# Topological computation of the first Milnor fiber cohomology of hyperplane
arrangements

Morihiko Saito  M. Saito : RIMS Kyoto University, Kyoto 606-8502 Japan
<EMAIL_ADDRESS>
###### Abstract.

We study a topological method to calculate the first Milnor fiber cohomology
of a defining polynomial of a reduced projective hyperplane arrangement $X$ of
degree $d$. We can show the vanishing of a monodromy eigenspace of the first
Milnor fiber cohomology with eigenvalue of order $m\geq 2$ if
$X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle})$ or more generally
$X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle}\cup X_{d})$ is connected. Here
$X^{[(m)]}$ is the set of points of $X$ with multiplicity divisible by $m$,
and $X^{\langle 3\rangle}:=\bigcup_{i,j,k}X_{i}\cap X_{j}\cap X_{k}$ with
$X_{i}$ the irreducible components of $X$, where the union is taken over
$i,j,k$ with ${\rm codim}\,X_{i}\cap X_{j}\cap X_{k}=3$. This hypothesis can
be relaxed to some extent. The assertion is reduced to the case of a line
arrangement in ${\bf P}^{2}$ by Artin’s vanishing theorem (where $X^{\langle
3\rangle}=\emptyset$), and we use a projection from ${\bf P}^{2}$ to ${\bf
P}^{1}$ with center a sufficiently general point of $X_{d}$. It may be
expected that the assumption of an improved assertion is always satisfied for
$m\geq 5$ (and also for $m=4$ except the Hessian arrangement). The resulting
vanishing of eigenspaces has been conjectured for $m\geq 5$.

Introduction

Let $f$ be a reduced polynomial defining a projective hyperplane arrangement
$X\subset Y:={\mathbb{P}}^{n-1}$ of degree $d$. For a calculation of monodromy
eigenspaces of its first Milnor fiber cohomology
$H^{1}(F_{\\!f},{\mathbb{C}})_{\lambda}$ with eigenvalue $\lambda$ of order
$m\,{\geqslant}\,2$, we may assume $d/m\in{\mathbb{Z}}$, since
$H^{j}(F_{\\!f},{\mathbb{C}})_{\lambda}\,{=}\,0$ otherwise, where $F_{\\!f}$
denotes the Milnor fiber of $f$. These eigenspaces are calculated by the
corresponding Aomoto complex  if some hypothesis from [ESV] is satisfied
(although the condition is rather restrictive, and is often unsatisfied), see
[BDS], [BSY], [Fa], [Sa3], [Sa4], etc. So we consider a topological way to
calculate it as in Section 1 below.

Define

$\displaystyle X^{[(m)]}$ $\displaystyle:=\\{P\in X\mid{\rm mult}_{P}X\in
m\hskip 1.0pt{\mathbb{Z}}\\},$ $\displaystyle X^{\langle 3\rangle}$
$\displaystyle:=\hbox{$\bigcup$}_{i,j,k}\,X_{i}\cap X_{j}\cap X_{k},$
$\displaystyle X^{\prime}_{\langle m\rangle}$
$\displaystyle:=X\,{\setminus}\,(X^{[(m)]}\cup X^{\langle 3\rangle}\cup
X_{d}).$

Here the $X_{i}\,\,(i\in[1,d])$ are irreducible components of $X$, and the
union in the definition of $X^{\langle 3\rangle}$ is taken over
$i,j,k\in[1,d]$ with ${\rm codim}_{Y}X_{i}\cap X_{j}\cap X_{k}\,{=}\,3$. There
is a partition

$\hbox{$\bigsqcup$}_{j=1}^{r^{\prime}_{m}}\,I^{\prime(j)}=\\{1,\dots,d{-}1\\},$

such that the $X^{\prime(j)}_{\langle m\rangle}:=\hbox{$\bigcup$}_{i\in
I^{\prime(j)}}\,X_{i}\cap X^{\prime}_{\langle m\rangle}$
($j\in[1,r^{\prime}_{m}]$) are connected components of $X^{\prime}_{\langle
m\rangle}$. Set

$I^{\prime(j)}_{Q}:=\\{i\in[1,d{-}1]\mid X_{i}\ni Q,\,i\in
I^{\prime(j)}\\}]\quad\quad(Q\in X^{\prime}),$

where $X^{\prime}:=X\,{\setminus}\,X_{d}$. We can show the following.

Theorem 1. We have $H^{1}(F_{\\!f},{\mathbb{C}})_{\lambda}\,{=}\,0$ for
$\,{\rm ord}\,\lambda\,{=}\,m\,{\geqslant}\,2$ if $X^{\prime}_{\langle
m\rangle}$ is connected, or if the $X^{\prime(j)}_{\langle m\rangle}$ are
irreducible for $j\in[2,r^{\prime}_{m}]$ with $r^{\prime}_{m}\,{\geqslant}\,2$
and ${\rm
codim}_{Y}\,\hbox{$\bigcap$}_{j=2}^{r^{\prime}_{m}}\,\overline{X^{\prime(j)}_{\langle
m\rangle}}\,{\leqslant}\,2$, assuming also that $X$ is essential $($that is,
it cannot be defined by a polynomial of $n{-}1$ variables$)$.

Theorem 2. Assume for any $k\in[2,r^{\prime}_{m}]$, there is

$Q_{k}\in X^{[(m)]}\cap\overline{X^{\prime(1)}_{\langle
m\rangle}}\cap\overline{X^{\prime(k)}_{\langle
m\rangle}}\,{\setminus}\,(X^{\langle 3\rangle}\cup X_{d}),$

such that $I^{\prime(j)}_{Q_{k}}\,{=}\,\emptyset$ for $j\notin\\{1,k\\}$, and
moreover $|I^{\prime(k)}_{Q_{k}}|$ is relatively prime to $\,m,$ and is at
most $4$ in the $m$ non-prime case. Then
$H^{1}(F_{\\!f},{\mathbb{C}})_{\lambda}\,{=}\,0$ for $\,{\rm
ord}\,\lambda\,{=}\,m\,{\geqslant}\,2$.

Theorem 1 is a special case of Theorem 2, see Remark 2.2 below. Theorems 1–2
are reduced to the case $n\,{=}\,3$ (where $X^{\langle
3\rangle}\,{=}\,\emptyset$) by Artin’s vanishing theorem [BBD] using an
iterated general hyperplane cut. We then study the direct image of a sheaf
complex calculating the vanishing cohomology under a projection to
${\mathbb{P}}^{1}$ from a blow-up of ${\mathbb{P}}^{2}$ along a sufficiently
general point of $X_{d}$, see (2.2), (2.5) and Section 1 below.

A variant of Theorem 1 has been known for instance in the case
$H^{1}(F_{\\!f},{\mathbb{C}})_{\lambda}$ is calculable by the method in [ESV]
mentioned above, see [BDS] and also [Ba], etc. (See also (2.6) below for some
explicit examples.) We have $r^{\prime}_{m}\,{=}\,r_{m}$ or $r_{m}\hskip
1.0pt{-}\hskip 1.0pt1$, changing appropriately the order of the $X_{i}$, where
$r_{m}$ is the number of connected components of $X_{\langle
m\rangle}:=X\,{\setminus}\,(X^{[(m)]}\cup X^{\langle 3\rangle})$. This can be
verified by using the dual $(m)$-graph  of $X$. (Its vertices correspond to
the irreducible components of $X$, and two vertices corresponding to
$X_{i},X_{j}$ are connected by an edge if and only if the multiplicity of $X$
at a general point of $X_{i}\cap X_{j}$ is not  divisible by $m$.) We may have
$r^{\prime}_{m}\,{=}\,r_{m}\hskip 1.0pt{-}\hskip 1.0pt1$ in the case
$X_{\langle m\rangle}$ has an irreducible connected component, see Remark 2.6
(ii) below.

There is no explicit example with hypothesis of Theorem 2 unsatisfied for
$m\,{\geqslant}\,4$ (with $d/m\in{\mathbb{Z}}$) for the moment except the
Hessian  arrangement case, where the dual $(4)$-graph has 4 connected
components (since it is a 4-net), see [BDS], [Di2], [Yu], and (2.6) below. By
Theorems 1–2, calculations of $H^{1}(F_{\\!f},{\mathbb{C}})_{\lambda}$ with
$m\,{=}\,{\rm ord}\,\lambda\,{\geqslant}\,4$ can be reduced effectively to
verifications of the hypothesis of Theorem 1 or 2 in almost all the cases. The
vanishing of $H^{1}(F_{\\!f},{\mathbb{C}})_{\lambda}$ with ${\rm
ord}\,\lambda\,{\geqslant}\,5$ has been conjectured in [PS, Conjecture 1.9],
see also [Sa4, Problem 2].

