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\title{Hypersphere Arrangement and Imaginary Cycles for Hypergeometric Integrals\\}
\author{Kazuhiko AOMOTO
\\
\footnotesize 
Department of Mathematics, Nagoya University, \\
\footnotesize 
Nagoya 464-8602, Japan
\\
\footnotesize 
\quad \tt kazuhiko@aba.ne.jp
\\}
\date{2009/Aug}



\begin{document}
\maketitle
\begin{abstract}
We construct imaginary cycles for hypergeometric integrals
associated with a hypersphere arrangement and discuss
the relation between the twisted rational de Rham cohomology.
We pose two geometric problems involved in it.
\end{abstract}

%{\scriptsize 2000 {\it Mathematical Subject Classification.} }

{\scriptsize  {\it Keywords.} 
Hypersphere Arrangement; Configuration Matrix, Hypergeometric Integrals; Twisted de Rham 
Cohomology, Singuralities, Real and Imaginary Cycles}

{\scriptsize  {\it Running title.} 
Hypersphere Arrangement and Imaginary Cycles }

%Intro
\section{Introduction}
First we want to illustrate in one dimensional case the main objective discussed in this article.
Let $Q$ be the complex circle :
$\{\xi=(\xi_1,\xi_2); \xi_1^2+\xi_2^2=1$ in the complex affine plane ${\bf C}^2.$ $Q$ is isomorphic to
${\bf C}^*$ by taking $\xi_1+\sqrt{-1}\xi_2=\zeta.$
Consider a family of $m$ complex lines $H_j: f_j=0$
where
\be
f_j=u_{j,0}+u_{j,1}\xi_1+u_{j,2}\xi_2
\ee
such that $u_{j,0},u_{j,1},u_{j,2}\in {\bf R}$ and that
$u_{j,1}^2+u_{j,2}^2-u_{j,0}^2=1.$ Denote
\be
a_{i,j}=u_{i,1}u_{j,1}+u_{i,2}u_{j,2}-u_{i.0}u_{j,0},\,
a_{i,0}=a_{0,i}=u_{i,0}
\ee

The intersection of
$Q$ and $H_j$ consisits of  $2$ different points
which we denote by $\zeta_j,\zeta_j^*$
such that $|\zeta_j|=|\zeta_j^*|=1.$ 
Let $R$ be the ${\bf C}[\xi_1,\xi_2]$ module 
\be
R=\sum_{\nu_1\ge0,\ldots,\nu_m\ge0}{\bf C}[\xi_1,\xi_2]\prod_{j=1}^m
f_j^{-\nu_j}
=\sum_{\nu_1\ge0,\ldots,\nu_m\ge0}{\bf C}[\zeta,\zeta^{-1}]\prod_{j=1}^m
\{(\zeta-\zeta_j)(\zeta-\zeta_j^*)\}^{-\nu_j}
\ee
because $f_j$ can be written as
\be
f_j=\sqrt{-1} \frac{(\zeta-\zeta_j)(\zeta-\zeta_j^*)}{(\zeta_j^*-\zeta_j)\zeta}
\ee
Consider the multiplicative function
\be
\Phi_0(\xi)=\prod_{j=1}^m f_j^{\lambda_j}\quad(\lambda_j\in {\bf R_{>0}})
\ee
and the associated rational de Rham cohomology on 
$Y=Q-\cup_{j=1}^m H_j$
\be
H^1(Y,\nabla_0)\cong R\tau_Q/\nabla_0(R) 
\ee
defined by the covariant differential $\nabla_0(\psi)=d\psi+d\log\Phi_0\psi$, where we denote
\be
\tau_Q=-\xi_1d\xi_2+\xi_2d\xi_1=\sqrt{-1}\frac{d\zeta}{\zeta}
\ee 
Suppose that $\zeta_1,\zeta_1^*,\ldots,\zeta_m,\zeta_m^*$ 
are different from each other. Then one can prove that
for generic $\lambda_j$
\be
H^1(Y,\nabla_0)\cong {\bf C}^{2m}
\ee
and it is spanned by
\be
&&\varphi_Q(\emptyset)=\tau_Q,\varphi_Q(j)=\frac{\tau_Q}{f_j}=d\log\frac{\zeta-\zeta_j}{\zeta-\zeta_j^*}\,(1\le j\le m),\\
&&\varphi_Q(j,k)=\frac{\tau_Q}{f_jf_k}\,(1\le j<k\le m)
\ee
These $1$ forms are not linearly independent on $Y.$
For any different $i,j,k$ there exists the fundamental linear relation
\beq
c_i\varphi_Q(i)+c_j\varphi_Q(j)+c_k\varphi_Q(k)+c_{j,k}\varphi_Q(j,k)+
c_{k,i}\varphi_Q(k,i)+
c_{i,j}\varphi_Q(i,j)=0\nonumber\\
\eeq
where $c_i,c_j,c_k,c_{j,k},c_{k,i},c_{i,j}$
can be written in terms of $a_{i,j},a_{k,0}$ as
\be
&&c_i=-\frac{A(0,i)}{
A
\left(
\begin{array}{ccc}
 0, &i,   &k   \\
 0, & i,  &j   \\
\end{array}
\right)
},\;
c_{j,k}=
\frac{A(j,k)}{
A
\left(
\begin{array}{ccc}
 i, &j,   &k   \\
 0, & j,  &k   \\
\end{array}
\right)
}
\ee
$c_j,c_k,c_{k,i},c_{i,j}$ being defined in the same way cyclically.
Moreover $A(0,i)=-1-a_{i,0}^2,A(j,k)=1-a_{j,k}^2$ and 
$
A
\left(
\begin{array}{ccc}
 i, &j,   &k   \\
 i^\prime, & j^\prime,  &k^\prime  \\
\end{array}
\right)
$ denotes the determinant of the matrix whose components
are $a_{p,q}\,(p=i,j,k;q=i^\prime,j^\prime,k^\prime).$

The twisted homology $H_1(Y,\hat{\cal L}_0)$
dual to $H^1(Y,\nabla_0)$ is spanned by the linearly independent cycles which are expressed by the closures 
(arcs) of the connected components of $\Re Y$.



Suppose now that for a fixed pair $i,j$,
one of $\zeta_i$ or $\zeta_i^*$
coincides with one of $\zeta_j$ or $\zeta_j^*.$
This occurs if and only if $A(i,j)=0$ i.e.,$a_{i,j}=\pm1.$
If $\zeta_j$ tends to the point $\zeta_i^*$, then the arc
connecting the points $\zeta_i^*,\zeta_j$ in $\Re Q$ 
reduces to the point. Hence if $\zeta_i^*=\zeta_j$, the dimension of $H_1(Y,\hat{\cal L}_0)$ decreses by one.
On the other hand one can show that $\varphi_Q(i,j)$ can be described homologically as a linear combination of $\varphi_Q(k,i),\varphi_Q(k,j),
\varphi_Q(k),\varphi_Q(\emptyset)$ :
\beq
&&2(\lambda_i+\lambda_j-1)\varphi_Q(i,j)\sim
-\sum_{k\ne i,j}\lambda_k
\left\{
\frac{A(k,i)}{a_{k,i}+a_{k,j}}\varphi_Q(k,i)
\right.\nonumber\\
&&\left.+\frac{A(k,j)}{a_{k,i}+a_{k,j}}\varphi_Q(k,j)
\right\}
+\sum_{k=0}^m \lambda_k a_{k,0}\varphi_Q(k)-
(\lambda_\infty-1)
\left\{\frac{A(0,i)}{a_{i,0}+a_{j,0}}\varphi_Q(i)\right.\nonumber\\
&&\left.+\frac{A(0,j)}{a_{i,0}+a_{j,0}}\varphi_Q(j)\right\}-\lambda_\infty\varphi_Q(\emptyset)
\eeq
where we denote $\lambda_\infty=\sum_{j=1}^m \lambda_j.$

In particular consider the case where $m=3$ and $A(1,2)=A(1,3)=A(2,3)=0,$ i.e., $\zeta_1^*=\zeta_2,\zeta_2^*=\zeta_3,\zeta_3^*=\zeta_1.$ Then (1.1) reduces to the only one identity :
\be
\varphi_Q(1)+\varphi_Q(2)+\varphi_Q(3)=0
\ee
and there are three identities of type (1.2) :
\be
&&2(\lambda_i+\lambda_j-1)\varphi_Q(i,j)
\sim\sum_{k=1}^3\lambda_k
a_{k,0}\varphi_Q(k)\\
&&+(\lambda_\infty-1)\left
\{\frac{1+a_{i,0}^2}{a_{i,0}+a_{j,0}}\varphi_Q(i)+\frac{1+a_{j,0}^2}{a_{i,0}+a_{j,0}}\varphi_Q(j)\right\}-\lambda_\infty\varphi_Q(\emptyset)
\ee
Hence $H^1(Y,\nabla_0)$ is of dimension $3$ and is spanned by
a basis of representatives $\varphi_Q(\emptyset), \varphi_Q(1),
\varphi_Q(2).$

In this article we want to extend the above observation to $n\, (n\ge1)$ dimensional cases. The twisted de Rham cohomology 
$H^n(Y,\nabla_0)$ can be formulated in the space $Y$, the complement of
a union of $n-1$ dimensional hyperspheres in the $n$ dimensional fundamental complex hypersphere 
$Q$.  There arise two problems.

In the first place, based on my preceding articles
(see [1],[4]),
we formulate the twisted de Rham cohomology presenting
explicitly its basis in terms of the invariants $a_{i,j}$ 
obtained by the Lorentz inner products of the coefficients of two linear functions defining hyperspheres, together with
$a_{k,0}$ which are the constant terms of a linear function.
The differential structures of the hypergeometric integrals are described by the invariants $a_{i,j}, a_{k,0}$
under Lorentz groups
or orthogonal groups. In the final section we show
how they should be modified in some degenerate cases.

In the second place the basis of the twisted homology $H_n(Y,\hat{\cal L}_0)$
cannot always be realized  by real domains in $\Re Y$
except in special domain of parameters. We must construct some of them
as imaginary cycles. We want to show that this can be 
done by deforming real cycles from a special domain of
parameters involved where all the cycles can be realized by real domains (see Theorem 13).




