Table Of ContentMilnor fibers of real line arrangements
Masahiko Yoshinaga
∗
3 December 11, 2013
1
0
2
n
a
Abstract
J
8
We study Milnor fibersof complexified real line arrangements. We
] give a new algorithm computing monodromy eigenspaces of the first
G
cohomology. The algorithm is based on the description of minimal
A
CW-complexes homotopic to the complements, and uses the real fig-
.
h ure, that is, the adjacency relations of chambers. It enables us to
t generalize a vanishing result of Libgober, give new upper-bounds and
a
m characterize the A -arrangement in terms of non-triviality of Milnor
3
[ monodromy.
1
v
0 1 Introduction
3
4
1 The Milnor fiber is a central object in the study of the topology of complex
1. hypersurface singularities. In particular, the monodromy action on its coho-
0 mologygroupshasbeen intensively studied. Monodromy eigenspaces contain
3
subtle geometric information. For example, for projective plane curves, the
1
: Betti numbers of Milnor fiber of the cone detect Zariski pairs [1]. In other
v
i words, Betti numbers of Milnor fiber of the cone of a plane curve are not in
X
general determined by local and combinatorial data of singularities.
r
a In the theory of hyperplane arrangements, one of the central problems
is to what extent topological invariants of the complements are determined
combinatorially. For example, the cohomology ring is combinatorially deter-
mined (Orlik and Solomon [16]), while the fundamental group is not (Ryb-
nikov [1, 11]). Between these two cases, local system cohomology groups
and monodromy eigenspaces of Milnor fibers recently received a considerable
amount of attention.
∗DepartmentofMathematics,HokkaidoUniversity,North10,West8,Kita-ku,Sapporo
060-0810,JAPAN E-mail: [email protected]
1
There are several ways to compute monodromy eigenspaces of the Milnor
fiber, especially for line arrangements. One is the topological method devel-
oped by Cohen andSuciu [4]. They first give a presentation of the fundamen-
tal group of the complement. Then, using Fox calculus, they compute the
monodromy eigenspaces. Another approach is the algebraic method, which
computesthemultiplicities ofmonodromyeigenvaluesasthesuperabundance
ofsingular points. This approachhasrecently beenwell developed, especially
for line arrangements having only double and triple points [14].
The purpose of this paper is to develop a topological method of comput-
ing Milnor monodromy for complexified real arrangements following Cohen
and Suciu. The new ingredient is a recent study of minimal cell structures
for the complements of complexified real arrangements [19, 22]. By using
the description of twisted minimal chain complexes, we obtain an algorithm
which computes monodromy eigenspaces directly from real figures without
passing through the presentations of π .
1
The paper is organized as follows. In 2 we recall a few results which are
§
used in this paper. 3 is the main section of the paper. First, in 3.1, we
§ §
introduce discrete geometric notions, the so-called k-resonant band and the
standing wave on this band. These notions are used in 3.2 for the computa-
§
tion of eigenspaces. Several consequences of our algorithm are discussed in
3.3, 3.4 and 3.5. Among other things, we prove that if the arrangement
§ § §
contains more than 6 lines and the cohomological monodromy action (of de-
gree one) is non-trivial, then each line has at least three multiple points (see
Corollary3.24foraprecisestatement). Sucharrangementshavebeenstudied
in discrete geometry as “configurations”, and several examples are provided
in [8, 9]. In 4, we apply our algorithm to arrangements appearing in papers
§
by Gru¨nbaum [8, 9]. We also present several examples and conjectures.
2 Preliminaries
2.1 Milnor fiber of arrangements
Let = H ,...,H be an affine line arrangement in R2 with the defining
1 n
A { }
equation Q (x,y) = n α , where α is a defining linear equation for H .
A i=1 i i i
In this paper, we assuQme that not all lines are parallel (or equivalently,
A
has at least one intersection). The coning c of is an arrangement of
A A
n+1 planes in R3 defined by the equation Q (x,y,z) = zn+1Q(x, y). The
cA z z
line z = 0 c is called the line at infinity and is denoted by H . The
∞
{ } ∈ A
space M( ) = C2 Q = 0 = P2 Q = 0 is called the complexified
A C cA
A \ { } \{ }
complement. In this article, always denotes a line arrangement in R2 and
A
2
c denotes a line arrangement in RP2. We call p RP2 a multiple point if
A ∈
the multiplicity of c at p (that is, the number of lines passing through p)
A
is greater than or equal to 3.