In Section 1 we study a topological method to describe the Milnor fiber
cohomology of line arrangements. In Section 2 we prove the main theorems.

This work was partially supported by JSPS Kakenhi 15K04816.

1\. Milnor fiber cohomology of line arrangements

In this section we study a topological method to describe the Milnor fiber
cohomology of line arrangements.

1.1. Calculation of the Milnor fiber cohomology. Let $L\subset
Y:={\mathbb{P}}^{2}$ be a reduced line arrangement with $f$ a defining
polynomial of degree $d$. It is well-known that there are local systems
${\mathcal{L}}^{k,}$ of rank 1 on $U:=Y\,{\setminus}\,L\,$ ($k\in[0,d{-}1]$)
such that

$None$ $H^{j}(F_{\\!f},{\mathbb{C}})_{\lambda}\cong
H^{j}(U,{\mathcal{L}}^{k,})\quad\quad\bigl{(}\lambda=\exp(-2\pi\sqrt{-1}\hskip
1.0ptk/d)\bigr{)},$

and the monodromy around any irreducible component of $L$ is given by
multiplication by $\lambda^{-1}=\exp(2\pi\sqrt{-1}\hskip 1.0ptk/d)$. This is
constructed by taking the direct image of the constant sheaf under the natural
morphism $f^{-1}(1)\to U$ which is an unramified covering of degree $d$, see
[Di1], [CS], [BS], etc. In particular, we have
${\mathcal{L}}^{(0)}={\mathbb{C}}_{U}$, and hence
$H^{j}(F_{\\!f},{\mathbb{C}})_{1}=H^{j}(U,{\mathbb{C}})$.

Let $\pi:{\mathcal{Y}}\to Y:={\mathbb{P}}^{2}$ be the blow-up along a
sufficiently general point $P\in L_{d}$ with
$j_{{\mathcal{Y}}}:U\hookrightarrow{\mathcal{Y}}$ a natural inclusion. Here
the $L_{i}$ ($i\in[1,d]$) are the irreducible components of $L$. Set

${\mathcal{F}}:={\mathbf{R}}(j_{{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}[2]\quad\,\,\,\hbox{with}\quad\,\,\,k^{\prime}\,{=}\,d/m\,\,\,\,\,\hbox{or}\,\,\,\,\,d-d/m.$

(Here we use the irreducibility of cyclotomic polynomials implicitly, see also
[Sa4].) This is a direct factor of the underlying ${\mathbb{C}}$-complex of a
mixed Hodge module, and has the weight filtration $W$, see [Sa2], [BS].

Lemma 1.1. We have the isomorphisms

$None$ ${\rm Gr}^{W}_{2}{\mathcal{F}}={\rm
IC}_{{\mathcal{Y}}}{\mathcal{L}}^{(k^{\prime})},\quad\quad{\rm
Gr}^{W}_{i}{\mathcal{F}}=0\,\,\,\,(i\,{\neq}\,2,3).$

Moreover $\,\dim_{{\mathbb{C}}}{\mathcal{H}}^{j}{\rm
Gr}^{W}_{i}\\!{\mathcal{F}}_{y}=km\hskip 1.0pt{-}\hskip 1.0pt2$ if
$\,\pi(y)\in L^{[km]}$ $(k\in{\mathbb{Z}}_{>0})\,$ with $\,j\,{=}\,i\hskip
1.0pt{-}\hskip 1.0pt3\,{=}\,{-}1\,$ or $\,0$, and it vanishes otherwise. Here
$L^{[r]}:=\\{P\in L\mid{\rm mult}_{P}L\,{=}\,r\hskip 1.0pt\\}$.

Proof. Let $\sigma:\widetilde{\mathcal{Y}}\to{\mathcal{Y}}$ be the blow-up
along all the points of $\pi^{-1}({\rm Sing}\,L)$ with
$j_{\widetilde{\mathcal{Y}}}:U\hookrightarrow\widetilde{\mathcal{Y}}$ the
inclusion. Then (1.1.2) holds with ${\mathcal{Y}}$, ${\mathcal{F}}$ replaced
by $\widetilde{\mathcal{Y}}$,
${\mathbf{R}}(j_{\widetilde{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}[2]$,
and we have the isomorphism

$None$ ${\rm
IC}_{\widetilde{\mathcal{Y}}}{\mathcal{L}}^{(k^{\prime})}[-2]=(j_{\widetilde{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}.$

Moreover the restriction of
$R^{j}(j_{\widetilde{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}$ to
$\sigma^{-1}(y)\cong{\mathbb{P}}^{1}$ with $\pi(y)\in{\rm Sing}\,L$ is the
direct image of a local system of rank 1 on the complement of $km$ points in
${\mathbb{P}}^{1}$ (whose local monodromies are non-trivial ) if $\pi(y)\in
L^{[km]}$ with $j\,{=}\,0$ or 1, and it vanishes otherwise. Since the Euler
number of the complement of $km$ points in ${\mathbb{P}}^{1}$ is $2\hskip
1.0pt{-}\hskip 1.0ptkm$, we then get the last assertions of Lemma 1.1 together
with the isomorphism

$None$ ${\mathbf{R}}\sigma_{*}{\rm
IC}_{\widetilde{\mathcal{Y}}}{\mathcal{L}}^{(k^{\prime})}={\rm
IC}_{{\mathcal{Y}}}{\mathcal{L}}^{(k^{\prime})},$

using the decomposition theorem [BBD] (see also [Sa1]). This finishes the
proof of Lemma 1.1.

1.2. Generic projection. There is a natural projection $\rho:{\mathcal{Y}}\to
C:={\mathbb{P}}^{1}$ so that ${\mathcal{Y}}$ is a ${\mathbb{P}}^{1}$-bundle
over $C$, and the exceptional divisor of the blow-up is the zero-section.
(Here $C$ is identified with the set of lines on ${\mathbb{P}}^{2}$ passing
through $P$.) There is $c_{\infty}\in C$ such that

$\pi\bigl{(}\rho^{-1}(c_{\infty})\bigr{)}=L_{d}.$

There are isomorphisms

$Y^{\prime}:=Y\,{\setminus}\,L_{d}\cong{\mathbb{C}}^{2},\quad\quad
C^{\prime}:=C\,{\setminus}\,\\{c_{\infty}\\}\cong{\mathbb{C}}.$

Let $\rho^{\prime}:Y^{\prime}\to C^{\prime}$ be the restriction of $\rho$ to
$Y^{\prime}\,\bigl{(}={\mathcal{Y}}\,{\setminus}\,\pi^{-1}(L_{d})\bigr{)}\cong{\mathbb{C}}^{2}$.
Then $\rho^{\prime}$ is identified with a linear function with a constant term
by the above isomorphisms.

Since $P\in L_{d}$ is sufficiently general, we may assume the following:

$None$ The restriction of $\rho^{\prime}$ to ${\rm
Sing}\,L\,{\setminus}\,L_{d}$ is injective.

Set

$C^{\prime\prime}:=C^{\prime}\,{\setminus}\,\rho^{\prime}({\rm
Sing}\,L\,{\setminus}\,L_{d}).$

Lemma 1.2. We have the vanishing

$None$ $R^{j}\rho_{*}{\rm Gr}^{W}_{i}\\!{\mathcal{F}}=0\,\,\,\,\,\hbox{unless
$\,(i,j)\,{=}\,(2,-1)\,$ or $\,(3,0)$}.$

For $c\in C^{\prime}\cong{\mathbb{C}}$, there are equalities

$None$ $\dim_{{\mathbb{C}}}(R^{-1}\rho_{*}{\rm
Gr}^{W}_{2}\\!{\mathcal{F}})_{c}=\begin{cases}d\hskip 1.0pt{-}\hskip
1.0pt2&\hbox{\it if}\,\,\,\,c\in C^{\prime\prime}\\\ d\hskip 1.0pt{-}\hskip
1.0pt3&\hbox{\it
if}\,\,\,\,c\in\rho^{\prime}(L^{[k]}),\,\,k/m\in{\mathbb{Z}},\\\ d\hskip
1.0pt{-}\hskip 1.0ptk\hskip 1.0pt{-}\hskip 1.0pt1&\hbox{\it
if}\,\,\,\,c\in\rho^{\prime}(L^{[k]}),\,\,k/m\notin{\mathbb{Z}},\end{cases}$

Proof. The assertion for ${\rm Gr}^{W}_{3}\\!{\mathcal{F}}$ follows from Lemma
1.1. For ${\rm Gr}^{W}_{2}\\!{\mathcal{F}}$, it is enough to calculate

$None$ $H^{j}({\mathcal{Y}}_{c},{\rm
IC}_{{\mathcal{Y}}}{\mathcal{L}}^{(k^{\prime})}|_{{\mathcal{Y}}_{c}})=H^{j}(\widetilde{\mathcal{Y}}_{c},{\rm
IC}_{\widetilde{\mathcal{Y}}}{\mathcal{L}}^{(k^{\prime})}|_{\widetilde{\mathcal{Y}}_{c}}),$

using (1.1.4) (since $\rho$, $\sigma$ are proper), where
${\mathcal{Y}}_{c}:=\rho^{-1}(c)$,
$\widetilde{\mathcal{Y}}_{c}:=\sigma^{-1}({\mathcal{Y}}_{c})$. We see that
(1.2.4) vanishes for $j\,{\neq}\,1$, and its dimension is as in (1.2.3) by an
argument similar to the proof of Lemma 1.1. This finishes the proof of Lemma
1.2.