%
\section{Basic Properties}
Let ${\cal A}$ be an arrangement of $m$ hyperplanes $H_j\;(1\le j\le m)$ 
defined over the real field of coefficients
in the $n+1$ dimensional complex affine space ${\bf C}^{n+1}$. 
Each hyperplane 
can be described as 
\be
H_j : u_{j,0}+\sum_{\nu=1}^n u_{j,\nu}\xi_\nu=0
\ee
for $\xi=(\xi_1,\ldots,\xi_{n+1})\in {\bf C}^{n+1}$ 
 

We denote by
$N(\cal A)$ the union of hyperplanes : $= \bigcup_{H_j\in {\cal A}}H_j $
and by $X=M({\cal A})$ the compliment of $N(\cal A)$
$: = {\bf C}^{n+1}-N({\cal A})$. Let $Q$ be the complex hypersphere $:  f_0=0$ defined by the quadratic polynomial$f_0=1-\sum_{\nu=1}^{n+1}\xi_\nu^2.$ Each intersection
$H_j\cap Q$ defines a hypersphere in $Q$ provided it does not reduce to a point i.e.,
$
-u_{j,0}^2+\sum_{\nu=1}^{n+1}u_{j,\nu}^2\ne0.
$
Throughout this article we shall assume this condition and
so may assume that
\be
-u_{j,0}^2+\sum_{\nu=1}^{n+1}u_{j,\nu}^2=1
\ee
The family ${\cal A}^\prime=\{H_j\cap Q\}_{1\le j\le m}$ defines a hypersphere arrangement
in $Q.$ We denote by $Y$ the intersection of $X$ and $Q$
$: Y=Q-\cup_{j=1}^m H_j\cap Q.$
We denote by $\Re Q$ the real part of $Q$ which is 
identified with the $n$ dimensional real hypersphere.
The real part $S_j=H_j\cap\Re Q$ is a real $(n-1)$
dimensional hypersphere in $\Re Q$.

We define the $(m+1)\times (m+1)$ configuration matrix $A=(a_{i,j})_{0\le i,j\le m}$ associated with ${\cal A}^\prime$, whose components are Lorentz inner products
\be
a_{i,j}=-u_{i0}u_{j0}+\sum_{\nu=1}^{n+1}u_{i,\nu}u_{j,\nu}\;(1\le i,j\le m);\;a_{i,0}=a_{0,i}=u_{i0};\;a_{0,0}=-1
\ee
so that $a_{i,i}=1$ for $1\le i\le m.$

For a set of indices $I=\{i_1,\ldots,i_p\}\subset\{0,1,2,\ldots,m\}$ the size $p$ will be denoted by $|I|$. We say that $I$ is admissible if $I\subset\{1,2,\ldots,m\}.$
For two sets of indices $I=\{i_1,i_2,\ldots,i_p\}$
and $J=\{j_1,\ldots,j_p\}$ we define the subdeterminant
\be
A
\left(
\begin{array}{ccc}
 I \\
 J 
  \end{array}
\right)
=
\left|
\begin{array}{cccc}
 a_{i_1,j_1} &a_{i_1,j_2}   &\ldots&  a_{i_1,j_p} \\
 a_{i_2,j_1} &a_{i_2,j_2}   &\ldots& a_{i_1,j_p}   \\
 \vdots & \vdots  &  &\vdots \\
 a_{i_p,j_1} &a_{i_p,j_2}   &\ldots& a_{i_p,j_p}
\end{array}
\right|
\ee
We abbreviate $A
\left(
\begin{array}{ccc}
 I \\
 I 
  \end{array}
\right)
$
by $A(I).$

For an admissible set $I=\{i_1,i_2,\ldots,i_p\}$ and a set
$J=\{j_1,j_2,\ldots,j_p\}\subset\{0,1,2,\ldots,n+1\}\,(p\le n+2)$ we denote the subdeterminant
\be
U\left(\begin{array}{ccc}
 i_1,i_2,\ldots,i_{p} \\
 j_1,\ldots,j_p 
  \end{array}
\right)=
\left|
\begin{array}{cccc}
 u_{i_1,j_1} &u_{i_1,j_2}   &\ldots&  u_{i_1,j_p} \\
 u_{i_2,j_1} &u_{i_2,j_2}   &\ldots& u_{i_2,j_p}   \\
 \vdots & \vdots  &  &\vdots \\
 u_{i_p,j_1} &u_{i_p,j_2}   &\ldots& u_{i_p,j_p}
\end{array}
\right|
\ee


\begin{re}
The matrix $A$ has at most rank $n+2.$ 
Assume that $I$ is admissible. Then $A(I)=0$ for $|I|\ge n+3$, and
\be
A(I)=-U^2\left(\begin{array}{ccc}
 i_1,i_2,\ldots,i_{n+2} \\
 0,1,\ldots,n+1 
  \end{array}
  \right)
\ee
for $|I|=n+2.$ On the other hand
$A(0,I)=0$ for $|I|\ge n+2$ and
\be
A(0,I)=-U^2\left(
\begin{array}{ccc}
 i_1,i_2,\ldots,i_{n+1} \\
 1,\ldots,n+1 
  \end{array}
  \right)
\ee
for $|I|=n+1.$
\end{re}

%lemma1
\begin{lm}
Fix an admissible set $I$
and consider the intersection subspace $V=\cap_{j\in I}H_j$
in ${\bf C}^{n+1}$.


{\rm(i)}
In case where $|I|\le n$, $A(I)=0$  if and only if $V$ has contact with $Q$ at one point.

{\rm(ii)} In case where $|I|=n+1$, $A(I)=0$ if and only if 
$V$ has a common point with $Q.$

{\rm(iii)} In case where $|I|=n+2,$ $A(I)=0$ if and only if 
$V$ is not empty.

\end{lm}
Assume that

%Cond
$({\cal H}1)$\quad:
$A(0,I)<0$ for an arbitrary admissible set $I$ such that $|I|\le n+1$, i.e., the homogeneous parts of $|I|$   
linear functions $f_j\,(j\in I)$ are linearly independent.

Assume further that

%Cond
$({\cal H}2)$\quad:
$A(I)\ne 0$
for an arbitrary admissible $I\,(2\le |I|\le n+2)$.

Then
for any $I\subset \{1,2,\ldots,m\}$ with $|I|=n+2$
the $(n+2)\times (n+2)$ symmetric submatrix $(a_{i,j})_{i,j\in I}$
has the signature of $n+1$ $(+)$sign and one $(-)$sign
so that $A(I)<0$ for $|I|=n+2$.
This is equivalent to say that for any sequence of increasing admissible
sets of indices
\be
I_1\subset I_2\subset \cdots \subset I_{n+1}\subset I_{n+2}
\ee
such that $|I_r|=r$, the signs of $A(I_r)A(I_{r+1})\;(1\le r\le n+1)$ are positive except for one. Hence
 $A(I)<0$ implies $A(J)<0$ if $I\subset J,\,|J|\le n+2$
(see [8]).
In particular the following two cases are  
interesting :

(${\cal H}$2a): $A(I)>0$ for all admissible $I$ with $2\le |I|\le n+1 $
and $A(I)<0$ for all admissible $I$ with $|I|=n+2.$

(${\cal H}$2b): $A(I)<0$ for all admissible $I$ with $2\le|I|\le n+2.$

Let $\tau$ be the $n+1$ form $d\xi_1\wedge\cdots\wedge d\xi_{n+1.}$ on ${\bf C}^{n+1}$. We denote the $n$ form $-\tau_Q$ on ${\bf C}^{n+1}$ such that its restriction to $Q$ is the standard volume form on $\Re Q$
 :
\be
-\tau_Q=\sum_{\nu=1}^{n+1}(-1)^{\nu-1}\xi_\nu
d\xi_1\wedge\cdots\wedge d\xi_{\nu-1}\wedge d\xi_{\nu+1}\wedge\cdots\wedge d\xi_{n+1}
\ee
such that 
$
df_0\wedge \tau_Q\equiv\tau\mod(f_0)
$.
We consider the multiplicative function on $X$
\be
\Phi_0(\xi)=\prod_{j=1}^m f_j(\xi)^{\lambda_j}
\ee
where we assume that every $\lambda_j\in {\bf R}$
is positive and generic.
We denote by $H^r(X-Y,\nabla_0)$ and $H^r(Y,\nabla_0)$
the $r$ dimensional twisted rational de Rham cohomologies on $X-Y$ and $Y$
associated with the covariant differentiation
$\nabla_0$ respectively :
\be
\nabla_0(\psi)=d\psi+d\log\Phi_0\wedge\psi
\ee
These cohomologies are defined in a standard way by using
${\bf C}[\xi_1,\ldots,\xi_{n+1}]$-module 
\be
R=\sum_{\nu_1\ge0,\ldots,\nu_m\ge0}{\bf C}[\xi_1,\ldots,\xi_{n+1}]\prod_{k=1}^{m}f_k({\xi})^{-\nu_k}
\ee
${\cal L}_0$ be the local systems
on $X-Y$ and $Y$ defined by $\Phi_0(\xi)$
respectively, and $\hat{\cal L}_0$
be their duals defined by $\Phi_0(\xi)^{-1}$. Then the $n+1$ and $n$ dimensional homologies $H_{n+1}(X-Y,\hat{\cal L}_0)$ and $H_n(Y,\hat{\cal L}_0)$
represented by twisted cycles are dual to the twisted rational de Rham cohomologies
$H^{n+1}(X-Y,\nabla_0)$ and $H^n(Y,\nabla_0)$ 
thorough the pairs of integrals respectively
\beq
&&H^{n+1}(X-Y,\nabla_0)\times H_{n+1}(X-Y,\hat{\cal L}_0)\ni
(\varphi,\mathfrak{c})\longrightarrow
\langle\varphi,\mathfrak{c}\rangle=\int_{\mathfrak{c}}\Phi\varphi\nonumber\\
\\
&&H^n(Y,\nabla_0)\times H_n(Y,\hat{\cal L}_0)\ni
(\varphi,\mathfrak{c})\longrightarrow
\langle\varphi,\mathfrak{c}\rangle=\int_{\mathfrak{c}}\Phi_0\varphi
\eeq
The following two Propositions have been proved in [2] and [3]
(see Proposition ${\rm 3.2}_p$, ${\rm 3.3}_p$
and Lemma 4.2 in [2]I,
and also [5]). 