Definition 2.1. F = (x,y,z) C3 Q (x,y,z) = 1) iscalledtheMilnor
A cA
{ ∈ | }
fiber of . The automorphism ρ : F F , (x,y,z) (ζx,ζy,ζz), with
A A
A −→ 7−→
ζ = exp(2πi/(n+1)), is called the monodromy action.
The automorphism ρ has order n + 1. It generates the cyclic group
ρ Z/(n+1)Z. Themonodromyρinducesalinearmapρ∗ : H1(F ,C)
A
h i ≃ −→
H1(F ,C). Since (ρ∗)n+1 is the identity, we have the eigenspace decompo-
A
sition H1(F ,C) = H1(F ,C) , where H1(F ,C) is the the set of
A λn+1=1 A λ A λ
λ-eigenvectors withLeigenvalue λ C∗. When λ = 1, H1(FA)1 = H1(FA)ρ∗ is
∈
the subspace of elements fixed by ρ∗, which is isomorphic to H1(F / ρ ). It
A
h i
is easily seen that the quotient by the monodromy action is F / ρ M( ).
A
h i ≃ A
Therefore, the 1-eigenspace of the first cohomology is combinatorially deter-
mined, H1(F ) H1(M( )) Cn. In general, let be a complex rank
A 1 λ
≃ A ≃ L
one local system associated with a representation
π (M( )) C∗, γ λ,
1 H
A −→ 7−→
where γ is a meridian loop of the line H. Then it is known that
H
H1(F ) H1(M( ), ). (1)
A λ λ
≃ A L
(See [4] for details.)
2.2 Multinets and Milnor monodromy
In this section, we recall a relation between the combinatorial structures
known as multinets and the eigenvalues of Milnor monodromy. We note that
a k-multinet gives a lower bound on the eigenspace.
Definition 2.2. A k-multinet on c is a pair ( , ), where is a partition
A N X N
of c into k 3 classes ,..., and is a set of multiple points such
1 k
A ≥ A A X
that
(i) = = ;
1 k
|A | ··· |A |
(ii) H and H′ (i = j) imply that H H′ ;
i j
∈ A ∈ A 6 ∩ ∈ X
(iii) for all p , H H p is constant and independent of i;
i
∈ X |{ ∈ A | ∋ }|
(iv) foranyH,H′ (i = 1,...,k),thereisasequenceH = H ,H ,...,H =
i 0 1 r
∈ A
H′ in such that H H / for 1 j r.
i j−1 j
A ∩ ∈ X ≤ ≤
3
The following is a consequence of [7, Theorem 3.11] and [6, Theorem 3.1
(i)]
Theorem 2.3. Suppose there exists a k-multinet on c for some k 3 and
A ≥
set λ = e2πi/k. Then
dimH1(F ) k 2.
A λ
≥ −
2.3 Twisted minimal cochain complexes
In this section, we recall the construction of the twisted minimal cochain
complex from [19, 20, 21], which will be used for the computation of the
right hand side of (1).
A connected component of R2 H is called a chamber. The set of
\ H∈A
all chambers is denoted by ch( ). ASchamber C ch( ) is called bounded
A ∈ A
(resp. unbounded) if the area is finite (resp. infinite). For an unbounded
chamber U ch( ), the opposite unbounded chamber is denoted by U∨ (see
∈ A
[21, Definition 2.1] for the definition; see also Figure 1 below).
Let be a generic flag in R2
F
: = −1 0 1 2 = R2,
F ∅ F ⊂ F ⊂ F ⊂ F
where k is a generic k-dimensional affine subspace.
F
Definition 2.4. For k = 0,1,2, define the subset chk( ) ch( ) by
F A ⊂ A
chk( ) := C ch( ) C k = ,C k−1 = .