1.3. Description of the first Milnor fiber cohomology. Let
$j_{C^{\prime}}:C^{\prime}\hookrightarrow C$ be a natural inclusion. Set

${\mathcal{G}}:={\mathbf{R}}\rho_{*}{\mathcal{F}}={\mathbf{R}}(j_{C^{\prime}})_{*}{\mathcal{G}}^{\prime}\quad\hbox{with}\quad{\mathcal{G}}^{\prime}:={\mathcal{G}}|_{C^{\prime}}.$

(This is equal to ${}^{\bf
p}{\mathcal{H}}^{0}{\mathbf{R}}\rho_{*}{\mathcal{F}}\,$ in the notation of
[BBD].) We have the weight filtration $W$ on ${\mathcal{G}}^{\prime}$ with

$None$ ${\rm Gr}^{W}_{i}{\mathcal{G}}^{\prime}=({\mathbf{R}}\rho_{*}{\rm
Gr}^{W}_{i}\\!{\mathcal{F}})|_{C^{\prime}}\quad\quad(i\,{=}\,2,3).$

We have

$None$ ${\rm Supp}\,{\rm Gr}^{W}_{3}{\mathcal{G}}^{\prime}\subset
C^{\prime}\,{\setminus}\,C^{\prime\prime},\quad{\mathcal{H}}^{j}{\rm
Gr}^{W}_{3}{\mathcal{G}}^{\prime}=0\,\,\,\,(j\,{\neq}\,0),$

and ${\rm Gr}^{W}_{2}{\mathcal{G}}^{\prime}$ is an intersection complex with
coefficients in a local system. More precisely, ${\rm
Gr}^{W}_{2}{\mathcal{G}}^{\prime}[-1]$ is a sheaf in the usual sense on
$C^{\prime}$, and is the open direct image of a local system
${\mathcal{G}}^{\prime\prime}$ of rank $d{-}2$ on $C^{\prime\prime}$, where
${\mathcal{G}}^{\prime\prime}:={\mathcal{G}}^{\prime}[-1]|_{C^{\prime\prime}}$.
(This can be shown using the decomposition theorem [BBD] or more directly by
Lemma 1.4 below.)

We thus get the following.

Proposition 1.3. There are isomorphisms

$None$ $(R^{-1}\rho_{*}{\rm
Gr}^{W}_{2}\\!{\mathcal{F}})|_{C^{\prime}}={\mathcal{H}}^{-1}{\mathcal{G}}^{\prime},$
$None$ $H^{1}(F_{\\!f},{\mathbb{C}})_{\lambda}\cong
H^{-1}(C^{\prime},{\mathcal{G}}^{\prime})=\Gamma(C^{\prime\prime},{\mathcal{G}}^{\prime\prime}),$

where $\lambda=\exp(\pm 2\pi\sqrt{-1}/m)$. $($Note that the monodromy is
defined over ${\mathbb{Q}}.)$

1.4. Description of the vanishing cycles. We have the following.

Lemma 1.4. Let
$c\in\rho^{\prime}(L^{[k]})\,{\setminus}\,\\{c_{\infty}\\}\,\subset\,C^{\prime}\cong{\mathbb{C}}$.
Then

$None$
$\dim_{{\mathbb{C}}}\varphi_{t-c,\,1}\,{\mathcal{G}}^{\prime\prime}=1,\,\,\,\,\dim_{{\mathbb{C}}}\varphi_{t-c,\,\neq
1}\,{\mathcal{G}}^{\prime\prime}=0\quad\hbox{if}\quad k/m\in{\mathbb{Z}}.$
$None$
$\dim_{{\mathbb{C}}}\varphi_{t-c,\,1}\,{\mathcal{G}}^{\prime\prime}=0,\,\,\,\,\dim_{{\mathbb{C}}}\varphi_{t-c,\,\neq
1}\,{\mathcal{G}}^{\prime\prime}=k{-}1\quad\hbox{if}\quad
k/m\notin{\mathbb{Z}}.$

Here $\,\varphi_{t-c,\,1}$ and $\,\varphi_{t-c,\,\neq 1}\,$ denote
respectively the unipotent and non-unipotent monodromy part of the vanishing
cycle functor $\varphi_{t-c}$ with $t$ the coordinate of
$C^{\prime}\cong{\mathbb{C}}$.

Proof. This is shown by calculating the vanishing cycles
$\varphi_{\widetilde{\rho}^{*}t-c}(j_{\widetilde{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}$
on $\widetilde{\mathcal{Y}}$ using (1.1.3–4) together with the commutativity
of the nearby and vanishing cycle functors with the direct image under a
proper morphism, where $\widetilde{\rho}:=\rho\,\raise
0.6458pt\hbox{${\scriptstyle\circ}$}\,\sigma$.

In the case $k/m\in{\mathbb{Z}}$, the direct image sheaf
$(j_{\widetilde{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}$ is a local
system on the complement of the proper transform of $L$ in an open
neighborhood of $\,\sigma^{-1}(y)$ with $\pi(y)\in L^{[k]}$. So (1.4.1)
follows.

In the case $k/m\notin{\mathbb{Z}}$, the restriction of
$\varphi_{\widetilde{\rho}^{*}t-c,\,\neq
1}(j_{\widetilde{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}$ to
$\,\sigma^{-1}(y)$ with $\pi(y)\in L^{[k]}$ is the direct image of a local
system of rank 1 on the complements of $k{+}1$ points in ${\mathbb{P}}^{1}$
(where the local monodromies are non-trivial), and we have the vanishing

$\varphi_{\widetilde{\rho}^{*}t-c,\,1}(j_{\widetilde{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}=0,$

since we can verify the isomorphism

$(j_{\widetilde{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}|_{\\{\widetilde{\rho}^{*}t=c\\}}\,\,\hbox
to0.0pt{\hskip 4.2679pt\raise
3.98337pt\hbox{$\sim$}\hss}\hbox{$\longrightarrow$}\,\,\psi_{\widetilde{\rho}^{*}t-c,\,1}(j_{\widetilde{\mathcal{Y}}})_{*}{\mathcal{L}}^{(k^{\prime})}.$

So we get (1.4.2). This finishes the proof of Lemma 1.4.

1.5. Description of a general stalk of ${\mathcal{G}}^{\prime\prime}$. Let
$c\in C^{\prime\prime}$. Set

$Z_{c}:=\pi\bigl{(}\rho^{-1}(c)\bigr{)}\subset Y,\quad
Z^{\prime}_{c}:=Z_{c}\,{\setminus}\,L,\quad{\mathcal{E}}^{\prime}_{c}:={\mathcal{L}}^{(k^{\prime})}|_{Z^{\prime}_{c}},$

so that

$None$
${\mathcal{G}}^{\prime\prime}_{c}=H^{1}(Z^{\prime}_{c},{\mathcal{E}}^{\prime}_{c}).$

Since $Z_{c}\cong{\mathbb{P}}^{1}$ and $d/m\in{\mathbb{Z}}$, we have a
ramified covering

$p_{c}:\widetilde{Z}_{c}\to Z_{c}$

of degree $m$, which is unramified over $Z^{\prime}_{c}$, and is totally
ramified  over

$\Xi_{c}:=Z_{c}\cap L,$

that is, $|p_{c}^{-1}(z)|=1$ if $z\in\Xi_{c}$. Here $\widetilde{Z}_{c}$ is
normal, and hence smooth. Let

$p^{\prime}_{c}:\widetilde{Z}^{\prime}_{c}\to Z^{\prime}_{c}$

be the restriction of $p_{c}$ over $Z^{\prime}_{c}$, which is an unramified
covering of degree $m$. We then get an isomorphism

$None$
${\mathcal{E}}^{\prime}_{c}\cong\bigl{(}(p^{\prime}_{c})_{*}{\mathbb{C}}_{\widetilde{Z}^{\prime}_{c}}\bigr{)}^{\gamma^{*}=\lambda^{\prime}},$

with right-hand side the direct factor of
$(p^{\prime}_{c})_{*}{\mathbb{C}}_{\widetilde{Z}^{\prime}_{c}}$ on which the
action of a generator $\gamma$ of the covering transformation group of $p_{c}$
coincides with multiplication by $\lambda^{\prime}:=\exp(2\pi\sqrt{-1}\hskip
1.0pt/m)$ (replacing $\gamma$ if necessary). Here $\gamma^{*}$ is semi-simple,
since it has finite order. (Note that a local system of rank 1 on a Zariski-
open subset of ${\mathbb{P}}^{1}$ is determined by the local monodromies.)