%prop2
\begin{prop}
Under the conditions $({\cal H}1),({\cal H}2)$
we have the isomorphism
\be
H^{n}(Y,\nabla_0)\cong {\bf C}^{\kappa_n}
\ee
where $\kappa_n=\sum_{\nu=0}^n {m\choose \nu}+{m-1\choose n}.$
$H^{n}(Y,\nabla_0)$ has a basis represented by the differential $n$ forms
\be
\varphi_Q(I)=\frac{\tau_Q}{f_{i_1}\ldots f_{i_p}}
\ee
where $I$ moves over the admissible sets $I$ of indices 
such that $0\le |I|\le n+1$. We denote $\varphi_Q(\emptyset)$
for $|I|=0$. There exist the fundamental
relations among them of the following type. 
For an arbitrary admissible set of indices $J$ with $|J|=n+2$
there exists the identity :
\beq
&&\frac{1}{2}\sum_{\mu\ne\nu}(-1)^{\mu+\nu}\varphi_Q(\partial_\mu\partial_\nu J)\frac{A(0,\partial_\mu\partial_\nu J)}
{A
\left(
\begin{array}{ccc}
 0,\partial_\mu J   \\
 0,\partial_\nu J    
\end{array}
\right)}
+\sum_{\mu=1}^{n+2}(-1)^{\mu-1}
\varphi_Q(\partial_\mu J)\frac{A(\partial_\mu J)}
{A
\left(
\begin{array}{ccc}
 0,\partial_\mu J   \\
  J    
\end{array}
\right)}\nonumber\\
&&=0
\eeq
where $\partial_\mu J$ denotes the subset of $J$ deleted by
the $\mu$th index $j_\mu.$
Further for $|I|=n+2$ a partial fraction gives
\beq
U\left(
\begin{array}{ccc}
 I   \\
 0,1,\ldots,n+1    
\end{array}
\right)
\varphi_Q(I)=\sum_{\mu=1}^{n+2}(-1)^{\mu-1}
U
\left(
\begin{array}{ccc}
 \partial_\mu I   \\
 1,\ldots,n+1    
\end{array}
\right)\varphi_Q(\partial_\mu I)
\eeq
\end{prop}

We denote by $\cal B$ a linear space spanned by the representatives $\varphi_Q(I),0\le|I|\le n+1.$ 

%prop3
\begin{prop}
Under the condition $({\cal H}1),({\cal H}2a)$
$H_n(Y,\hat{\cal L}_0)$ has a basis represented by 
the closures of all the connected components of 
$\Re Y=\Re Q\cap Y.$
Their number is equal to $\kappa_n.$
In other words, $H_n(Y,\hat{\cal L}_0)$ is spanned by only real twisted cycles defined by connected components of $\Re Y$.
\end{prop}
For example we have 
$
\kappa_1=2m,\,\kappa_2=m^2-m+2,\,\kappa_3=\frac{1}{3}m^3-m^2+\frac{8}{3}m
$.
\begin{re}
The number $\kappa_n$ is also equal to the number of non-compact connected components of ${\bf R}^{n+1}-N(\cal A).$ 
\end{re}


%section3
\section{Twisted imaginary cycles}


We may assume without losing generality 
\beq
u_{j,0}\le0\quad{\rm for\; all}\;j,1\le j\le m
\eeq
Define the set 
\be
S_{j,+}: \{\xi\in \Re Q;  f_j(\xi)>0 \} 
\ee
as the inside of the real hypersphere $S_j=\Re Q\cap H_j.$
We denote by ${\boldsymbol\nu}_j$ the unit normal of
$\Re{H}_j$ :
\beq
{\boldsymbol \nu }_j=\frac{(u_{j,1},u_{j,2},\ldots,u_{j,n+1})}{\sqrt{\sum_{\nu=1}^{n+1}u_{\j,\nu}^2}}
\eeq 
Remark that ${\boldsymbol \nu}_j\in S_{j,+}.$

First notice the following:
%Lemma4
\begin{lm}
Suppose $I$ is admissible. The real affine subspace $\bigcap_{j\in I}\Re H_j$ is disjoint with $\Re Q$ if and only if
$A(I)<0.$
\end{lm}

%proof
\begin{pf}
In fact the square of the distance between the subspace
$\bigcap_{j\in I}\Re H_j$ and the origin is equal to
$\{A(I)+A(0,I)\}/A(0,I)$. It is bigger than $1$ if and only if
$A(I)<0$ because $A(0,I)<0.$
\end{pf}
%corollary
\begin{co}
Suppose that $A(i,j)<0$, i.e., $a_{i,j}^2>1$ for every pair $i,j\in \{1,2,\ldots,m\},i\ne j$
then every $S_{j}$ is disjoint with each other. 
In this case, $S_{i,+},S_{j,+}$ are disjoint, or the one is included in the other, according as 
$a_{i,j}<-1$ or $a_{i,j}>1.$ 
\end{co}



%prop6
\begin{prop}
Under the condition $({\cal H}2)$ consider
an admissible set $I$ such that $2\le |I|\le n+1.$ 
Suppose further $A(I)$ is a positive number near $0$ and that $A(J)<0$ for any admissible $J\supset I,\;
|J|>|I|.$ Then the compact domain
\be
{\mathfrak l}(I) : =\{\xi\in \Re Q; f_j(\xi)\ge0\quad(j\in I)\}
\ee
gives a twisted real cycle representing an element $H_n(Y,\hat{\cal L}_0).$ This cycle vanishes if it is deformed in an isotopic
way
by the matrix $A$ as 
$A(I)$ tend to $0$, any other $A(K)$ being never equal to $0.$  
\end{prop}
\begin{pf}
Since $A(I)>0$ and near $0$, ${\mathfrak l}(I)$ 
is one of the compact components of $\Re Y.$ This reduces to
a point for $A(I)\to0$ as is seen from Lemma 4.
\end{pf}

%def7
\begin{df}
The cycle ${\mathfrak l}(I)$ mentioned in Proposition 6  is called 
\underline{the twisted} \underline{vanishing cycle
(Lefschetz cycle) at the singularity} $A(I)=0.$
\end{df}

Assume now the conditions $({\cal H}2b)$
together with $({\cal H}1)$. Then each $n-1$ dimensional hypersphere $\Re Q\cap H_j$ is disjoint with
each other. This means that $\Re Y$ has only $m+1$ 
connected components which make only a part of the basis
of $H_n(Y,\hat{\cal L}_0)$. We want to construct a basis of
$H_n(Y,\hat{\cal L}_0)$ represented by imaginary cycles
in addition to real ones.


%def8
\begin{df}
We start from an admissible $I=\{i,j\},|I|=2$.
By hypothesis we have $A(I)<0$ i.e.,
$a_{i,j}<-1.$ By an orthogonal transformation we may
choose the new coordinates $\xi=(\xi_1,\ldots,\xi_{n+1})$ 
such that ${\boldsymbol \nu}_i$ coincides with the positive $\xi_1$-axis
and ${\boldsymbol \nu}_j$ lies in the $(\xi_1,\xi_2)$-plane, i.e.,
\be
&&f_i(\xi)=f_i(\xi_1)=u_{i,0}+u_{i,1}\xi_1,\quad (u_{i,1}>0)\\
&&f_j(\xi)=f_j(\xi_1,\xi_2)=u_{j,0}+u_{j,1}\xi_1+u_{j,2}\xi_2
\quad(u_{j,2}>0)
\ee
The set of all points $\xi=(\xi_1,\xi_2,\sqrt{-1}\xi_3^*,\ldots,\sqrt{-1}\xi_{n+1}^*)\in({\bf R}^2\times(\sqrt{-1}{\bf R})^{n-1})\cap Q$
which is a piece of an ultra hyperboloid
\be
\xi_1^2+\xi_2^2-{\xi_3^*}^2-\cdots-{\xi_{n+1}^*}^2=1; f_i(\xi_1)\le0,\,f_j(\xi_1,\xi_2)\le0\}
\ee
is denoted by ${\mathfrak l}^*(I)$.
More generally let $I$ be an admissible set such that $|I|=p,
2\le p\le n.$ We have $A(I)<0.$
We may assume without losing generality
$I=\{1,2,\ldots,p\}$ and choose the new coordinates 
$\xi=(\xi_1,\ldots,\xi_{n+1})$ of ${\bf R}^{n+1}$ such that
${\boldsymbol \nu}_1$ coincides with the positive $\xi_1$-axis and that ${\boldsymbol \nu}_r$ lies in
the $r$ dimensional $(\xi_1,\ldots,\xi_r)$-subspace $(1\le r\le p)$ such that
\beq
f_r(\xi)=f_r(\xi_1,\ldots,\xi_r)=u_{r,0}+\sum_{\nu=1}^r u_{r,\nu}\xi_\nu,\;(u_{r,r}>0)\quad(1\le r\le p)
\eeq
We denote by $\Delta^*(I)$ the set of all points $\xi=(\xi_1,\ldots,\xi_p,\sqrt{-1}\xi_{p+1}^*,\ldots,\sqrt{-1}{\xi_{n+1}^*})\\
\in({\bf R}^p\times( \sqrt{-1}{\bf R})^{n+1-p})\cap Q$
which is a piece of an ultra hyperboloid
\beq
&&\xi_1^2+\cdots+\xi_p^2-{\xi_{p+1}^*}^2-\cdots-{\xi_{n+1}^*}^2=1\\
&&f_r(\xi_1,\ldots,\xi_r)\le 0\quad(1\le r\le p)
\eeq 
and by $\partial_{p,+}\Delta^*(I)$ the set of all points
$\xi=(\xi_1,\ldots,\xi_{p-1},\sqrt{-1}\xi_{p}^*,\ldots,\sqrt{-1}\xi_{n+1}^*)\\
\in({\bf R}^{p-1}\times( \sqrt{-1}{\bf R})^{n+2-p})\cap Q$
which is a piece of an ultra hyperboloid :
\beq
&&\xi_1^2+\cdots+\xi_{p-1}^2-{\xi_{p}^*}^2-\cdots-{\xi_{n+1}^*}^2=1\\
&&f_r(\xi_1,\ldots,\xi_r)\le 0\quad(1\le r\le p-1),\xi_p^*\ge0
\eeq
Remark that 
\be
\Delta^*(I)\cap\partial_{p,+}\Delta^*(I)=
\{\xi_p^*=0\}\cap\partial_{p,+}\Delta^*(I)=\Delta^*(I)\cap
\Delta^*(\partial_pI)
\ee