F A { ∈ A | ∩F 6 ∅ ∩F ∅}
Thesetofchambersdecomposesintoadisjointunionasch( ) = ch0 ( )
A F A ⊔
ch1 ( ) ch2 ( ). The cardinality of chk( ) is equal to b (M( )) for k =
F A ⊔ F A F A k A
0,1,2.
We further assume that the generic flag satisfies the following condi-
F
tions:
1 does not separate intersections of ,
• F A
0 does not separate n-points 1.
• F A∩F
Then we can choose coordinates x ,x so that 0 is the origin (0,0), 1
1 2
F F
is given by x = 0, all intersections of are contained in the upper-half
2
A
plane (x ,x ) R2 x > 0 and 1 is contained in the half-line
1 2 2
{ ∈ | } A ∩ F
(x ,0) x > 0 .
1 1
{ | }
We set H 1 to have coordinates (a ,0). By changing the numbering
i i
∩F
of lines and the signs of the defining equation α of H we may assume
i i
∈ A
that
4
0 < a < a < < a ,
1 2 n
• ···
the origin 0 is contained in the negative half-plane H− = α < 0 .
• F i { i }
We set chF( ) = U and chF( ) = U ,...,U ,U∨ so that U 1 is
0 A { 0} 1 A { 1 n−1 0 } p ∩F
equal to the interval (a ,a ) for p = 1,...,n 1. It is easily seen that the
p p+1
−
chambers U ,U ,...,U and U∨ have the following expression:
0 1 n−1 0
n
U = α < 0 ,
0 i
{ }
i\=1
p n
U = α > 0 α < 0 , (p = 1,...,n 1), (2)
p i i
{ }∩ { } −
i\=1 i=\p+1
n
U∨ = α > 0 .
0 { i }
i\=1
The notations introduced to this point are illustrated in Figure 1.
❍❍❍ U4∨ U2∨ U3∨ U1∨ ✟✟✟ ch0F(A) = {U0}
❍❍ ✟✟ ch1 ( ) = U∨,U ,...,U
❍❍❍ ✟✟✟ chF2 (A) = {U0∨,..1.,U∨,C4},C
U0 C✟1 ✟❍✟❍❍C2 U0∨ F A { 1 4 1 2}
✟ ❍
✟ ❍
0(0,0) a ✟✟ a a a❍❍ a 1
F♣♣♣♣♣♣♣s♣♣♣♣♣♣♣✟♣♣♣✟r1♣♣♣♣♣♣♣♣♣♣♣♣♣r♣♣♣♣2♣♣♣♣♣♣r♣♣♣♣3♣♣♣♣♣♣r♣♣♣♣4♣♣♣♣♣♣❍♣♣♣♣r❍♣♣♣5♣♣♣♣♣♣♣♣♣♣♣♣F♣✲♣♣♣
✟ ❍
✟ ❍
✟ U U U U ❍
1 2 3 4
H H H H H
1 2 3 4 5
Figure 1: Numbering of lines and chambers.
Let beacomplex rank-onelocal system onM( ). Thelocal system is
L A L
determined by non-zero complex numbers (monodromy around H ) q C∗,
i i
∈
i = 1,...,n. Fix a square root q1/2 C∗ for each i.
i ∈
Definition 2.5. (1) For C,C′ ch( ), let us denote by Sep(C,C′) the set
∈ A
of lines H which separate C and C′.
i
∈ A
(2) Define the complex number ∆(C,C′) C by
∈
∆(C,C′) := q1/2 q−1/2.
i − i
Hi∈SYep(C,C′) Hi∈SYep(C,C′)
Now we construct the cochain complex (C[ch• ( )],d ).
F A L
(i) The map d : C[ch0 ( )] C[ch1 ( )] is defined by
L F A −→ F A
n−1
d ([U ]) = ∆(U ,U∨)[U∨]+ ∆(U ,U )[U ].
L 0 0 0 0 0 p p
Xp=1
5
(ii) d : C[ch1 ( )] C[ch2 ( )] is defined by
L F A −→ F A
d ([U ]) = ∆(U ,C)[C]+ ∆(U ,C)[C], (for p = 1,...,n 1),
L p p p
− −
C∈Xch2F(A) C∈Xch2F(A)
αp(C)>0 αp(C)<0
αp+1(C)<0 αp+1(C)>0
d ([U∨]) = ∆(U∨,C)[C].