Since $p_{c}$ is totally ramified at $\Xi_{c}$, we get by (1.5.1–2) the
following.

Proposition 1.5. We have the isomorphism

$None$ ${\mathcal{G}}^{\prime\prime}_{c}\cong
H^{1}(\widetilde{Z}_{c},{\mathbb{C}})^{\gamma^{*}=\lambda^{\prime}}.$

(The right-hand side denotes the direct factor of
$H^{1}(\widetilde{Z}_{c},{\mathbb{C}})$ on which the action of $\gamma^{*}$
coincides with multiplication by $\lambda^{\prime}$. Note that the
isomorphisms in (1.5.2–3) are canonical up to constant multiplication by
${\mathbb{C}}^{*}$.)

2\. Proof of Theorems 1 and 2

In this section we prove the main theorems.

2.1. Description of the right-hand side of (1.5.3). We can describe the right-
hand side of (1.5.3) using the isomorphism by Poincaré duality:

$None$
$H^{1}(\widetilde{Z}_{c},{\mathbb{C}})=H^{1}(\widetilde{Z}_{c},{\mathbb{C}})^{\vee}=H_{1}(\widetilde{Z}_{c},{\mathbb{C}}).$

Note that for $\tau,\tau^{\prime}\in H^{1}(\widetilde{Z}_{c},{\mathbb{C}})$,
$\sigma\in H_{1}(\widetilde{Z}_{c},{\mathbb{C}})$, we have

$None$
$\langle\gamma^{*}\tau,\gamma^{*}\tau^{\prime}\rangle=\langle\tau,\tau^{\prime}\rangle,\quad\langle\gamma^{*}\tau,\sigma\rangle=\langle\tau,\gamma_{*}\sigma\rangle.$

Let $z_{i}\in Z_{c}$ with $\\{z_{i}\\}=Z_{c}\cap L_{i}$ for each line
$L_{i}\subset L$ (where $z_{d}=P$). Note that
$\Xi_{c}=\\{z_{1},\dots,z_{d}\\}$. Choose a path $\xi_{i}$ between $z_{d}$ and
$z_{i}$, which is contained in $Z^{\prime}_{c}$ except the both ends. Here we
may assume $\xi_{i}\cap\xi_{j}=\\{z_{d}\\}$ ($i\,{\neq}\,j$). We have the
following.

Lemma 2.1. Choosing $z_{\infty}\in Z_{c}$, its complement
$Z_{c}\,{\setminus}\,\\{z_{\infty}\\}$ is identified with ${\mathbb{C}}$, and
the $\xi_{i}$ can be defined as the closed line segment with endpoints
$z_{i}$, $z_{d}$ in the $2$-dimensional real affine space ${\mathbb{C}}$, if
$P,c$ are sufficiently general.

Proof. Let $\ell_{i}$ be the half line in the real 2-dimensional affine space
$Z_{c}\,{\setminus}\,\\{z_{\infty}\\}\cong{\mathbb{C}}$ with endpoint $z_{d}$
and containing $z_{i}$. (Recall that any algebraic automorphism of the complex
algebraic variety ${\mathbb{C}}\subset{\mathbb{P}}^{1}$ is given by
$\alpha\hskip 1.0ptx\hskip 1.0pt{+}\hskip 1.0pt\beta$ with
$\alpha\in{\mathbb{C}}^{*},\beta\in{\mathbb{C}}$, where $x$ is an algebraic
coordinate of ${\mathbb{C}}\subset{\mathbb{P}}^{1}$. This says also that the
ambiguity of algebraic coordinate of ${\mathbb{C}}\subset{\mathbb{P}}^{1}$ is
given by it.) We have to show that $\ell_{i}\,{\neq}\,\ell_{j}$
($i\,{\neq}\,j$) if $P,c$ are sufficiently general.

Take coordinates $x,y$ of
${\mathbb{C}}^{2}\cong{\mathbb{P}}^{2}\,{\setminus}\,L_{d}$ such that the
equation of $L^{\prime}_{i}:=L_{i}\,{\setminus}\,L_{d}\subset{\mathbb{C}}^{2}$
($i\in[1,d{-}1]$) can be written as

$y=a_{i}x-b_{i}\quad(a_{i},b_{i}\in{\mathbb{C}}),$

that is, $L^{\prime}_{i}\cap\\{x\,{=}\,0\\}\,{\neq}\,\emptyset$. We consider
only $P,c$ such that the equation of
$Z_{c}\,{\setminus}\,\\{z_{d}\\}=\rho^{\prime}{}^{-1}(c)$ is written as

$y=\alpha\hskip 1.0ptx-\beta\quad(\alpha,\beta\in{\mathbb{C}}).$

Here the ambiguity (or freedom) of $P,c$ corresponds to that of $\alpha,\beta$
respectively. We will show that $\ell_{i}\,{\neq}\,\ell_{j}$ ($i\,{\neq}\,j$)
if $\beta$ is sufficiently general for each $\alpha$ with (1.2.1) satisfied.

We define $z_{\infty}$ to be the intersection of
$Z_{c}\,{\setminus}\,\\{z_{d}\\}$ with
$\\{x\,{=}\,0\\}\subset{\mathbb{C}}^{2}$. Then $x^{-1}$ can be extended to a
complex coordinate of $Z_{c}\,{\setminus}\,\\{z_{\infty}\\}\cong{\mathbb{C}}$
with origin $z_{d}$. So it is enough to show that

$None$ $\arg x(z_{i})\,{\neq}\,\arg x(z_{j})\quad(i\,{\neq}\,j),$

where $x(z_{i})$ is the value of the coordinate $x$ at $z_{i}$. Put

$g_{i}(\beta):=\arg(\beta\hskip 1.0pt{-}\hskip 1.0ptb_{i}),\quad
h_{i}(\alpha):=\arg(\alpha\hskip 1.0pt{-}\hskip 1.0pta_{i}).$

Since $\\{z_{i}\\}=L_{i}\cap Z_{c}$, we get

$x(z_{i})=(\beta\hskip 1.0pt{-}\hskip 1.0ptb_{i})/(\alpha\hskip 1.0pt{-}\hskip
1.0pta_{i})$

from the above equations. So condition (2.1.3) is equivalent to that

$None$ $g_{i}(\beta)\hskip 1.0pt{-}\hskip 1.0pth_{i}(\alpha)\neq
g_{j}(\beta)\hskip 1.0pt{-}\hskip 1.0pth_{j}(\alpha)\quad(i\,{\neq}\,j),$

assuming that $\beta$ is sufficiently general. Here “sufficiently general”
means that the set of $\beta$ satisfying (2.1.4) is a dense open subset of
${\mathbb{C}}$. (Recall that $\,\arg x\,$ for $x\in{\mathbb{C}}$ is the
imaginary part of a brach of $\log x$ contained in $[0,2\pi)$.)

Set $\,\Sigma_{i,r_{i}}:=\\{g_{i}(\beta)\,{=}\,r_{i}\\}\subset{\mathbb{C}}$
for $r_{i}\in[0,2\pi)$. (Note that $\Sigma_{i,r_{i}}$ is a real half line with
endpoint $b_{i}$.) Let $U_{\beta}\subset{\mathbb{C}}$ be the complement of the
union of real lines $\ell_{i,j}$ in ${\mathbb{C}}$ passing through
$b_{i},b_{j}$ ($i\,{\neq}\,j$). Then $\Sigma_{i,r_{i}}\cap\Sigma_{j,r_{j}}\cap
U_{\beta}$ is discrete, and $\Sigma_{j,r_{j}}\cap U_{\beta}$ is not contained
in $\Sigma_{i,r_{i}}\cap U_{\beta}$ for $r_{i},r_{j}\in[0,2\pi)$. This implies
that we can change the left-hand side of (2.1.4) slightly by changing $\beta$
with right-hand side unchanged. We can then get (2.1.4) by induction of the
number of the $i$ such that (2.1.4) does not hold for some $j$. Each change of
$\beta$ becomes much smaller and smaller as induction goes up so that once we
get (2.1.4) for some $i,j$, we never loose it by changing $\beta$ further.
This finishes the proof of Lemma 2.1.