One can define similarly the chains $\partial_{j,+}\Delta^*(I)$ for $1\le j\le p-1$ by exchange of coordinates
and can have the identities
\be
\Delta^*(I)\cap\partial_{j,+}\Delta^*(I)=\Delta^*(I)\cap\Delta^*(\partial_jI)
\ee
 Then the $n$-chain
\beq
{\mathfrak l}^*(I)=\Delta^*(I)+\sum_{j=1}^p(-1)^{j-1}\partial_{j,+}\Delta^*(I)
\eeq
defines an $n$-cycle in $H_n(Y,\hat{L_0})$.
In fact 
\be
&&\partial{\mathfrak l}^*(I)=\sum_{k=1}^p(-1)^{k-1}\Delta^*(\partial_kI)\cap\Delta^*(I)-\sum_{j=1}^p(-1)^{j-1}\Delta^*(\partial_jI)
\cap \Delta^*(I)\\
&&+
\sum_{k=1,k\ne j}^p(-1)^{k-1}\sum_{j=1}^p
(-1)^{j-1}\partial_{k,+}\partial_{j,+}\Delta^*(I)=0
\ee
since $\partial_{k,+}\partial_{j,+}+\partial_{j,+}\partial_{k,+}=0.$ 

Note that ${\mathfrak l}^*(I)$ coincides with $\Delta^*(I)$
if $|I|=2.$
${\mathfrak l}^*(I)$ is called \underline{twisted}
\underline{ Lefschetz cycle} associated with $I$. If $A(I)$
is near $0$, this is
a deformation of ${\mathfrak l}(I)$ as the matrix $A$ moves from
the part $A(I)>0$ to the one $A(I)<0$ being detoured from
the singularity $A(I)=0$.
\end{df}

The following lemma immediately follows from the above Definition.
%lemma9
\begin{lm}
The number of Lefschetz cycles is equal to
$\sum_{\nu=2}^{m}{m\choose \nu}$.
\end{lm}

Now we construct the $m-1\choose n$ remaining imaginary cycles. Since this number vanishes unless $m-1\ge n,$
we may assume $m\ge n+1.$ 

First consider the case
where $n=1,\,m\ge2.$ By hypothesis every $a_{i,j}^2>1.$
The $2m$ points $\cup_{j=1}^m\Re Q\cap H_j$ are different from each other. Therefore $\Re Y$ consists of $2m$ connected components which make a basis of $H_1(Y,\hat{\cal L}_0).$
None of imaginary cycles occur.
Next consider the case where $n=2,\,m\ge3.$
Suppose $I$ is admissible with $|I|=3.$
Let $\Delta(I)$ be the geodesic triangle in $\Re Q$ with the vertices ${\boldsymbol \nu}_i,{\boldsymbol \nu}_j,{\boldsymbol \nu}_k$. Define the chain $\Delta^*(I)$ as
\be
\Delta^*(I)=\overline{\Delta(I)-S_{i,+}-S_{j,+}-S_{k,+}}
\ee
where the overline $\overline{\Delta-\cdots}$ denotes the closure.

To each geodesic $\widehat{{\boldsymbol \nu}_i,{\boldsymbol \nu}_j}$ going through ${\boldsymbol \nu}_i,{\boldsymbol \nu}_j$, by an orthogonal transformation
there exist the new coordinates $\xi_1,\xi_2,\xi_3$
such that 
${\boldsymbol \nu}_i$ coincides with the $\xi_1$-axis,
the geodesic lies in the $\xi_1,\xi_2$-plane and the inner normal
of the geodesic in $\Delta(I)$ coincides with the positive
$\xi_3$-axis. Then the Lefschetz cycle ${\mathfrak l}^*(\{i,j\})$
is defined as the chain
\be
\{\xi=(\xi_1,\xi_2,\sqrt{-1}\xi_3^*)\in Q\cap({\bf R}^2\times{\sqrt{-1}{\bf R}}), f_i(\xi_1)\le0, f_j(\xi_1,\xi_2)\le0
\}
\ee
The $2$ dimensional cell
\be
&&{\partial}_{k,+}\Delta^*(\{i,j,k\})=\{\xi=(\xi_1,\xi_2,\sqrt{-1}\xi_3^*)\in Q\cap({\bf R}^2\times{\sqrt{-1}{\bf R}}),\\
&&\xi_3^*\ge0,\,f_i(\xi_1)\le0,f_j(\xi_1,\xi_2)\le0\}
\ee 
has the intersection 
$
\overline{\Delta(\{i,j\})-S_{i,+}-S_{j,+}}
$
with $\Delta^*(I).$ ${\partial_{i,+}}\Delta^*(\{i,j,k\}),\\{\partial_{j,+}}\Delta^*(\{i,j,k\})$ can be constructed similarly by exchange of coordinates. We can choose the unique orientations of ${\partial_{i,+}}\Delta^*(\{i,j,k\}),{\partial_{j,+}}\Delta^*(\{i,j,k\}),\\{\partial_{k,+}}\Delta^*(\{i,j,k\})$ such that
the boundaries satisfy
\be
&&\partial\{\Delta^*(I)+{\partial_{i,+}}\Delta^*(\{i,j,k\})-{\partial}_{j,+}\Delta^*(\{i,j,k\})+{\partial_{k,+}}\Delta^*(\{i,j,k\})\}\\
&&\equiv 0\mod S_i\cup S_j\cup S_k
\ee
Hence we have constructed the new $2$ dimensional cycle
\be
{\mathfrak a}^*(\Delta)=\Delta^*(I)+{\partial_{i,+}}\Delta^*(\{i,j,k\})-{\partial}_{j,+}\Delta^*(\{i,j,k\})+{\partial_{k,+}}\Delta^*(\{i,j,k\})
\ee
The above construction can be extended to higher dimensional
cases by induction on dimensions as follows.


Consider now the case where $n\ge3.$
$I$ be an admissible set such that $|I|=n+1$. We may assume
for simplicity that $I=\{1,2,\ldots,n+1\}$.
We have the $n$ dimensional geodesic simplex $\Delta(I)$ with the vertices ${\boldsymbol \nu}_j\,(1\le j\le n+1)$.
in $\Re Q$
such that its edges are geodesic segments and its higher dimensional faces are all totally geodesic.
Define the subdomain $\Delta^*(I)$ as
\be
\Delta^*(I)=\overline{\Delta(I)-\sum_{j=1}^{n+1}S_{j,+}}
\ee
Remark that 
\be
\partial\Delta^*(I)=\sum_{j=1}^{n+1}(-1)^{j-1}\{\Delta(\partial_j I)-\bigcup_{k\ne j}S_{k,+}\}
\ee
For each $j,\,(1\le j\le n+1)$, by an orthogonal transformation there exists the new
coordinates system $\xi=(\xi_1,\ldots,\xi_{n+1})$
such that the $j$ th face 
$\Delta(\partial_jI)$ spanned by the normals
${\boldsymbol \nu}_k,\,(k\ne j)$ included in
the $\xi_1,\ldots,\xi_{n}$-coordinate subspace and that
its inner normal to $\Delta(\partial_jI)$ coincides with the positive 
$\xi_{n+1}$-axis. Suppose for simplicity that $j=n+1$ and 
the subspace spanned by ${\boldsymbol \nu}_1,\ldots,{\boldsymbol \nu}_r$
coincides with the $\xi_1,\ldots,\xi_r$-subspace, i.e.,
\beq
f_r(\xi)=u_{r,0}+\sum_{k=1}^r u_{r,k}\xi_k,\;(u_{r,r}>0)\quad(1\le r\le n)
\eeq
The $n$ dimensional cell
\beq
&&{\partial_{n+1,+}}\Delta^*(I)=\{\xi=(\xi_1,\ldots,\xi_n,\sqrt{-1}\xi_{n+1}^*)\in
Q\cap({\bf R}^n\times \sqrt{-1}{\bf R}),\xi_{n+1}^*\ge0, \nonumber\\
&&f_r(\xi_1,\ldots,\xi_r)\le 0\,(1\le r\le n)\}
\eeq
whose intersection with $\Delta^*(I)$ coincides with
$\overline{\Delta(\partial_{n+1} I)-\bigcup_{k=1}^nS_{k,+}}$. Similarly by exchange of coordinates we can construct the
cell ${\partial}_{j,+}\Delta^*(I))\,(1\le j\le n)$ whose intersection
with $\Delta^*(I)$ coincides with $\overline{\Delta(\partial_{j} I)-\bigcup_{k\ne j}S_{k,+}}$.
Hence there exist a suitable orientation for each
${\partial}_{j,+}\Delta^*(I)$ such that the boundary
vanishes:
\be
\partial\{\Delta^*(I)+\sum_{j=1}^{n+1}(-1)^{j-1}
{\partial}_{j,+}\Delta^*(I)\}\equiv0 \mod \bigcup_{j=1}^{n+1}S_j
\ee
In the same way as above one can prove that the following
chain
\be
{\mathfrak a}^*(I)=\Delta^*(I)+\sum_{j=1}^{n+1}(-1)^{j-1}
{\partial}_{j,+}\Delta^*(I)\in H_n(Y,\hat{\cal L}_0)
\ee
is an $n$ dimensional cycle.

%def10
\begin{df}
The ${\mathfrak a}^*(I)\,(|I|=n+1)$ will be called \underline{twisted adjacent cycle} associated with $I$.
\end{df}
All the cycles thus constructed are not necessarily linearly
independent in $H_n(Y,\hat{\cal L}_0).$ In fact we have
%lemma11
%lemma
\begin{lm}
For an admissible $I$ with $|I|=n+2,$ the following identity
holds:
\beq
\sum_{j=1}^{n+2}(-1)^{j-1}{\mathfrak a}^*(\partial_j I )=0
\eeq
\end{lm}
%proof
\begin{pf}
We may assume that $I=\{1,2,\ldots,n+2\}$. Then
\be
{\mathfrak a}^*(\partial_j I )=\Delta^*(\partial_j I)
+\sum_{k=1}^{j-1}(-1)^{k-1}{\partial}_{k,+}\Delta^*(\partial_j I)
+\sum_{k=j+1}^{n+2}(-1)^{k}{\partial}_{k,+}\Delta^*(\partial_j I)
\ee
On the other hand by definition for $j\ne k$
\be
\partial_{j,+}\Delta^*(\partial_{k}I)=\partial_{k,+}\Delta^*(\partial_{j}I)
\ee
also
\be
\sum_{j=1}^{n+2}(-1)^{j-1}\Delta^*(\partial_j I)=0
\ee
Hence the {\rm LHS} of {\rm (3.11)} equals
\be
\sum_{j=1}^{n+2}(-1)^{j-1}\{\sum_{k=1}^{j-1}(-1)^{k-1}{\partial_{k,+}}\Delta^*(\partial_j I)
+\sum_{k=j+1}^{n+2}(-1)^{k}{\partial_{k,+}}\Delta^*(\partial_j I)\}
=0
\ee
which implies Lemma 11.\quad Q.E.D.