L 0 − 0
X
αn(C)>0
Example 2.6. Let = H ,...,H , and let the flag be as in Figure 1.
1 5
A { } F
Then
1/2 −1/2
q q
1 − 1
q1/2 q−1/2
12 − 12
d ([U ]) = ([U ],[U ],[U ],[U ],[U∨]) q1/2 q−1/2 ,
L 0 1 2 3 4 0 11/223 − 1−231/2
q q
1234 − 1234
q1/2 q−1/2
12345 − 12345
d ([U ],[U ],[U ],[U ],[U∨]) = ([U∨],[U∨],[U∨],[U∨],[C ],[C ])
L 1 2 3 4 0 1 2 3 4 1 2
1/2 −1/2 1/2 −1/2
q q 0 0 0 (q q )
12345 − 12345 − 1 − 1
1/2 −1/2 1/2 −1/2 1/2 −1/2 1/2 −1/2
q q (q q ) 0 q q (q q )
125 − 125 − 15 − 15 1345 − 1345 − 134 − 134
1/2 −1/2 1/2 −1/2 1/2 −1/2 1/2 −1/2
q q 0 (q q ) q q (q q )
1235 − 1235 − 15 − 15 145 − 145 − 14 − 14 .
× 0 0 0 q1/2 q−1/2 (q1/2 q−1/2)
12345 − 12345 − 1234 − 1234
1/2 −1/2 1/2 −1/2
q q (q q ) 0 0 0
12 − 12 − 1 − 1 1/2 −1/2 1/2 −1/2 1/2 −1/2
0 0 (q q ) q q (q q )
− 5 − 5 45 − 45 − 4 − 4
Theorem 2.7. Under the above notation, (C[ch• ( )],d ) is a cochain com-
F A L
plex and
Hk(C[ch• ( )],d ) Hk(M( ), ).
F A L ≃ A L
See [19, 20, 21] for details.
3 Resonant band algorithm
Let = H ,...,H be an arrangement of affine lines in R2, and let be
1 n
A { } F
a generic flag as in 2.3.
§
Fix an integer k > 1 with k (n+1), and set λ = e2πi/k. In this section, we
|
will give an algorithm for computing the λ-eigenspace H1(F ) of the first
A λ
cohomology of a Milnor fiber.
6
3.1 Resonant bands and standing waves
Definition 3.1. A band B is a region bounded by a pair of consecutive
parallel lines H and H .
i i+1
Each band B includes two unbounded chambers U (B),U (B) ch( ).
1 2
∈ A
By definition, U (B) and U (B) are opposite each other, U (B)∨ = U (B)
1 2 1 2
and U (B)∨ = U (B).
2 1
Define the adjacency distance d(C,C′) between two chambers C and C′
to be the number of lines H that separate C and C′, that is,
∈ A
d(C,C′) = Sep(C,C′) .
| |
The distance d(U (B),U (B)) is called the length of the band B.
1 2
Remark 3.2. Let B be the closure of B in the real projective plane RP2. B
intersects H in one point, B H . Each line H H either passes
∞ ∞ ∞
∩ ∈ A∪{ }
B H or separates U (B) and U (B). Therefore the length of B is equal
∞ 1 2
∩
to n+1 mult(B H ).
∞
− ∩
Definition 3.3. A band B is called k-resonant if the length of B is divisible
by k. We denote the set of all k-resonant bands by RB ( ).
k
A
To a k-resonant band B RB ( ), we can associate a standing wave
k
∈ A
(B) C[ch( )] on the band B as follows:
∇ ∈ A
(B) = eπid(U1k(B),C) e−πid(U1k(B),C) [C]
∇ − ·
C∈Xch(A),(cid:16) (cid:17)
C⊂B
= λd(U1(2B),C) λ−d(U1(2B),C) [C]
− ·
C∈Xch(A),(cid:16) (cid:17) (3)
C⊂B
πd(U (B),C)
1
= 2i sin [C].