Using $\xi_{i}$ in Lemma 2.1 together with a sufficiently small circle around
$z_{i}$, we get an element

$None$
$\eta_{i}\in\pi_{1}(Z_{c}\,{\setminus}\,\Xi^{\prime}_{c},z_{d})\quad\hbox{with}\quad\Xi^{\prime}_{c}:=\Xi_{c}\,{\setminus}\,\\{z_{d}\\}.$

This can be lifted non-canonically to an element

$None$
$\widetilde{\eta}_{i}\in\pi_{1}(\widetilde{Z}_{c}\,{\setminus}\,\widetilde{\Xi}^{\prime}_{c},\widetilde{z}_{d})\quad\hbox{with}\quad
p_{c}(\widetilde{\Xi}^{\prime}_{c})=\Xi_{c},\,\,p_{c}(\widetilde{z}_{d})=z_{d}.$

Its ambiguity is given by the action of the covering transformation group
$\Gamma_{p_{c}}\cong\mu_{m}$ of $p_{c}$. Set
$\lambda^{\prime\prime}:=\lambda^{\prime}{}^{-1}$. We denote by $\sigma_{i}$
the image of $\widetilde{\eta}_{i}$ in $H_{1}(\widetilde{Z}_{c},{\mathbb{C}})$
(viewed as a topological cycle), and by $\sigma^{\lambda^{\prime\prime}}_{i}$
its image in
$H_{1}(\widetilde{Z}_{c},{\mathbb{C}})^{\gamma_{*}=\lambda^{\prime\prime}}$
using the eigenvalue decomposition by the action of $\gamma_{*}$ on
$H_{1}(\widetilde{Z}_{c},{\mathbb{C}})$. The ambiguity of
$\sigma^{\lambda^{\prime\prime}}_{i}$ is then given by the natural action of
$\mu_{m}\subset{\mathbb{C}}^{*}$ (by scalar multiplication), since the
isomorphism $\Gamma_{p_{c}}\cong\mu_{m}$ is given by the generator
$\gamma\in\Gamma_{p_{c}}$. We have the following.

Proposition 2.1. For $\,j\in[1,d{-}1]$, set
$\,I_{(j)}:=\\{1,\dots,d{-1}\\}\,{\setminus}\,\\{j\\}$. Then the
$\,\sigma^{\lambda^{\prime\prime}}_{i}\,\,(i\in I_{(j)})$ form a
${\mathbb{C}}$-basis of
$H_{1}(\widetilde{Z}_{c},{\mathbb{C}})^{\gamma_{*}=\lambda^{\prime\prime}}\,(=H^{1}(\widetilde{Z}_{c},{\mathbb{C}})^{\gamma^{*}=\lambda^{\prime}})$.

Proof. Since $\dim
H_{1}(\widetilde{Z}_{c},{\mathbb{C}})^{\gamma_{*}=\lambda^{\prime\prime}}\,{=}\,d\hskip
1.0pt{-}\hskip 1.0pt2$ by (1.5.3), the assertion is reduced to showing that
the $\sigma^{\lambda^{\prime\prime}}_{i}$ ($i\,{\neq}\,j$) span
$H_{1}(\widetilde{Z}_{c},{\mathbb{C}})^{\gamma_{*}=\lambda^{\prime\prime}}$
for any $j\in[1,d{-}1]$. For this it is enough to prove that the $\sigma_{i}$
($i\,{\neq}\,j$) generate $H_{1}(\widetilde{Z}_{c},{\mathbb{C}})$ over the
group ring ${\mathbb{C}}[\Gamma_{p_{c}}]$. (Note that
$\Gamma_{p_{c}}\cong\mu_{m}$ is generated by $\gamma$.) Here we have the
surjection

$None$
$H_{1}(\widetilde{Z}_{c}\,{\setminus}\,\\{\widetilde{z}_{j}\\},{\mathbb{C}})\twoheadrightarrow
H_{1}(\widetilde{Z}_{c},{\mathbb{C}}).$

We see that $\bigcup_{i\neq j}\xi_{i}$, $\bigcup_{i\neq j}\widetilde{\xi}_{i}$
are deformation retracts of $Z_{c}\,{\setminus}\,\\{z_{j}\\}$ and
$\widetilde{Z}_{c}\,{\setminus}\,\\{\widetilde{z}_{j}\\}$ respectively, where
$\widetilde{\xi}_{i}:=p_{c}^{-1}(\xi_{i})$. (Indeed, choosing $z_{\infty}\in
Z_{c}$, its complement $Z_{c}\,{\setminus}\,\\{z_{\infty}\\}$ is identified
with ${\mathbb{C}}$, and the $\xi_{i}$ can be obtained as the convex hull of
$z_{i}$ and $z_{d}$ in the real vector space ${\mathbb{C}}$ by Lemma 2.1.
Consider the real half line $\ell_{j}$ in ${\mathbb{C}}$ which contains
$z_{j}$ as the endpoint and such that the real full line containing $\ell_{j}$
contains $z_{d}$ although $z_{d}\notin\ell_{j}$. Then the complement in
$Z_{c}$ of an appropriate open neighborhood of the closure of $\ell_{j}$ in
$Z_{c}$ is a deformation retract of $Z_{c}\,{\setminus}\,\\{z_{j}\\}$, and
$\bigcup_{i\neq j}\xi_{i}$ is a deformation retract of the complement. The
assertion for $\bigcup_{i\neq j}\widetilde{\xi}_{i}$,
$\widetilde{Z}_{c}\,{\setminus}\,\\{\widetilde{z}_{j}\\}$ follows from this,
since $\widetilde{Z}_{c}$ is an unramified covering over $Z^{\prime}_{c}$.)
The assertion then follows, since $\mu_{m}$ acts as the covering
transformation group of the unramified covering

$\widetilde{\xi}_{i}\,{\setminus}\,\\{\widetilde{z}_{i},\widetilde{z}_{d}\\}\to\xi_{i}\,{\setminus}\,\\{z_{i},z_{d}\\}.$

This finishes the proof of Proposition 2.1.

Corollary 2.1. The $\sigma^{\lambda^{\prime\prime}}_{i}\,\,(i\in[1,d{-}1])$
span
$H_{1}(\widetilde{Z}_{c},{\mathbb{C}})^{\gamma_{*}=\lambda^{\prime\prime}}$,
and there are $\alpha_{i}\in{\mathbb{C}}^{*}$ with

$None$ $\hbox{$\sum$}_{i=1}^{d-1}\,\alpha_{i}\hskip
1.0pt\sigma^{\lambda^{\prime\prime}}_{i}=0.$

Hence we get the isomorphism

$None$
$H_{1}(\widetilde{Z}_{c},{\mathbb{C}})^{\gamma_{*}=\lambda^{\prime\prime}}=\bigl{(}\hbox{$\bigoplus$}_{i=1}^{d-1}\,{\mathbb{C}}\hskip
1.0pte_{i}\bigr{)}/{\mathbb{C}}\bigl{(}\hbox{$\sum$}_{i=1}^{d-1}\,\alpha_{i}\hskip
1.0pte_{i}\bigr{)},$

where $(e_{1},\dots,e_{d-1})$ is the basis of the complex vector space
${\mathbb{C}}^{d-1}$, and $\sigma^{\lambda^{\prime\prime}}_{i}$ is identified
with the image of $e_{i}$ in the right-hand side of $(2.1.9)$ for
$i\in\\{1,\dots,d{-}1\\}$.

2.2. Proof of Theorem 1 in the $X^{\prime}_{\langle m\rangle}$ connected case.
The assertion is reduced to the line arrangement case by Artin’s vanishing
theorem [BBD] using an iterated general hyperplane cut. So $X$ will be denoted
by $L$ and the notation in Section 1 will be used. By Proposition 1.3, it is
enough to show that

$None$ $\Gamma(C^{\prime\prime},{\mathcal{G}}^{\prime\prime})=0,\,\,\,\hbox{or
equivalently,}\,\,\,({\mathcal{G}}^{\prime\prime}_{c})^{G}=0.$

Here $c\in C^{\prime\prime}$ as in (1.5), and $G$ is the monodromy group of
the local system ${\mathcal{G}}^{\prime\prime}$ on $C^{\prime\prime}$ with
base point $c$.

