of an arbitrary arrangement.
Let us say that a hyperplane $H\in\CH$ {\em separates} two chambers $C, C'$
if they lie in different halfspaces with respect to $H_{\BR}$.
Let us define numbers
\begin{equation}
\label{qsep}
\bq(C,C')=\prod\bq(H),
\end{equation}
the product over all hyperplanes $H\in \CH$ separating $C$ and $C'$.
In particular, $\bq(C,C)=1$.

\subsection{Lemma.} {\em Let $F$ be a face, $C\in\Ch(F)$. The canonical mapping
$$
m:\Phi_F(\CL_*)^*\lra\Phi_F(\CL_!)^*
$$
is given by the formula
\begin{equation}
\label{sform}
m(c(\CL_*)_{F<C})=\sum_{C'\in\Ch(F)}\bq(C,C')c(\CL_!)_{F<C'}
\end{equation}}

\subsubsection{} Since $\Phi_F(\CL_*)$ is dual to $\Phi_F(\CL_!)$, we may
view $m$ as  a {\em bilinear form} on $\Phi_F(\CL_!)$. By ~(\ref{sform})
it is {\em symmetric}.

{\bf Proof} of lemma. We generalize the argument of the previous example.
First consider the case of zero-dimensional $F$.
Given a chain $c(\CL_!^{-1})_{F<C}$, the corresponding dual chain may be
taken as
$$
Dc_{F<C}= \epsilon\cdot\ ^Cw\oplus i\cdot\BA_{\BR},
$$
were $\epsilon$ is a sufficiently small positive real.
Next, to get the dual chain $c(\CL_*)_{F<C}$, we should make a deformation
similar to the above one, and a rotation by $\frac{\pi}{2}$. It is convenient
to make the rotation first. After the rotation, we get a chain
$\BA_{\BR} -\epsilon i\cdot\ ^Cw$. The value of
$m(c(\CL_*)_{F<C})$ is given by the projection of this chain to $\BA_{\BR}$.

The coefficient at $c(\CL_!)_{F<C'}$ is given by the monodromy of $\CL^{-1}$
along the following path from $C$ to $C'$. First, go "down" from $^Cw$
to $-\epsilon i\cdot\ ^Cw$; next, travel in
$\BA_{\BR} -\epsilon i\cdot\ ^Cw$ along the straight line from
$-\epsilon i\cdot\ ^Cw$ to $-\epsilon i\cdot ^{C'}w$, and then go "up" to
$^{C'}w$. Each time we are passing under a hyperplane $H_{\BR}$
separating $C$ and $C'$, we gain a factor $\bq(H)$. This gives
desired coefficient for the case $\dim\ F=0$.

For an arbitrary $F$ we use the same argument by considering the intersection
of our picture with a transversal slice.   $\Box$

\subsection{} Let $E$ be a facet which is a component of a
real edge $L_{E,\BR}$; as usually $L$ will denote the complexification.
Let $\CH_L\subset \CH$ be a subset consisting of all hyperplanes
containg $L$. If we assign to a chamber $C\in\Ch(E)$ a unique chamber of the
subarrangement $\CH_L$ comtaining $C$, we get a bijection of $\Ch(E)$ with
the set of {\em all} chambers of $\CH_L$.

Let $F<E$ be another facet. Each chamber $C\in\Ch(F)$ is contained in a unique
chamber of $\CH_L$. Taking into account a previous bijection, we get a mapping
\begin{equation}
\label{pi}
\pi^F_E:\Ch(F)\lra\Ch(E)
\end{equation}

\subsection{Lemma.}{\em Let $F$ be a facet, $C\in\Ch(F)$. We have
\begin{equation}
\label{u*}
u^*(c(\CL_*)_{F<C})=\sum\sgn(F,E)\bq(C,\pi^F_E(C))c(\CL_*)_{E<\pi^F_E(C)},
\end{equation}
the summation over all facets $E$ such that $F<E$ and $\dim\ E=\dim\ F+1$.}

(Signs $\sgn(F,E)$ have been defined in ~\ref{sign}.)

{\bf Proof.} Again, the crucial case is $\dim\ F=0$ --- the case of arbitrary
dimension is treated using a transversal slice. So, let us suppose that
$F$ is zero-dimensional. In order to compute the coefficient of
$u^*(c(\CL_*)_{F<C})$ at $c(\CL^*)_{E<C'}$ where $E$ is a one-dimensional
facet adjacent to $F$ and $C'\in\Ch(E)$, we have to do the following.