· (cid:18) k (cid:19)·
C∈Xch(A),
C⊂B
Remark 3.4. Since the length d(U (B),U (B)) of the band B is divisible by
1 2
k, the coefficients of [U (B)] and [U (B)] in the linear combination in (3) are
1 2
zero. Hence the chambers in the summations in (3) run only over bounded
chambers contained in B. We also note that exchanging of U (B) and U (B)
1 2
affects at most the sign of (B).
∇
Remark 3.5. To indicate the choice of U (B) and U (B), we always put the
1 2
name B of the band in the unbounded chamber U (B) (see Figure 2).
1
7
3.2 Eigenspaces via resonant bands
The map B (B) can be naturally extended to the linear map
7−→ ∇
: C[RB ( )] C[ch( )]. (4)
k
∇ A −→ A
Theorem 3.6. The kernel of is isomorphic to the λ-eigenspace of the
∇
Milnor fiber monodromy, that is,
Ker( : C[RB ( )] C[ch( )]) H1(F ) .
k A λ
∇ A −→ A ≃
In particular, dimH1(F ) is equal to the number of linear relations among
A λ
the standing waves (B), B RB ( ).
k
∇ ∈ A
Proof. Let be the rank-one local system on M( ) defined by q = =
λ 1
L A ···
q = λ C∗ (see 2.1 and 2.3). In this case, ∆(C,C′) depends only on the
n
∈ § §
adjacency distance d(C,C′), or more precisely,
′ ′
∆(C,C′) = λd(C2,C ) λ−d(C2,C ).
−
Now, weconsiderthefirstcohomologygroupH1(C[ch• ( )],d )ofthetwisted
F A L
minimal cochain complex. The image d : C[ch0 ( )] C[ch1 ( )] is gen-
L F A −→ F A
erated by
n−1
dL([U0]) = (λp2 −λ−2p)[Up]+ λn2 −λ−n2 [U0∨].
Xp=1 (cid:0) (cid:1)
Since λ = e2πi/k with k > 1 and k (n+1), we have λn λ−n = λ−n(λn 1) =
2 2 2
| − − 6
0. Thus the coefficient of [U∨] in d ([U ]) is non-zero. Define the subspace
0 L 0
V of C[ch1 ( )] by
F A
n−1
V = C [U ]
p
· (5)
Mp=1
( Coker d : C[ch0 ( )] C[ch1 ( )] ).
≃ L F A −→ F A
(cid:0) (cid:1)
Then H1(C[ch• ( )],d ) is isomorphic to Ker d : V C[ch2 ( )] . It
F A L L|V −→ F A
is sufficient to show that Ker(d ) Ker , (cid:0)which will be done in sev(cid:1)eral
L V
steps. Suppose that ϕ = n−1c| [U≃] K∇er(d ).
p=1 p · p ∈ L|V
P
(i) If H and H are not parallel, then c = 0.
i i+1 i
8
Note that if j = i, then the chamber [U∨] does not appear in d ([U ]). Thus
6 i L j
the coefficient of [U∨] in
i
n−1
d (ϕ) = c d ([U ])
L p L p
·
Xp=1
is c ∆(U ,U∨) = c (λn λ−n). This equals zero if and only if c = 0.
i · i i i 2 − 2 i
Now we may assume that ϕ = c [U ] Ker(d ) is a linear combina-
p p· p ∈ L
tion of [Up]s such that Hp and Hp+P1 are parallel. Suppose that Hi and Hi+1
are parallel and denote by B the band determined by these lines.
i
(ii) If B is not k-resonant, then c = 0.
i i
∨ ∨
In this case, ∆(U ,U∨) = λd(Ui,Ui ) λ−d(Ui,Ui ). By the assumption that
i i 2 − 2
d(U ,U∨) is not divisible by k, we have ∆(U ,U∨) = 0. Since ϕ is a linear
i i i i 6
combination of [U ]s with parallel boundaries H and H , the term [U∨]
p p p+1 i
appears only in d ([U ]), which is equal to c ∆(U ,U∨)[U∨]. Therefore
L i i · i i i
c = 0.
i
Finally we may assume that ϕ is a linear combination of [U ]s such that
p
the boundaries H and H are parallel and the length of the corresponding
p p+1
band B is divisible by k. In this case, it is straightforward to check that the
p
maps d and are identical. This completes the proof.