By Proposition 1.5 and Corollary 2.1 together with Poincaré duality as in
(2.1.1–2), any element of $({\mathcal{G}}^{\prime\prime}_{c})^{G}$ can be
expressed by

$\hbox{$\sum$}_{i=1}^{d-1}\,\beta_{i}\hskip
1.0pt\sigma^{\lambda^{\prime\prime}}_{i},$

where the $\beta_{i}\in{\mathbb{C}}$ are unique up to addition by
$\varepsilon\hskip 1.0pt\alpha_{i}$ with $\varepsilon\in{\mathbb{C}}$.
Assuming $X^{\prime}_{\langle m\rangle}$ is connected, we have to show that

$None$ $\beta_{i}/\alpha_{i}\,\,\,\hbox{is independent
of}\,\,\,i\in\\{1,\dots,d{-}1\\}.$

Here $\\{1,\dots,d{-}1\\}$ is identified with the vertices of the dual
$(m)$-graph. Since the latter is connected by hypothesis, it is enough to show
(2.2.2) in the case $i$ belong to a subset $I_{z}$ such that the $L_{i}$
($i\in I_{z}$) are the lines in $L$ passing through some $z\in
L^{[k]}\,{\setminus}\,L_{d}$ with $k/m\notin{\mathbb{Z}}$, where
$|I_{z}|\,{=}\,k$.

Choosing a path between $c$ and $\rho^{\prime}(z)$, and taking a sufficiently
small circle around $\rho^{\prime}(z)$, we get
$\zeta\in\pi_{1}(C^{\prime\prime},c)$ going around $\rho^{\prime}(z)$.
Consider the short exact sequence

$None$
$0\to({\mathcal{G}}^{\prime\prime}_{c})^{\zeta}\buildrel\iota\over{\to}{\mathcal{G}}^{\prime\prime}_{c}\to{\rm
Coker}\,\iota\to 0,$

where the first term is the invariant part of
${\mathcal{G}}^{\prime\prime}_{c}$ under the monodromy action by $\zeta$.
Using (1.2.3) with (1.3.3) and (1.4.2), the dimensions of the three terms are
given respectively by

$d\hskip 1.0pt{-}\hskip 1.0ptk\hskip 1.0pt{-}\hskip 1.0pt1,\quad d\hskip
1.0pt{-}\hskip 1.0pt2,\quad k\hskip 1.0pt{-}\hskip 1.0pt1.$

We can verify that the $\sigma^{\lambda^{\prime\prime}}_{i}$ for $i\in
I_{z}^{c}:=\\{1,\dots,d{-}1\\}\,{\setminus}\,I_{z}\,$ belong to
$({\mathcal{G}}^{\prime\prime}_{c})^{\zeta}$. Then they span it using
Proposition 2.1, since $|I_{z}^{c}|\,{=}\,d\hskip 1.0pt{-}\hskip 1.0ptk\hskip
1.0pt{-}\hskip 1.0pt1$. We thus get by Corollary 2.1 the isomorphism

$None$ ${\rm Coker}\,\iota=\bigl{(}\hbox{$\bigoplus$}_{i\in
I_{z}}\,{\mathbb{C}}\hskip
1.0pte_{i}\bigr{)}/{\mathbb{C}}\bigl{(}\hbox{$\sum$}_{i\in
I_{z}}\,\alpha_{i}\hskip 1.0pte_{i}\bigr{)},$

such that the image of $\sigma^{\lambda^{\prime\prime}}_{i}$ in ${\rm
Coker}\,\iota$ is identified with the image of $e_{i}$ in the right-hand side
of (2.2.4) for $i\in I_{z}$.

Since $\hbox{$\sum$}_{i=1}^{d-1}\,\beta_{i}\hskip
1.0pt\sigma^{\lambda^{\prime\prime}}_{i}\in({\mathcal{G}}^{\prime\prime}_{c})^{G}$
belongs to $({\mathcal{G}}^{\prime\prime}_{c})^{\zeta}$, its image in ${\rm
Coker}\,\iota$ vanishes. This means that

$\hbox{$\sum$}_{i\in I_{z}}\,\beta_{i}\hskip
1.0pte_{i}\in{\mathbb{C}}\bigl{(}\hbox{$\sum$}_{i\in I_{z}}\,\alpha_{i}\hskip
1.0pte_{i}\bigr{)}.$

So we get (2.2.2) for $i\in I_{z}$. This finishes the proof of Theorem 1 in
the $X^{\prime}_{\langle m\rangle}$ connected case.

Remark 2.2. Theorem 1 is a special case of Theorem 2. Indeed, this can be
verified easily for $r^{\prime}_{m}\,{\leqslant}\,2$, since $X$ is essential.
In the case $r^{\prime}_{m}\,{\geqslant}\,3$, the latter property of $X$
implies that there is a hyperplane
$X_{i}\subset\overline{X^{\prime(1)}_{\langle m\rangle}}$ not containing the
projective subspace
$\hbox{$\bigcap$}_{j=2}^{r^{\prime}_{m}}\,\overline{X^{\prime(j)}_{\langle
m\rangle}}$ of codimension 2 so that
$X_{i}\cap\overline{X^{\prime(j)}_{\langle m\rangle}}$ does not coincide with
it for each $j\in[2,r^{\prime}_{m}]$. The assumption of Theorem 2 is then
satisfied.

2.3. Construction of the covering space $\widetilde{Z}_{c}$. Let $\xi_{i}$ be
as in Lemma 2.1. We may assume

$None$ The $\arg z(z_{i})$ are strictly increasing ($i\in[1,d{-}1]$).

Here $z$ is a complex coordinate of
$Z_{c}\,{\setminus}\,\\{z_{\infty}\\}\cong{\mathbb{C}}$ with
$z(z_{d})\,{=}\,0$. Set

$E_{k}:=Z_{c}\,{\setminus}\,\hbox{$\bigcup$}_{i=1}^{d-1}\,\xi_{i}\quad(k\in{\mathbb{Z}}/m\hskip
1.0pt{\mathbb{Z}}).$

There is a compactification ${}\,\overline{\\!E}{}_{k}$ of $E_{k}$ together
with a surjection $\phi_{k}:{}\,\overline{\\!E}{}_{k}\to Z_{c}$ such that the
boundary is as follows:

${}\,\overline{\\!E}{}_{k}\,{\setminus}\,E_{k}=\hbox{$\bigcup$}_{i=1}^{d-1}\,\xi^{k,+}_{i}\,\cup\,\hbox{$\bigcup$}_{i=1}^{d-1}\,\xi^{k,-}_{i},$

where $\xi^{k,+}_{i}$, $\xi^{k,-}_{i}$ are isomorphic to $\xi_{i}$ by
$\phi_{k}$. Moreover

$\xi^{k,+}_{i}\cap\xi^{k,-}_{i}=\\{p^{k}_{i}\\},\quad\xi^{k,+}_{i}\cap\xi^{k,-}_{i+1}=\\{q^{k}_{i}\\},$

with $\phi_{k}(p^{k}_{i})\,{=}\,z_{i}$, $\phi_{k}(q^{k}_{i})\,{=}\,z_{d}$ for
$i\in\\{1,d{-}1\\}\cong{\mathbb{Z}}/(d{-}1){\mathbb{Z}}$. The picture of the
images of $\xi^{k,\pm}_{i}$, $p^{k}_{i}$, etc. by $\phi_{k}$ is as follows
(where $\phi_{k}$ is omitted to simplify the picture):

We get the covering space $\widetilde{Z}_{c}$ topologically by identifying
$\,\xi^{k,+}_{i}$ with $\,\xi^{k+1,-}_{i}$; in particular, $p^{k}_{i}$ with
$p^{k}_{i}$, and $q^{k}_{i}$ with $q^{k+1}_{i-1}$. (Related to the latter,
note that $m$ and $d{-}1$ are mutually prime, hence
$(1,-1)\in({\mathbb{Z}}/m\hskip
1.0pt{\mathbb{Z}}){\times}({\mathbb{Z}}/(d{-}1)\hskip 1.0pt{\mathbb{Z}})$ is a
generator of the finite cyclic group.)