As an immediate consequence we have
\end{pf}
%corollary
\begin{co}
Among ${\mathfrak a}^*(I)\,(|I|=n+1)$, there exist ${m-1\choose n}$ linearly independent ones, say ${\mathfrak a}^*(I), 1\in I$ such that all the others are linear combination
of the latter.
\end{co}



Summing up the above we have proved the following :
%theorem13
\begin{thm}
As a basis of $H_n(Y,\hat{\cal L}_0)$, one can choose the representatives of twisted cycles of the following kinds:

{\rm(i)} Real cycles. This can be realized by
the real chambers which are the closures of the connected components of $\Re Y.$ Their numbers are $1+m.$

{\rm(ii)} Imaginary Lefschetz cycles $\mathfrak l^*(I)$.
Their numbers are equal to $\sum_{\nu=2}^n{m\choose \nu}$.

{\rm(iii)} Adjacent cycles ${\mathfrak a}^*(I)$ such that $1\in I.$ Their numbers
are equal to ${m-1\choose n}.$
\end{thm}


%section4
\section{Stereographic projection}

The cycles defined in the previous section can 
also be described in the $n$ dimensional Euclidean space
as below. 
The compliment of the south pole,
$Q-\{(-1,0,\ldots,0)\}$, is isomorphic to
${\bf R}^n$ through the stereopgraphic projection
\beq
\eta_1=\frac{\xi_2}{1+\xi_1},\ldots,\eta_n=\frac{\xi_{n+1}}{1+\xi_1}
\eeq
which is a conformal transformation.
Then a hypersphere $S$ in $\Re Q$ corresponds to a hypersphere or a hyperplane $\tilde{S}$ in ${\bf R}^{n}$ :
\be
\sum_{\nu=1}^n(\eta_\nu-v_\nu)^2=r^2\quad(r>0)
\ee
where a hyperplane can be regarded as a limiting case
for $r=\infty.$ Denote the center of 
$\tilde{S}$ by ${\bf v}=(v_1,\ldots,v_n)$ and its length by
$||{\bf v}||=\sqrt{\sum_{\nu=1}^nv_\nu^2}.$
Then we have
\be
S: u_0+\sum_{\nu=0}^{n+1}u_\nu \xi_\nu=0,\;(u_{0}\le0)
\ee
\be
u_{0}=\frac{r^2-1-||{\bf v}||^2}{2r},
u_{1}=\frac{r^2+1-||{\bf v}||^2}{2r},
u_{\nu+1}=\frac{v_{\nu}}{r}\;(1\le\nu\le n)
\ee
or
\be
-u_{0}=\frac{r^2-1-||{\bf v}||^2}{2r},
-u_{1}=\frac{r^2+1-||{\bf v}||^2}{2r},
-u_{\nu+1}=\frac{v_{\nu}}{r}\;(1\le\nu\le n)
\ee
according as $r^2-1-||{\bf v}||^2\le0$ or $>0$,
namely
%lemma14
\begin{lm}
$S_{+}$ corresponds to the inside or
the outside of $\tilde{S}$ 
according as  $r^2-1-||{\bf v}||^2<0$ or
$>0.$
\end{lm}

As for $a_{i,j}$
\be
a_{i,j}=\frac{
r_i^2+r_j^2-||{\bf v}^{(i)}-{\bf v}^{(j)}||^2
}
{2r_ir_j}
\ee
where $r_i,r_j,{\bf v}^{(i)},{\bf v}^{(j)}$
denote the radii and the centers of $\tilde{S}_i,\tilde{S}_j$
respectively.
Hence
%lemma15
\begin{lm}
We have
\be
A(i,j)=\frac{(r_i-r_j+a)(-r_i+r_j+a)(r_i+r_j+a)(r_i+r_j-a)}{4r_i^2r_j^2}
\ee
where we put $a=||{\bf v}^{(i)}-{\bf v}^{(j)}||$. This implies
$a_{i,j}>1$ if and only if $|r_i-r_j|>a$. $a_{i,j}<-1$ if and only if $r_i+r_j<a.$
\end{lm}
In the same way
\begin{lm}
Suppose that $|a_{i,j}|<1$ for an admissible $I=\{i,j,k\}$
and put $-\cos \alpha_{i,j}=a_{i,j}$ such that
$0<\alpha_{i,j}<\pi.$ Then
\be
&&A(i,j,k)=-4\cos\frac{\alpha_{i,j}+\alpha_{j,k}+\alpha_{i,k}}{2}\cdot\cos\frac{-\alpha_{i,j}+\alpha_{j,k}+\alpha_{i,k}}{2}\cdot\cos\frac{\alpha_{i,j}-\alpha_{j,k}+\alpha_{i,k}}{2}\\
&&\cdot
\cos\frac{\alpha_{i,j}+\alpha_{j,k}-\alpha_{i,k}}{2}
\ee
The three hyperspheres $\tilde{S}_i,\tilde{S}_j,\tilde{S}_k$ intersect each other. $\pi -\alpha_{i,j}$ is equal to the angle subtended
by the tangents of $\tilde{S}_i,\tilde{S}_j$ at an intersection point
of $\tilde{S}_i\cap \tilde{S}_j$.
$A(1,2,3)=0$ if and only if $\alpha_{i,j}+\alpha_{j,k}+\alpha_{i,k}=\pi,$ or $-\alpha_{i,j}+\alpha_{j,k}+\alpha_{i,k}=\pi,$
or $\alpha_{i,j}-\alpha_{j,k}+\alpha_{i,k}=\pi,$ or
$\alpha_{i,j}+\alpha_{j,k}-\alpha_{i,k}=\pi.$
\end{lm}

%lemma17
\begin{lm}
For an arbitrary admissible $I, |I|\le n+1$
there exist the new coordinates $\eta=(\eta_1,\ldots,\eta_n)$
of ${\bf R}^n$ such that ${\bf v}^{(i_1)}=0$ and
${\bf v}^{(i_{r+1})}$ lies in the $\eta_1,\ldots,\eta_{r}$-subspace $2\le r\le |I|-1$.
\end{lm}
%proof
\begin{pf}
We may assume that $I=\{1,2,\ldots,p\}.$
By the change of coordinates {(4.1)}, there exist the coordinates $\xi_1,\ldots,\xi_{n+1}$
such that ${\boldsymbol \nu}_{r+1}$ 
lies in the $\xi_1,\ldots,\xi_{r+1}$-subspace $(1\le r\le p)$.
Since $u_{1,\nu}=0$ for $\nu\ge2$, ${\bf v}^{(1)}=0.$
And $u_{r,\nu}=0$ for $\nu\ge r+1,$ ${\bf v}^{(r+1)}$ lies in the $\eta_1,\ldots,\eta_r$-subspace.  
\end{pf}
The cycles equivalent to the one constructed in section 3 
are described as follows:
 
Consider the case where $m=2.$ Let $O_1,O_2$ be the centers of
$\tilde{S}_1,\tilde{S}_2$ and the insides of $\tilde{S}_1,\tilde{S}_2$ be denoted by $\tilde{S}_{1,+}\tilde{S}_{2,+}$  respectively. 
Suppose first that $|a_{1,2}|<1.$ Then $\tilde{S}_{1,+}\cap\tilde{S}_{2,+}$ is a non-empty domain so that
${\bf R}^n-\tilde{S}_1\cup\tilde{S}_2$ consists of 4 connected components:
\be
{\bf R}^n-\tilde{S}_{1+}\cup\tilde{S}_{2,+}, 
\tilde{S}_{1,+}-\tilde{S}_{2,+},\tilde{S}_{2,+}-\tilde{S}_{1,+},\tilde{S}_{1,+}\cap\tilde{S}_{2,+}
\ee
 Their closures make
the representatives of a basis of $H_n(Y,\hat{\cal L}_0).$

Suppose that $a_{1,2}<-1.$ Then
$\tilde{S}_{1,+}$ is disjoint with $\tilde{S}_{2,+}.$
We have three real domains $\tilde{S}_{1,+}, \tilde{S}_{2,+},
{\bf R}^n-\tilde{S}_{1+}\cup\tilde{S}_{2,+}.$ On the other hand suppose that $a_{1,2}>1.$ Then $\tilde{S}_{1,+}$
includes or is included in $\tilde{S}_{2,+}.$ 
Assume for example that $\tilde{S}_{1,+}\supset\tilde{S}_{2,+}.$ Then
there are three real domains ${\bf R}^n-\tilde{S}_{1,+},
\tilde{S}_{1,+}-\tilde{S}_{2+},\tilde{S}_{2,+}.$

There is the Lefschetz 
cycle enclosed by two pieces of hyperboloids 
\be
&&\tilde{{\mathfrak l}}(\{1,2\}) : \{\eta=(\eta_1,\sqrt{-1}\eta_2^*,\ldots,\sqrt{-1}\eta_n^*)\in{\bf R}\times(\sqrt{-1}{\bf R})^{n-1};\\
&&\eta_1^2-\sum_{\nu=2}^n{\eta_\nu^*}^2\ge r_1^2, 
(\eta_1-v_1^{(2)})^2-\sum_{\nu=2}^n {\eta_\nu^*}^2\ge r_2^2\}
\ee