L
∇
Example 3.7. (A -arrangement, (6,1) or ) The three arrangements in
3 6
A B
Figure2areprojectivelyequivalent, andarerespectivelycalledA -arrangement,
3
(6,1) or . (See 4 for the latter two notations.) We use the left figure
6
A B §
to compute dimH1(F ) . (The symbol indicates that the line at infinity
A λ
∞
is an element of .) Since c = n + 1 = 6, k 2,3,6 and we have
A | A| ∈ { }
RB ( ) = RB ( ) = , RB ( ) = B ,B . By definition, we have
2 6 3 1 2
A A ∅ A { }
(B ) = √ 3 [C ]+√ 3 [C ]
1 1 2
∇ − · − ·
(B ) = √ 3 [C ]+√ 3 [C ].
2 1 2
∇ − · − ·
Hence we have a linear relation (B B ) = 0 and dimH1(F ) = 1 for
1 2 A λ
∇ −
λ = e2πi/3. (Hence the A -arrangement is pure-tone; see Definition 4.1.)
3
Example 3.8. ( (12,2) from [8]) Let be the line arrangement in Figure
A A
3 (together with the line at infinity). Then c = n + 1 = 12. There are
| A|
seven bands, B ,...,B . Among them, B ,B and B have length 7 which
1 7 5 6 7
is coprime with 12 so we can ignore them. We have RB ( ) = B ,B and
3 1 4
A { }
RB ( ) = RB ( ) = B ,B . First consider the case k = 3. Then
2 4 2 3
A A { }
(B ) = √ 3 [C ]+...,
1 1
∇ − ·
(B ) = √ 3 [C ]+....
4 6
∇ − ·
9
B1 (cid:0)
∞
(cid:0)
(cid:0) ❅ (cid:0)
(cid:0) ❅ (cid:0)
U (B )
1 1 (cid:0) ❅ (cid:0)
(cid:0) ❅(cid:0)
C (cid:0) (cid:0)❅
1
B2 U1(B2) (cid:0)C U2(B2) (cid:0) ❅
(cid:0) 2 (cid:0) ❅
(cid:0) (cid:0) ❅
(cid:0)
U (B )
(cid:0) 2 1
(cid:0)
(cid:0)
(cid:0)
Figure 2: The A -arrangement (= (6,1) = )
3 6
A B
Since the chamber C is not contained in the band B , it does not appear
6 1
in the linear combination for (B ). Hence (B ) and (B ) are linearly
1 1 4
∇ ∇ ∇
independent. We conclude that H1(F ) = 0 for λ = e2πi/3. The cases k = 2
A λ
and k = 4 are similar. More precisely, since B ,B RB ( ) = RB ( ) are
2 3 2 4
∈ A A
parallel and they do not overlap, (B ) and (B ) are linearly independent.
2 3
∇ ∇
Consequently we have H1(F ) = 0 and so the cohomology does not have
A 6=1
non-trivial eigenvalues.
❩ ✚
B ❩ ✚
❩ 1 ❩ ✚ ✚
❩ ❩ ✚ ✚
❩ C ❩ ✚ ✚
1 ∞
❩ ❩ ✚ ✚
❩ C ❩ ✚ ✚
B2 C❩3❩2 ✚❩✚❩ ✚✚
❩✚ ❩✚
C ✚❩ ✚❩
4✚ ❩✚ ❩
B
3 ✚ C ✚❩ ❩
✚ 5 ✚ ❩ ❩
✚ C ✚ ❩ ❩
✚ 6✚ ❩ ❩
✚ ✚ ❩ ❩
✚ ✚ ❩ ❩
B ✚ ❩
4✚ B B B ❩
H 5 6 7
1
Figure 3: (12,2)
A
The argument used in Example 3.8 is generalized in the next section. See
4 for further examples.
§
Remark 3.9. The cohomology of the Milnor fiber H1(F ) depends only on
A
the projective arrangement H . The change of the line at infinity H
∞ ∞
A∪{ }
sometimes makes the structure of resonant bands RB simpler. This fact will
k
be used in Corollary 3.16.
10