Remark 2.3. By the above construction, we can choose the
$\widetilde{\eta}_{i}$ in (2.1.6) appropriately so that

$None$
$\hbox{$\sum$}_{i=1}^{d-1}\,\sigma_{i}=0\quad\hbox{in}\,\,\,H_{1}(\widetilde{Z}_{c},{\mathbb{C}}),$

since the $E_{k}$ are contractible. In (2.1.8) we may then assume

$None$ $\alpha_{i}\,{=}\,1\quad(\forall\,i\in[1,d{-}1]).$

2.4. Calculation of local monodromies of ${\mathcal{G}}^{\prime\prime}$. Let
$Q\in{\rm Sing}\,L\setminus L_{d}$. There are coordinates
$x^{\prime},y^{\prime}$ of $Y\setminus L_{d}\cong{\mathbb{C}}^{2}$ with origin
$Q$ such that $x^{\prime}\,{=}\,x\hskip 1.0pt{-}\hskip 1.0pta$ with $x$ as in
the proof of Lemma 2.1 and $y^{\prime}$ is the pull-back of a coordinate of
$C^{\prime}\cong{\mathbb{C}}$ by $\rho^{\prime}$ in (1.2). Then the
$L_{i}\subset L$ ($i\in I_{Q}$) passing through $Q$ can be described as

$None$ $x^{\prime}=a^{\prime}_{i}\hskip
1.0pty^{\prime}\quad(a^{\prime}_{i}\in{\mathbb{C}},\,\,i\in I_{Q}).$

We assume $c\in C^{\prime\prime}$ is sufficiently near $\rho^{\prime}(Q)$ so
that

$None$ $\max(I_{Q})-\min(I_{Q})=|I_{Q}|\hskip 1.0pt{-}\hskip 1.0pt1,$

that is, $I_{Q}={\mathbb{Z}}\cap[\min(I_{Q}),\max(I_{Q})]$. (Note that (2.3.1)
is assumed.)

Substituting $y^{\prime}\,{=}\,\varepsilon e^{i\theta}$ in (2.4.1) with
$0\,{<}\,\varepsilon\,{\ll}\,1$ fixed and $\,\theta\in[0,2\pi]$ varying, we
can verify that, under the local monodromy around $\rho^{\prime}(Q)\in
C^{\prime}$, the loop
$\,\eta_{i}\in\pi_{1}(\widetilde{Z}_{c}\,{\setminus}\,\Xi^{\prime}_{c},z_{d})\,$
in (2.1.5) is changed as below (where we assume $\,i\,{=}\,2\,$ with
$I_{Q}=\\{1,2,3\\}\,$ for simplicity):

$\scriptstyle z_{3}$$\scriptstyle z_{2}$$\scriptstyle z_{1}$$\scriptstyle
z_{d}$$\Longrightarrow$$\scriptstyle z_{3}$$\scriptstyle z_{2}$$\scriptstyle
z_{1}$$\scriptstyle z_{d}$ $\sim$ $\scriptstyle z_{3}$$\scriptstyle
z_{2}$$\scriptstyle z_{1}$$\scriptstyle z_{d}$

Here a thick path means a “double path” consisting of two paths with opposite
directions. These have two different liftings to the covering space, since
they are connected to a circle around $z_{2}$.

Crossing the images of $\xi_{i}^{k,+}$, $\xi_{i}^{k,-}$ from left to right
corresponds to the action of a generator $\widetilde{\gamma}$ of the covering
transformation group. We thus get the following.

Lemma 2.4. Let $i_{0}:=\min I_{Q}$, $q:=|I_{Q}|$. The image of
$\sigma_{i_{0}+i}\,\,(i\in[0,k-1])$ by the local monodromy around
$\rho^{\prime}(Q)$ is given by

$None$
$\widetilde{\gamma}_{*}^{q}\sigma_{i_{0}+i}+\widetilde{\gamma}_{*}^{i}(1\hskip
1.0pt{-}\hskip
1.0pt\widetilde{\gamma}_{*})\bigl{(}\hbox{$\sum$}_{j=0}^{q-1}\,\sigma_{i_{0}+j}\bigr{)}.$

2.5. Proof of Theorem 2. Assume $r^{\prime}_{m}\,{\geqslant}\,2$. For
$k\in[2,r^{\prime}_{m}]$, let $Q_{k}$, $I^{\prime(j)}_{Q_{k}}$ be as in
Theorem 1. By assumption, we have

$None$
$I^{\prime(j)}_{Q_{k}}\,{=}\,\emptyset\,\,\,\,\,\hbox{if}\,\,\,\,\,j\,{\neq}\,1,k,$
$None$ $|I^{\prime(k)}_{Q_{k}}|$ is relatively prime to $m$, and is at most 4
in the $m$ non-prime case.

Let $c\in C^{\prime\prime}$ be sufficiently near $\rho^{\prime}(Q_{k})$ as in
(2.4). Construct the $\sigma_{i}$ as in (2.3) so that (2.3.3) holds. Assume
$\hbox{$\sum$}_{i=1}^{d-1}\beta_{i}\sigma_{i}^{\lambda^{\prime\prime}}\in({\mathcal{G}}^{\prime\prime}_{c})^{G}$
as in (2.2). By the same argument as there, there are
$\beta^{(j)}\in{\mathbb{C}}$ ($j\in[1,r^{\prime}_{m}]$) such that

$None$ $\beta_{i}=\beta^{(j)}\quad\hbox{if}\quad i\in
I^{\prime(j)}\quad(j\in[1,r^{\prime}_{m}]),$

where the $I^{\prime(j)}$ are as in the introduction. Note that
$\alpha_{i}\,{=}\,1$ ($\forall\,i)$ by (2.3.3).

Replacing $\beta_{i}$ with $\beta_{i}\hskip 1.0pt{-}\hskip 1.0pt\beta^{(1)}$
($\forall\,i\in[1,d{-}1]$), we may assume

$None$ $\beta_{i}\,{=}\,\beta^{(1)}\,{=}\,0\quad\hbox{if}\quad i\in
I^{\prime(1)}.$

Using Lemma 2.4 and (2.5.1), the image of
$\hbox{$\sum$}_{i=1}^{d-1}\beta_{i}\sigma_{i}^{\lambda^{\prime\prime}}$ under
the local monodromy around $\rho^{\prime}(Q_{k})$ is then given by

$None$
$\hbox{$\sum$}_{i=1}^{d-1}\beta_{i}\sigma_{i}^{\lambda^{\prime\prime}}+\beta^{(k)}\bigl{(}\hbox{$\sum$}_{i\in
I^{\prime(k)}_{Q_{k}}}\,\widetilde{\lambda}^{i}\bigr{)}(1\hskip 1.0pt{-}\hskip
1.0pt\widetilde{\lambda})\bigl{(}\hbox{$\sum$}_{j=0}^{q-1}\,\sigma_{i_{0}+j}^{\lambda^{\prime\prime}}\bigr{)},$

where $\widetilde{\lambda}$ is a primitive $m$ th root of unity. Note that the
$\sigma_{i}$ ($i\notin I_{Q_{k}}$) are invariant by the local monodromy around
$\rho^{\prime}(Q_{k})$.

Assume $\beta^{(k)}\,{\neq}\,0$. Since
$\hbox{$\sum$}_{i=1}^{d-1}\beta_{i}\sigma_{i}^{\lambda^{\prime\prime}}$ is
monodromy invariant by assumption, we can then deduce the following from
(2.5.5) using Proposition 2.1:

$None$ $\hbox{$\sum$}_{i\in
I^{\prime(k)}_{Q_{k}}}\,\widetilde{\lambda}^{i}=0.$

However, this contradicts (2.5.2) by Remark 2.5 below. This finishes the proof
of Theorem 2.

Remark 2.5. Let $\lambda$ be a primitive $m$ th root of unity. Let $J$ be a
subset of $\\{0,\dots,m{-}1\\}$ such that $|J|$ is relatively prime to $m$,
and is at most 4 in the $m$ non-prime case. Then we have $\hbox{$\sum$}_{j\in
J}\,\lambda^{j}\,{\neq}\,0$. This does not hold without the last hypothesis,
for instance in the case $J=\\{0,3,4,8,9\\}$ with $m\,{=}\,12$, $|J|\,{=}\,5$.
(In the $m$ prime case, $\hbox{$\sum$}_{i=0}^{m-1}\,x^{i}\in{\mathbb{Q}}[x]$
is an irreducible polynomial by the theory of cyclotomic polynomials.)