More generally suppose that $A(I)<0$ for $|I|=p\,(2\le p\le n).$ 
We may assume that $I=\{1,2,\ldots,p\}.$ 
There exist the new coordinates $\eta=(\eta_1,\ldots,\eta_{n})$ such that $\tilde{S_j}\,(1\le j\le p)$
are defined by
\be
\sum_{\nu=1}^{j-1}(\eta_\nu-v_\nu^{(j)})^2+\sum_{\nu=j}^n\eta_\nu^2=r_j^2
\ee
We define the chain enclosed by $p$ pieces of ultra hyperboloids
\be
&&\tilde{\Delta}^*(I)=\{\eta=(\eta_1,\ldots,\eta_{p-1},
\sqrt{-1}\eta_p^*,\ldots,\sqrt{-1}\eta_{n}^*)\in
{\bf R}^{p-1}\times(\sqrt{-1}{\bf R})^{n-p+1};\\
&&
\sum_{\nu=1}^{j-1}(\eta_\nu-v_\nu^{(j)})^2+\sum_{\nu=j}^{p-1}\eta_\nu^2-\sum_{\nu=p}^n {\eta_\nu^*}^2\ge r_j^2\,(1\le j\le p)\}
\ee
Further we put 
\be
&&\partial_{p,+}\tilde{\Delta^*}(I)=\{\eta=(\eta_1,\ldots,\eta_{p-2},
\sqrt{-1}\eta_{p-1}^*,\ldots,\sqrt{-1}\eta_{n}^*)\in
{\bf R}^{p-2}\\
&&\times(\sqrt{-1}{\bf R})^{n-p+2};
\sum_{\nu=1}^{j-1}(\eta_\nu-v_\nu^{(j)})^2+\sum_{\nu=j}^{p-2}\eta_\nu^2-\sum_{\nu=p-1}^n {\eta_\nu^*}^2\ge r_j^2\,(1\le j\le p-1),\\
&&\eta_{p-1}^*\ge0\}
\ee
which is the chain enclosed by $p-1$ pieces of ultra hyperboloids
and the hyperplane $\eta_{p-1}^*=0$. By exchange of coordinates
one can similarly define the chains $\partial_{k,+}\tilde{\Delta}^*(I)\,(1\le k\le p-1)$. Then 
the Lefschetz cycle $\tilde{{\mathfrak l}}^*(I)$ is defined to be
\be
&&\tilde{{\mathfrak l}}^*(I) =\tilde{\Delta}^*(I)+\sum_{j=1}^p
(-1)^{j-1}\partial_{j,+}\tilde{\Delta}^*(I)
\ee 
Finally suppose $A(I)<0$ for $|I|=n+1.$ 
We may assume $I=\{1,2,\ldots,n+1.\}$ Denote by $\tilde{\Delta}$ the Euclidean $n$-simplex with the vertices ${\bf v}^{(j)}.$
We want to construct a series of chains $\partial_{j,+}\tilde{\Delta}^*(I)$ associated with each face $\tilde{\Delta}(\partial_j I)$ as follows. 
For simplicity we may assume $j=n+1.$
There exist
the coordinates $\eta=(\eta_1,\ldots,\eta_n)$ of ${\bf R}^n$ such that
${\bf v}^{(1)}=0,$ and ${\bf v}^{(j)}$ lies in the $\eta_1,\ldots,\eta_{j-1}\,(2\le j\le n.)$ so that the face $\partial_{n+1}\tilde{\Delta}$ lie in the $\eta_1,\ldots,\eta_{n-1}$-subspace. Define
\be
&&\partial_{n+1,+}\tilde{\Delta}^*(I)=
\{\eta=(\eta_1,\ldots,\eta_{n-1}),\sqrt{-1}\eta_n^*)\in{\bf R}^{n-1}\times\sqrt{-1}{\bf R};\\
&&
\sum_{\nu=1}^{j-1}(\eta_\nu-v_\nu^{(j)})^2+\sum_{\nu=j}^{n-1}\eta_\nu^2-{\eta_n^*}^2\ge r_j^2\,(1\le j\le n),\eta_n^*\ge0
\}
\ee
In the same way one can construct the chains $\partial_{j,+}
\tilde{\Delta}^*(I)\,(1\le j\le n)$
and finally put
\be
\tilde{{\mathfrak a}}^*(I)=\tilde{\Delta}^*(I)+\sum_{j=1}^{n+1}
(-1)^{j-1}\partial_{j,+}\tilde{\Delta}^*(I)
\ee
Then we have
\be
\partial(\tilde{{\mathfrak a}}^*(I))\equiv 0\mod\bigcup_{j=1}^{n+1}\tilde{S}_j
\ee
i.e., $\tilde{{\mathfrak a}}^*(I)\in H_n(Y,\hat{\cal L}_0)$.
Furthermore we have as Lemma 11
%lemma18
\begin{lm}
For any admissible $I$ such that $|I|=n+2,$ 
the identity holds
\be
\sum_{j=1}^{n+2}(-1)^{j-1} \partial(\tilde{{\mathfrak a}}^*(\partial_jI))=0
\ee
\end{lm}
In conclusion we may choose admissible $I$ with $|I|=n+1,$ such that
$1\in I$ so that any other can be a linear combination of them. We have the same conclusion as Theorem 13.

%section5
\section{Degenerate cases}

In Section 3, and 4 we have assumed $({\cal H}1)$ and
$({\cal H}2).$ In this section we discuss the cases
where these conditions are not necessarily satisfied.

First note the following (for example, see [2]I, Lemma 4.2)
:


%
\newcommand{\mapright}[1]{%
\smash{\mathop{%
\hbox to 1cm{\rightarrowfill}}\limits^{#1}}}
\newcommand{\mapleft}[1]{%
\smash{\mathop{%
\hbox to 1cm{\leftarrowfill}}\limits^{#1}}}
%
\newcommand{\mapupdown}[1]{\Big\updownarrow\llap
{$\vcenter{\hbox{$\scriptstyle#1\,$}}$}}

\begin{lm}
We have the commutative diagram :
\[
\begin{array}{ccc}
 H^{n+1}(X-Y;\nabla_0) &\mapright{\rm Res} & H^n(Y;\nabla_0)  \\
 \mapupdown{}&{}&\mapupdown{}   \\
H_{n+1}(X-Y;\hat{L}_0)&
 \mapleft{\delta} 
 & H_n(Y;\hat{L}_0)
\end{array}
\]
where ${\rm Res}$ denotes the Residue along $Y$, and $\delta$
means the boundary operation {\rm(}Leray map{\rm)} into a tubular neighborhood of $Y$ in $X-Y$.
\end{lm}

For an arbitrary $\varphi(\xi)\tau\in R\tau$
such that its representative $\in H^{n+1}(X-Y,\nabla_0),$
denote
\be
\varphi^{(1)}\tau_Q= {\rm Res}\left(\frac{\varphi}{f_0}\tau\right)=\left[\frac{\varphi\tau}{df_0}\right]_Y,\;
\varphi^{(2)}\tau_Q={\rm Res}\left(\frac{\varphi}{f_0^2}\tau\right)
\ee

Then
$\varphi^{(1)}(I)$ is equal to the restriction of
\be
\varphi(I)=\frac{1}{f_{i_1}\cdots f_{i_p}}
\ee
to $Q.$ As for $\varphi^{(2)}(I)$ the following two  recurrence
relations play an important role in the sequel:

%lemma20
\begin{lm}
For an admissible $I$ with $|I|=p\,(0\le p\le n+1)$
\beq
&&A(I)\varphi^{(2)}(I)
\sim\sum_{k\notin I}\lambda_k
A\left(
\begin{array}{ccc}
 0, &I      \\
 k, &I    
\end{array}
\right)\varphi_Q(k,I)
+(\lambda_{\infty}+n-p-1)A(0,I)\varphi_Q(I)\nonumber\\
&&-
\sum_{\nu=1}^p(-1)^{\nu-1} 
A\left(
\begin{array}{ccc}
 I      \\
0,\partial_\nu I    
\end{array}
\right)\varphi^{(2)}(\partial_\nu I)
\eeq
In particular 
\be
&&\varphi^{(2)}(\emptyset)\sim\sum_{k=1}^m \lambda_ka_{k,0}\varphi_Q(k)-(\lambda_\infty+n-1)\varphi(\emptyset),\\
&&\varphi^{(2)}(j)\sim\sum_{k\ne j}\lambda_k
A\left(
\begin{array}{ccc}
 0, &j      \\
 k, &j    
\end{array}
\right)
\varphi_Q(k,j)
-\sum_{k=1}^m \lambda_k
a_{j,0}a_{k,0}
\varphi_Q(k)\\
&&+(\lambda_\infty+n-2)A(0,j)\varphi_Q(j)
+(\lambda_\infty+n-1)a_{j,0}\varphi_Q(\emptyset)
\ee
Therefore $\varphi^{(2)}(I)$ can be described as
a linear combination of $\varphi_Q(J)$ such that
$|J-J\cap I|\le1$ with the coefficients of rational
functions of $a_{i,j},\,a_{k,0}$ whose denominators
are products of $A(K)$ for $K\subset I.$
\end{lm}
For the proof see [2]I,Proposition 4.2.

%lemma21
\begin{lm}
Fix an admissible $I$ with $p=|I|\le n+1.$
Then an arbitrary $\mu,1\le \mu\le p$
\beq
&&(-1)^{\mu-1}(\lambda_{i_\mu}-1)A(I)\frac{\varphi_Q(I)}{f_{i_\mu}}
\sim -\sum_{k\notin I}\lambda_k
A\left(
\begin{array}{ccc}
 I     \\
 k, \partial_\mu I    
\end{array}
\right)
\varphi_Q(k,I)\nonumber\\
&&-(\lambda_\infty+n-p-1)
A\left(
\begin{array}{ccc}
 I     \\
 0, \partial_\mu I    
\end{array}
\right)\varphi_Q(I)
+\sum_{\nu=1}^p (-1)^{\nu-1}
A\left(
\begin{array}{ccc}
 \partial_\mu I     \\
 \partial_\nu I    
\end{array}
\right)\varphi^{(2)}(\partial_\nu I)\nonumber\\
\eeq
\end{lm}
For the proof see [1]I, Proposition 4.2.

Owing to Lemma 20 and 21 an arbitray form $\frac{\tau_Q}{\prod_{k=1}^m
f_k^{\nu_k}}\,(\nu_k\ge0)$ can be described explicitly as a
linear combination of the representatives
of admissible forms $\varphi_Q(I)\tau_Q.$



%prop22
\begin{prop}
In addition to $({\cal H}1)$ suppose the following condition :

$({\cal H}IV(p))$ \quad
For a fixed admissible $I$ with $p=|I|\le n$, $A(I)=0$. But for any other
admissible $J$ such that $|J|\le n+2$ $A(J)\ne0$. 