Consider the intersection of a real affine subspace
$\BA_{\BR}-\epsilon\cdot i\cdot\ ^Cw$ (as in the proof of the previous lemma)
with a complex hyperplane $M_E$ passing through $^Ew$ and transversal
to $E$. The intersection will be homotopic to a certain chain
$c(\CL_*)_{E<C''}$ where $C''$ is easily seen to be equal to $\pi^F_E(C)$,
and the coefficient is obtained by the same rule as described
in the previous proof. The sign will appear in accordance with
compatibility of orientations of $D_F$ and $D_E$. $\Box$

\subsection{} Let us collect our results. Let us denote by
$\{ b(\CL_?)_{F<C}\}_{C\in\Ch(F)}$ the basis in $\Phi_F(\CL_?)$ dual
to $\{ c(\CL_?)_{F<C}\}$, where $?=!$ or $*$.

\subsection{Theorem.}{\em (i) The complex $C^{\bullet}_{\CH}(\BA;\CL_!)$
is described as follows. For each $p,\ 0\leq p\leq N$, the space
$C^{-p}_{\CH}(\BA;\CL_!)$ admits a basis consisting of all cochains
$b(\CL_!)_{F<C}$ where $F$ runs through all facets of $\CH_{\BR}$ of dimension
$p$, and $C$ through $\Ch(F)$. The differential
$$
d:C^{-p}_{\CH}(\BA;\CL_!)\lra C^{-p+1}_{\CH}(\BA;\CL_!)
$$
is given by the formula
\begin{equation}
\label{dl!}
d(b(L_!)_{F<C})=\sum_{E:E<F,\ \dim\ E=\dim\ F-1} \sgn(E,F)b(L_!)_{E<C}
\end{equation}

(ii) The complex $C^{\bullet}_{\CH}(\BA;\CL_*)$
is described as follows. For each $p,\ 0\leq p\leq N$, the space
$C^{-p}_{\CH}(\BA;\CL_*)$ admits a basis consisting of all cochains
$b(\CL_*)_{F<C}$ where $F$ runs through all facets of $\CH_{\BR}$ of dimension
$p$, and $C$ through $\Ch(F)$. The differential
$$
d:C^{-p}_{\CH}(\BA;\CL_*)\lra C^{-p+1}_{\CH}(\BA;\CL_*)
$$
is given by the formula
\begin{equation}
\label{dl*}
d(b(L_*)_{F<C})=\sum \sgn(E,F)\bq(C,C')b(L_*)_{E<C'},
\end{equation}
the summation over all facets $E<F$ such that $\dim\ E=\dim\ F-1$ and
all chambers $C'\in\Ch(E)$ such that $\pi^E_F(C')=C$.

(iii) The natural map of complexes
\begin{equation}
\label{mapc}
m:C^{\bullet}_{\CH}(\BA;\CL_!)\lra C^{\bullet}_{\CH}(\BA;\CL_*)
\end{equation}
induced by the canonical map $\CL_!\lra\CL_*$, is given by the formula
\begin{equation}
\label{formb}
m(b(\CL_!)_{F<C})=\sum_{C'\in\Ch(F)}\bq(C,C')b(\CL_*)_{F<C'}
\end{equation}}

All statements have already been proven.

\subsubsection{} {\bf Corollary.} {\em The complexes
$C^{\bullet}_{\CH}(\BA;\CL_!)$
and $C^{\bullet}_{\CH}(\BA;\CL_*)$ described explicitely in the above theorem,
compute the relative cohomology $H^{\bullet}(\BA,\ _{\CH}H;\CL)$ and the
cohomology of the open stratum $H^{\bullet}(\BAO,\CL)$ respectively,
and the map $m$ induces the canonical map in cohomology.}

{\bf Proof.} This follows immediately from ~\ref{rgamma}. $\Box$

This corollary was proven in ~\cite{v}, Sec. 2, by a different argument.

\subsection{Theorem.}
\label{inters} {\em The complex $C^{\bullet}_{\CH}(\BA;\CL_{!*})$
is canonically isomorphic to the image of ~(\ref{mapc}).}

{\bf Proof.} This follows from the previous theorem and the exactness
of the functor $\CM\mapsto C^{\bullet}_{\CH}(\BA;\CM)$,
cf. Thm. ~\ref{rgamma} (i). $\Box$

The above description of cohomology is analogous to ~\cite{sv}, p. I,
whose results may be considered as a "quasiclassical" version
of the above computations.