2.6. Generalized Hessian arrangements. For an integer $b\,{\geqslant}\,2$, set
$L:=\\{f\,{=}\,0\\}\subset{\mathbb{P}}^{2}$ with

$f=xyz\,\hbox{$\prod$}_{i,j=0}^{b-1}\,(\zeta^{i}x\hskip 1.0pt{+}\hskip
1.0pt\zeta^{j}y\hskip 1.0pt{+}\hskip 1.0ptz).$

with $\zeta:=e^{2\pi\sqrt{-1}/b}$. Here ${\rm Sing}\,L\,{=}\,L^{[b+1]}\cup
L^{[2]}$ with $L^{[b+1]}\,{=}\,{\rm Sing}\,L\cap\\{xyz\,{=}\,0\\}$, and the
singular points of $\\{xyz\,{=}\,0\\}$ are disjoint from $L^{[b+1]}$, see
[Sa4, 3.4–5]. Assuming $b{+}1\in m\hskip 1.0pt{\mathbb{Z}}$, we have
$d\,{=}\,b^{2}\hskip 1.0pt{+}\hskip 1.0pt3\in m\hskip 1.0pt{\mathbb{Z}}$ if
and only if $4\in m\hskip 1.0pt{\mathbb{Z}}$, that is, $m\,{=}\,2$ or $4$. In
the case $m\,{=}\,4$, we have $b\,{=}\,4a\hskip 1.0pt{-}\hskip 1.0pt1$ with
$a\in{\mathbb{Z}}_{>0}$. For $a\,{=}\,1$, this arrangement is called the
Hessian  arrangement, see [BDS], [Di2], [Sa4], [Yu], etc. It may be called a
generalized Hessian  arrangement if $a\,{\geqslant}\,2$. The dual $(4)$-graph
has two or four connected components depending on whether $a\,{\geqslant}\,2$
or $a\,{=}\,1$. One connected component is always given by
$\\{xyz\,{=}\,0\\}$, and the hypothesis of Theorem 2 is satisfied for
$a\,{\geqslant}\,2$. However, this fails in Theorem 1, since the connected
component has three irreducible components.

Remark 2.6 (i). There are many other examples where the hypothesis of Theorem
2 is satisfied with $r^{\prime}_{m}\,{\geqslant}\,2$. For instance, let
$L_{1},L_{2}$ be two lines in ${\mathbb{P}}^{2}$. Take $a(m{-}1)$ points

$P_{i,j}\in L_{1}\,{\setminus}\,L_{2},\quad Q_{i,j}\in
L_{2}\,{\setminus}\,L_{1}\quad(i\in[1,a],\,j\in[1,m{-}1]).$

Let $L_{i,j,j^{\prime}}$ be the line passing through $P_{i,j}$,
$Q_{i,j^{\prime}}$ for $i\in[1,a],\,j,j^{\prime}\in[1,m{-}1]$. Set

$L=L_{1}\cup L_{2}\cup\hbox{$\bigcup$}_{i,j,j^{\prime}}\,L_{i,j,j^{\prime}}.$

Assume the $P_{i,j}$, $Q_{i,j^{\prime}}$ are sufficiently general so that
${\rm Sing}\,L=L^{[m]}\cup L^{[2]}$ and $L^{[m]}\subset L_{1}\cup L_{2}$,
where $m\,{\geqslant}\,3$. Put $a\,{=}\,m{-}2$. Then
$d=(m{-}2)(m{-}1)^{2}+2\in m\hskip 1.0pt{\mathbb{Z}}$, and
$r^{\prime}_{m}\,{=}\,2$. One connected component of $L^{\prime}_{\langle
m\rangle}$ is given by its intersection with $L_{1}\cup L_{2}$. Here $L_{d}$
can be $L_{1}$ or $L_{2}$, and the assumption of Theorem 1 is also satisfied
in this case.

Remark 2.6 (ii). We have the following example where the hypothesis of Theorem
1 is satisfied with $r^{\prime}_{m}\,{=}\,m\,{=}\,4$. Take sufficiently
general real numbers $a_{i},b_{i}$ ($i\in[1,4]$) and a sufficiently small
positive number $c$ (in particular, $a_{i}\hskip 1.0pt{+}\hskip
1.0ptb_{j}\,{\neq}\,a_{i^{\prime}}\hskip 1.0pt{+}\hskip 1.0ptb_{j^{\prime}}$
if $(i,j)\,{\neq}\,(i^{\prime},j^{\prime}))$. Consider the lines $L_{i,j,k}$
defined in ${\mathbb{P}}^{2}$ by

$y=(a_{i}\hskip 1.0pt{-}\hskip 1.0ptb_{j}\hskip 1.0pt{+}\hskip 1.0ptk\hskip
1.0ptc)x+(a_{i}\hskip 1.0pt{+}\hskip 1.0ptb_{j})z,$

for $i,j\in[1,4],\,k\in\\{-1,0,1\\}$ with $i\,{\neq}\,j$, and also the lines
$L_{l}$ ($l\,{=}\,{-1},0,1$), $L_{\infty}\,(=L_{d})$ which are defined
respectively by

$x=l\hskip 1.0ptz,\quad z\,{=}\,0.$

They form a line arrangement $L$ of degree 40 with
$r^{\prime}_{m}\,{=}\,m\,{=}\,4$, and the closure of three connected
components of $L^{\prime}_{\langle m\rangle}$ with $m\,{=}\,4$ are given by
the $L_{l}$ ($l\,{=}\,{-1},0,1$). Note that $L$ does not support a 4-net. We
may get $r^{\prime}_{m}\,{=}\,3$ replacing the order of $\\{1,\dots,d\\}$
appropriately, since $L_{\langle m\rangle}$ contain irreducible connected
components and $r_{m}\,{=}\,4$ in the notation of the introduction with $X$
replaced by $L$, see also Remark 2.6 (iii) just below.

Remark 2.6 (iii). We can modify the example in Remark 2.6 (ii) by considering
the lines $L_{i,j,k,k^{\prime}}$ (instead of $L_{i,j,k}$) defined by

$y=(a_{i}\hskip 1.0pt{-}\hskip 1.0ptb_{j}\hskip 1.0pt{+}\hskip 1.0ptk\hskip
1.0ptc)x+(a_{i}\hskip 1.0pt{+}\hskip 1.0ptb_{j}\hskip 1.0pt{+}\hskip
1.0ptk^{\prime}c^{\prime})z,$

for $i,j\in[1,4],\,k,k^{\prime}\in\\{-1,0,1\\}$ with $i\,{\neq}\,j$, where
$c^{\prime}$ is sufficiently large compared with the $a_{i},b_{j}$.

We have $d\,{=}\,36\cdot 3\hskip 1.0pt{+}\hskip 1.0pt4\,{=}\,112$, and
$r^{\prime}_{m}\,{=}\,4$ with $m\,{=}\,4$. However, $r_{m}\,{=}\,5$, and we
may get $r^{\prime}_{m}\,{=}\,5$ replacing the order of $\\{1,\dots,d\\}$
appropriately. (In particular, this does not support a 4-net.) The assumption
of Theorem 1 may be satisfied with $r^{\prime}_{m}\,{=}\,5$.

## References

  * [Ba] Bailet, P., On the monodromy of Milnor fibers of hyperplane arrangements, Canad. Math. Bull. 57 (4) (2014), 697–707.
  * [BBD] Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, Astérisque 100, Soc. Math. France, Paris, 1982.
  * [BDS] Budur, N., Dimca, A. and Saito, M., First Milnor cohomology of hyperplane arrangements, Contemp. Math., 538, Amer. Math. Soc., Providence, RI, 2011, pp. 279–292.
  * [BS] Budur, N. and Saito, M., Jumping coefficients and spectrum of a hyperplane arrangement, Math. Ann. 347 (2010), 545–579.
  * [BSY] Budur, N., Saito, M. and Yuzvinsky, S., On the local zeta functions and the $b$-functions of certain hyperplane arrangements, J. London Math. Soc. 84 (2011), 631–648.
  * [CS] Cohen, D.C. and Suciu, A., On Milnor fibrations of arrangements, J. London Math. Soc. 51 (1995), 105–119.
  * [Di1] Dimca, A., Singularities and Topology of Hypersurfaces, Universitext, Springer, Berlin, 1992.
  * [Di2] Dimca, A., Hyperplane Arrangements, Universitext, Springer, 2017.
  * [ESV] Esnault, H., Schechtman, V. and Viehweg, E., Cohomology of local systems on the complement of hyperplanes, Inv. Math. 109 (1992), 557–561.
  * [Fa] Falk, M., Arrangements and cohomology, Ann. Combin. 1 (1997), 135–157.
  * [PS] Papadima, S. and Suciu, A.I., The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy, Proc. London Math. Soc. (3) 114 (2017), 961–1004.
  * [Sa1] Saito, M., Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ. 24 (1988), 849–995.
  * [Sa2] Saito, M., Mixed Hodge modules, Publ. RIMS, Kyoto Univ. 26 (1990), 221–333.
  * [Sa3] Saito, M., Rank one local systems on complements of hyperplanes and Aomoto complexes (arxiv:1807.00333).
  * [Sa4] Saito, M., Efficiency and complexity of hyperplane arrangements (arXiv:2103.13734).
  * [Yu] Yuzvinsky, S., Resonance varieties of arrangement complements, in Arrangements of Hyperplanes-Sapporo 2009, Adv. Study Math., 62 (2012), pp. 553–570.