Then ${\mathfrak l}^*(I)$ vanishes. The dimension of
$H_n(Y,\hat{\cal L}_0)$ decreases by one and is equal to $\kappa_n-1.$
On the other hand the representatives $\varphi_Q(I)$ in {\rm Proposition 2}
does not make a basis of $H^n(Y,\nabla_0).$
We have a linear relation 
\beq
&&(\sum_{j\in I}\lambda_j+\lambda_\infty+n-p-1)
A(0,I)\varphi_Q(I)
+\sum_{k\notin I}\lambda_k A
\left(
\begin{array}{ccc}
 0, &I     \\
 k, &I    
\end{array}
\right)\varphi_Q(k,I)\nonumber\\
&&-
\sum_{k\notin I}\lambda_k\sum_{\nu=1}^p
(-1)^{\nu-1}
\frac{A
\left(
\begin{array}{ccc}
 I &   \\
 0, & \partial_\nu I    
\end{array}
\right)
A\left(
\begin{array}{ccc}
 k, & \partial_\nu I   \nonumber\\
 0, & \partial_\nu I    
\end{array}
\right)}
{A(\partial_\nu I)}
\varphi_Q(k,\partial_\nu I)
\equiv0\nonumber\\
&&\mod{\cal B}(I)
\eeq
where ${\cal B}(I)$ denotes a linear space spanned by
$\varphi_Q(J)$ such that $|J-J\cap I|\le 1$ and $|J|< |I|.$
$H^n(Y,\nabla_0)$ is of dimension $\kappa_n-1$ and is spanned by
$\varphi_Q(J)$ such that $J\ne I$ and $|J|\le n+1$
with the fundamental relations {\rm(2.3), \rm(2.4).} 
\end{prop}
\begin{pf}
In fact since $A(I)=0$, the {\rm LHS} of {\rm(5.1)} vanishes.
A repeated application of {\rm(5.1)} to $\varphi^{(2)}(\partial_\nu I)$ shows the RHS
of {\rm(5.1)} equals the {\rm RHS} of {\rm(5.3)}
in view of the Jacobi identities
\be
A^2\left(
\begin{array}{ccc}
 I &    \\
 0, & \partial_\nu I    
\end{array}
\right)=-A(0,I)A(\partial_\nu I)
\ee
\end{pf}

%prop23
\begin{prop}
In addition to $({\cal H}1)$ suppose the following condition :

${\cal H}IV(n+1)$ \quad
For a fixed admissible $I$ with $|I|=n+1$, $A(I)=0$. But for any other
admissible $J$ such that $|J|\le n+2$ $A(J)\ne0$. Then
${\mathfrak l}^*(I)$ vanishes and $\dim H_n(Y,\hat{L}_0)=\kappa_n-1.$
We have a linear relation
\beq
&&
2(\sum_{j\in I}\lambda_j-1)A(0,I)\varphi_Q(I)
\equiv
\sum_{k\notin I}\sum_{\nu=1}^{n+1}(-1)^{\nu-1}\lambda_k\nonumber\\
&&\cdot\frac{A(0,I)
A\left(
\begin{array}{ccc}
 I  \\
 k, \partial_\nu I    
\end{array}
\right)
A(k,\partial_\nu I)}
{A(k,I)A(\partial_\nu I)}
\varphi_Q(k,\partial_\nu I)
\mod
{\cal B}(I)
\eeq
$H^n(Y,\nabla_0)$ is of dimension $\kappa_n-1$ and is spanned by
$\varphi_Q(J)$ such that $J\ne I$ and $|J|\le n+1$
with the fundamental relations {\rm(2.3), \rm(2.4).} 
\end{prop}
%proof
\begin{pf}
Since $A(I)=0$ the {\rm LHS} of {\rm(5.1)} vanishes.
Applying repeatedly {\rm(2.4)} to $\varphi_Q(k,I)$ and {\rm(5.1)}
to $\varphi^{(2)}(\partial_\nu I)$ one sees that the {\rm RHS} of {\rm(5.1)} equals
\be
&&-2(\sum_{j\in I}\lambda_j-1)A(0,I)\varphi_Q(I)\\
&&+\sum_{k\notin I}\sum_{\nu=1}^{n+1}(-1)^{\nu-1}\lambda_k
\frac
{A\left(\begin{array}{ccc}
0, I  \\
 k, I    
\end{array}
\right)
A\left(\begin{array}{ccc}
 I  \\
 0, \partial_\nu I    
\end{array}
\right)A(k,\partial_\nu I)}
{A(k,I)A(\partial_\nu I)}\varphi_Q(k,\partial_\nu)\mod{\cal B}(I)
\ee
Hence {\rm(5.4)} follows owing to the identities
\be
&&A(k,I)=
-U^2
\left(
\begin{array}{ccc}
 k, I    \\
 0,1,\ldots,n+1   \\   
\end{array}
\right),\\
&&A\left(
\begin{array}{ccc}
 0,I   \\
 k, I    
\end{array}
\right)A\left(
\begin{array}{ccc}
 0,\partial_\nu I  \\
 I    
\end{array}
\right)
=A(0,I)A\left(
\begin{array}{ccc}
 I   \\
 k,  \partial_\nu I    
\end{array}
\right)
\ee
\end{pf}
%corollary24
\begin{co}
Suppose that $m=n+2,n\ge1$ and that $A(I)=0$ for all admissible $I$ with $|I|=n+1.$
Then $H^n(Y,\nabla_0)$ is of dimension 
$\kappa_n-(n+2)=2^{n+2}-n-4$ and is spanned by the representatives
$\varphi_Q(I)$ with $|I|\le n$ with the one fundamental relation: For $J=\{1,2,\ldots,n+2\}$
\beq
\sum_{\mu\ne\nu}(-1)^{\mu+\nu}\varphi_Q(\partial_\nu\partial_\nu J)\frac{A(0,\partial_\mu\partial_\nu J)}
{A
\left(
\begin{array}{ccc}
 0, &\partial_\mu J    \\
 0, &\partial_\nu J     
\end{array}
\right)
}=0
\eeq 
$\varphi_Q(I)\,(|I|=n+1)$ can be expressed as 
\beq
2(\sum_{j\in I}\lambda_j-1)
A(0,I)\varphi_Q(I)\equiv0\mod{\cal B}(I)
\eeq
over the coefficients of rational functions of
$a_{i,j},a_{k,0}$ with the denominators
$A(K)\,(|K|\le n).$
This identity is just an $n$ dimensional version
of {\rm(1.2)}.
\end{co}
%proof
\begin{pf}
{\rm(5.5)} is a special case of {\rm(2.3)}
since $A(\partial_\mu J)=0.$ On the other hand
{\rm(5.6)} is a special case of {\rm(5.4)}
since $A(k,\partial_\nu I)=0.$
\end{pf}
%prop25
\begin{prop}
In addition to $({\cal H}1)$ suppose the following condition :

$({\cal H}IV(n+2))$ \quad
For a fixed admissible $I$ with $|I|=n+2$, $A(I)=0$. But for any other
admissible $J$  such that $|J|\le n+2$
$A(J)\ne0$. 
Then there is no vanishing of Lefschetz cycles and 
$\dim H_n(Y,\hat{\cal L}_0)=\kappa_n.$
On the other hand, for any fixed $\mu$
\beq
(\sum_{\nu\in I}\lambda_\nu-1)
U
\left(
\begin{array}{ccc}
 \partial_\mu I   \\
 1,\ldots,n+1  \\  
\end{array}
\right)
\varphi_Q(I)\equiv0\mod
{\cal B}(I)
\eeq
$H^n(Y,\nabla_0)$ is of dimension $\kappa_n$ and is spanned by
$\varphi_Q(J)$ such that $|J|\le n+1$
with the fundamental relations {\rm(2.3), \rm(2.4).}
\end{prop}
\begin{pf}
By hypothesis the {\rm LHS} of {\rm(2.4)} vanishes. 
By a multiplication
by $f_{i_\mu}$ of both sides of {\rm(2.4)} one sees that $\varphi_Q(I)$ is
linearly dependent on $\frac{\varphi_Q(\partial_\nu I)}{f_{i_\mu}}\,(\nu\ne\mu).$ On the other hand due to {\rm(5.2)}
each $\frac{\varphi_Q(\partial_\nu I)}{f_{i_\mu}}$ is linearly dependent on admissible $\varphi_Q(J)$ with $|J|\le n+1$.
Hence the Proposition follows.
\end{pf}


Finally we consider the special case where $n\ge2, m=n+2$ 
and $A(i,j)=0$, i.e., $a_{i,j}=\pm1$ for all $i,j\in\{1,2,\ldots,n+2\}\,(i\ne j).$
Since the signature of $A$ is of type $(n+1,1)$, we have
$A(1,2)=0, A(1,2,\ldots,p)<0$ if $3\le p\le n+2.$


By a suitable Lorentz transformation we may assume that
$a_{i,j}=-1$ for all $i,j\,(i\ne j).$ In fact
%Lemma26
\begin{lm}
There
exist a diagonal matrix $P$ with diagonal elements equal
to $\pm1$ such that $B=P\cdot A\cdot{}^tP$ is the matrix
with diagonal elements $1$ and off-diagonal elements $-1.$ 
\end{lm}
\begin{pf}
Denote by $B_r$ the matrix of size $r+2$ with diagonal
elements $1$ and off-diagonal elements $-1.$
Let $A_r$ be the matrix with the $(i,j)$ elements
$a_{i,j}\,(1\le i,j\le r+2).$ For $r=0$ the Lemma is
trivial. Suppose that the Lemma is true for $A_{r-1}.$
There exists a diagonal matrix $P_{r-1}$ with diagonal elements
$\pm1$ such that $B_{r-1}=P_{r-1}\cdot A_{r-1}\cdot{}^tP_{r-1}.$
Let $\tilde{P}_r$ be the diagonal matrix of size $r+2$
such that the first $r+1$ diagonal elements coincides with 
the ones of $P_{r-1}$ and the last one equal to $1.$
Then $\tilde{P}_r\cdot A_{r}\cdot{}^t\tilde{P}_r$
has the same components as $B_{r}$ except for
the off-diagonal components in the last column or row.
Denote these components by $\varepsilon_1,\ldots,\varepsilon_{r+1.}$
Then we have
\be
{\rm det}(\tilde{P}_r\cdot A_{r}\cdot ^t\tilde{P}_r)
=(1-r)2^{r}+(r-2)2^{r-1}\sum_{k=1}^{r+1}\varepsilon_k^2
-2^r\sum_{1\le i<j\le r+1}\varepsilon_i\varepsilon_j<0
\ee 
But this inequality goes to a contradiction except for
the case where all $\varepsilon_j$ equal $1$ or 
all $\varepsilon_j$ equal $-1.$ One sees that the first case
is equivalent to $B_{r}$, while the last one coincides with $B_r$. Q.E.D. 
\end{pf}

%lemma27
\begin{lm}
Suppose $I=\{1,2,\ldots,n+2.\}$
The matrix $A$ for all off diagonal elements $a_{i,j}=-1$
defines the hypersphere arrangement 
${\cal A}^\prime$ if and only if 
$\{a_{j,0}\}_j$ satisfy the quadratic relation
\be
(n-1)\sum_{j=1}^{n+2}a_{j,0}^2-2\sum_{1\le j\le k\le n+2}a_{j,0}a_{k,0}+2n=0
\ee
\end{lm}
%proof
\begin{pf}
First remark that if $A(0,I)\le0$ then
$A(0,J)<0$ for $J\subset I,J\ne I.$ In fact 
it is sufficient to show this in case $J=\{1,\ldots,r\}\,(3\le r\le n+1.)$ This follows by lowering induction from the identity
\be
&&A(0,1,\ldots,r)A(1,\ldots,r+1)-A^2
\left(
\begin{array}{ccc}
 0,1,\ldots,r  \\
 r+1, 1,\ldots,r   \\
\end{array}
\right)\\
&&=A(0,1,\ldots,r+1)A(1,\ldots,r)
\ee
because $A(1,\ldots,r),A(1,\ldots,r+1)$ are both negative.
On the other hand
\be
A(0,I)=2^n\{(n-1)\sum_{j=1}^{n+2}a_{j,0}^2-2\sum_{1\le j\le k\le n+2}a_{j,0}a_{k,0}+2n\}
\ee
Hence the Lemma.
\end{pf}
We now apply to it the formula (5.3) for $p=2.$


For $I=\{i,j\}$ we have
\be
A(0,i,j)=-(a_{i,0}+a_{j,0})^2, 
A\left(
\begin{array}{ccc}
 0,i,j   \\
 k,i,j    
\end{array}
\right)=2(a_{i,0}+a_{j,0})
\ee
Hence (5.3) and {\rm Lemma 20} give 
\beq
&&(\lambda_\infty+n-3+\lambda_i+\lambda_j)(a_{i,0}+a_{j,0})\varphi_Q(i,j)+\sum_{k\ne i,j}\lambda_k(a_{k,0}+a_{i,0})\varphi_Q(k,i)\nonumber\\
&&+\sum_{k\ne i,j}\lambda_k(a_{k,0}+a_{j,0})\varphi_Q(k,j)
\sim w_{i,j}
\eeq
where we put
\be
&&w_{i,j}=2\sum_{k\ne i,j}\lambda_k\varphi_Q(k,i,j)
+(a_{i,0}+a_{j,0})\sum_{k=1}^{n+2}\lambda_ka_{k,0}\varphi_Q(k)
-(\lambda_\infty+n-1)\\
&&\cdot(a_{i,0}+a_{j,0})\varphi_Q(\emptyset)
-(\lambda_\infty+n-2)\{A(0,i)\varphi_Q(i)+A(0,j)\varphi_Q(j)\}
\ee
To solve (5.4) with respect to $\varphi_Q(i,j)$ we put $v_{i,j}=(a_{i,0}+a_{j,0})\varphi_Q(i,j)$ and
\be
V_i=\sum_{k\ne i}\lambda_k v_{k,i},\;
V_\infty=\sum_{i\ne j}\lambda_i\lambda_j v_{i,j},\;
W_i=\sum_{k\ne i}\lambda_k w_{k,i},\;
W_\infty=\sum_{i\ne j}\lambda_i\lambda_j w_{i,j}
\ee
Then (5.8) is equivalent to
\beq
(\lambda_\infty+n-3)v_{i,j}+V_i+V_j\sim w_{i,j}
\eeq
(5.9) can be uniquely solved for $v_{i,j}$:
\beq
&&(\lambda_\infty+n-3)v_{i,j}\sim w_{i,j}+
V_\infty\left(\frac{1}{2\lambda_\infty+n-3-2\lambda_i}+
\frac{1}{2\lambda_\infty+n-3-2\lambda_j}\right)\nonumber\\
&&-\left(\frac{W_i}{2\lambda_\infty+n-3-2\lambda_i}+
\frac{W_j}{2\lambda_\infty+n-3-2\lambda_j}\right)
\eeq
where $V_i$ and $V_\infty$ are uniquely determined by
\be
&&(2\lambda_\infty+n-3-2\lambda_i)V_i\sim W_i-V_\infty,\\
&&\left(1+\sum_{k=1}^{n+2}\frac{\lambda_k}{2\lambda_\infty+n-3-2\lambda_k}\right)V_\infty\sim 
\sum_{k=1}^{n+2}\frac{\lambda_k W_k}{2\lambda_\infty+n-3-2\lambda_k}
\ee
provided none of $2\lambda_\infty+n-3-2\lambda_k$
or the symmetric polynomial
\be
G(\lambda)=\prod_{k=1}^{n+2}(2\lambda_\infty+n-3-2\lambda_k)
+\sum_{k=1}^{n+2}\lambda_k\prod_{j\ne k}(2\lambda_\infty+n-3-2\lambda_j)
\ee
vanishes. In this way we can conclude
%prop28
\begin{prop}
For $m=n+2,n\ge2,$ suppose that in addition to $({\cal H}1),$
$a_{i,j}=-1$ for all $i,j\,(i\ne j)$, and $A(I)<0$
for all admissible $I$ with $3\le |I|\le n+2.$
Suppose further that neither of $2\lambda_\infty+n-3-2\lambda_k$ or $\lambda_\infty+n-3$ or $G(\lambda)$
vanish. Then
all the Lefschetz cycles ${\mathfrak l}^*(I)\,(|I|=2)$
vanish.
$H^n(Y,\nabla_0)$ is of dimension $\kappa_n-{n+2\choose2}$
and has a basis of representatives
$\varphi_Q(I)$ with $0\le|I|\le n+1,|I|\ne2$
satisfying the fundamental relations {\rm(2.3),(2.4)}.
$\varphi_Q(i,j)$ can be described as a linear combination
of these representatives as in {\rm(5.10).}
\end{prop}

\section{2 problems}
As is seen from Propositon 2, 
$H^n(Y,\nabla_0)$ is spanned by the representatives
$\varphi_Q(I).$ The result due to Orlik-Terao (see [9]) suggests that this fact still holds
in general in the following sense :

\begin{con}. Without any of the hypotheses  $({\cal H}1)$
or $({\cal H}2)$, $H^n(Y,\nabla_0)$ is spanned by the 
representatives $\varphi_Q(I),\,I\subset\{1,2,\ldots,m\}$ 
including $\varphi_Q(\emptyset).$ 
\end{con}

The complex hypersphere $Q$ has the K\"ahler metric
\be
ds^2=\sum_{\nu=1}^{n+1}|d\xi_\nu|^2=\sum_{\mu,\nu=1}^ng_{\mu,\overline{\nu}}d\zeta^\mu d\overline{\zeta^\nu}
\ee
with respect to local coordinates $\zeta=(\zeta^\nu)_{1\le\nu\le n}.$
We put
\be
\lambda_j=Nl_j+\lambda_j^\prime\quad(N\in {\bf Z}_{>0})
\ee
for fixed $l=(l_j)_j\in ({\bf Z}_{>0})^m, \lambda^\prime=(\lambda_j^\prime)_j\in{\bf R}^m$. For a large $N$ the asymptotic behavior
of the integral (2.2) can be explicitly evaluated 
if the cycle $\bf c$ is a stable cycle defined by the gradient vector field on $Q$ :
\be
d\zeta^\mu=\sum_{\nu=1}^n\frac{\partial}{\partial\overline{\zeta_\nu}}(\sum_{j=1}^m l_j\log|f_j|) g^{\mu,\overline{\nu}}dt
\ee
where $g^{\mu,\overline{\nu}}$ denotes the inverse matrix of
the metric tensor $g_{\mu,\overline{\nu}}$. The critical points are determined by the equations on $Q$
\be
\sum_{j=1}^m l_j\,d\log f_j =0
\ee
Every cycle mentioned in Theorem 13 seems to have one-to-one relation with a stable cycle corresponding to these critical points. This fact suggests :
\begin{con}
All the critical points of the gradient vector field 
lie in $\Re Q$.
\end{con} 


%
\begin{thebibliography}{99}
\bibitem{Ao1} K.Aomoto, On vanishing of cohomology attached to
certain many valued meromorphic functions, Jour. Math. Soc. Japan, 27(1975), 248-255.


\bibitem{Ao1} K.Aomoto, Configuration and invariant
Gauss-Manin connections of integrals I, Tokyo J. Math., 
5(1982), 249-287;  II, {\it ibid}, 6(1983), 1-24.

\bibitem{Ao2} K.Aomoto, Errata to "Configuration and invariant
Gauss-Manin connections of integrals I,II",
Tokyo J. Math.22(1999), 511-512.

\bibitem{Ao3} K.Aomoto, Vanishing of certain 1-form attached to a configuration, Tokyo J. Math., 9(1986), 453-454.



\bibitem{Ao4} K.Aomoto, Gauss-Manin connections of Schl\"afli
type for hypersphere arrangements, Ann.Inst.Fourier, 53(2003),977-995.

\bibitem{AF} K.Aomoto and P.Forrester, On a Jacobian identity
associated with real hyperplane arrangements, Compositio Math. 121(2000), 263-295.

\bibitem{AKOT} K.Aomoto, M.Kita, P.Orlik, and H.Terao,
Twisted de Rham cohomology groups of logarithmic forms,
Ad in Math., 128(1997), 119-152.

\bibitem{Ga} F.Gantmacher, The Theory of Matrices I, Chelsea,
1959. 

\bibitem{OT1} P.Orlik and H.Terao, Commutative algebra for
arrangements, Nagoya Math. J., 134(1994), 65-73.

\bibitem{OT2} P.Orlik and H.Terao, The number of critical points of a product of powers of linear functions, Invent.Math. 120(1995), 1-14.

\bibitem{OT3} P.Orlik and H.Terao, Arrangements and Hypergeometric Integrals, MSJ Memoirs, 9(2001).

\bibitem{Ph} F.Pham, Introduction $\grave{\rm a}$ L'\'etude Topologique
des Singularit\'es de Landau, Gauthiers Villars, 1967. 

\bibitem{Ro} C.A.Rogers, Packing and Covering, Cambridge,
1964. 

\bibitem{Va} A.N.Varchenko, Critical points of the product of powers of linear functions and families of bases of singular vectors, Compositio Math. 97(1995), 385-401.

\end{thebibliography}


\end{document}