MATSUNO, TAKANORI

Matsuno, Takanori

Institutional Repositories DataBase (IRDB)

On a theorem of Zariski-van Kampen type and its applications

Osaka City University Repository

Matsuno, T.
Osaka J. Math.
32 (1995), 645-658
ON A THEOREM OF ZARISKI-VAN KAMPEN
TYPE AND ITS APPLICATIONS
TAKANORI MATSUNO
(Received May 11, 1994)
1. Introduction. Zariski constructed a method to calculate m(P2—C),
where P2 is the complex projective plane and C is a curve on it. In this paper,
following the ideas of Zariski [5] and Van Kampen [4], we give a method to
calculate m(E — S), where E is a holomorphic line bundle over a complex
manifold M and 5 is a hypersurface of E under certain conditions. Applying our
method and the Reidemeister-Schreier method (see Rolfsen [3]), we can calculate
the fundamental groups of regular loci of certain normal complex spaces. We give
a few concrete examples in the final section.
This paper is a revised version of the author's master thesis [ l ] . The author
would like to express his thanks to Professor M. Namba for his useful suggestions
and encouragements and to Professor M. Sakuma whose suggestions about Lemma
1 (see section 2) was a great help to prove Main Theorem. He also expresses his
thanks to the referee for useful comments.
2. Statement of Main Theorem. Let M be a connected -dimensional
complex manifold and μ : E^>M be a holomorphic line bundle over M and 5 be
a hypersurface of E. We assume that E and 5 satisfy the following conditions :
(1) μ : S—>M is a finite proper holomorphic map, where μf is the ristriction
of μ to S(μ' = μ\s).
(2) There is a hypersurfase B of M such that μf\
s
-μ-HB): S~ μ~\B)->M-B
is an unbranched covering of degree .
(3) (dμ')p '. T(S — μ~1(B))p-^>T(M — B)μ'(p) is isomorphic for every point p
Then we have a following lemma whose proof is given in section 4.
Lemma 1. μlε-s-μ-HB): E — S — μ'1^) is a continuous fiber bundle.
We denote a standard fiber od μ : E-^M by F and that of M\E-S-M-HB) : E — S
— μ~\B)-^M — B by F. We assume that there is a continuous section ξ : M-^E
of μ: E^M such that ξ(M)Γ\S = φ (see Figure 1).
646 T. MATSUNO
E
μ
M
ξ(M)
Figure 1
REMARK. Such a continuous sectinuous does not always exist. For example,
if E is a negative line bundle and S is the image of the zero section, then there
exists no such a continuous section.
In order to describe Main Theorem, we must prepare some more symbols. We
choose JPΠ ξ( * ) as a base point bo and we omit the base point hereafter. Since F
can be identified with C — {n points}, πι(F) is isomorphic to the n-ih free group
F
n
 = iγu ", 7n> (see Figure 2).
- n points
Figure 2
Let Q be the kernel of the surjective homomorphism
induced from the injection i : M — B^>M. We assume that Q has a finite presenta-
tion as follows :
Q = <βi, " , βt\Π = l, •", D = l (some relations) >.
THEOREM OF ZARISKI-VAN KAMPEN TYPE 647
Let θ : π\{M — B)-^Bn be the braid monodromy representation of the contin-
uous fiber bundle μ\E-s-μ~
l(B) in Lemma 1, where B
n
 is the n-th braid group :
Bn = <σi, •••, σ
n
-i\[0i, σj] = l(\i-j\>2), Λ Λ + I Λ = Λ - + I Λ Λ + I ( Z = 1 > •••, n-2)>.
We define a homomorphism φ : B
n
—*Aut(π\{F)) as follows :
Then we have the following theorem of Zariski-Van Kampen type:
Main Theorem. If there is a continuous section ξ of μ: E^>M such that
ξ(M)ΠS = φ, then
m(E-S) = <γh - , γn\γj=φ(θ(βk))(γj)(l<j<n, l<k<t)>Xm(M).
(a semi-direct product)
Here βi, •••, βu generate the kernel of the homomorphism j * \ m(M
— B)-^πi(M) and γi, •••, γ
n
 generate the image of the homomorphism i* :
πι(F)-*πι(E — S — μ~ι(B)), where i* is induced from injection i: F^->E—Q
-\B\
REMARK. In Main Theorem, the relations 7j=φ(θ(βk))(γj) are same as the
usual monodromy relations, so it is not essential to facter the homomorphism
m(M-B) >Aut(m(F))
through the braid group.
Corollary. Under the same assumptions in Main Theorem, assume more-
over that M is simply connected (i.e.π\{M) = {\)), then
πi(E-S) = <γh - , 7n\γj=φ(θ(βk))(γj)(l<j<n,
3. Proof of Main Theorem. Since μ : E — S — μ~ι{B)-^M — B is a continu-
ous fiber bundle, there is the following exact sequence:
—• rb(F)—•••• ( e x a c t ) ,
where μ* and <?* are the homomorphisms induced by μ and ξ respectively.
μ*oξ* = id\π2{M-B), since μoξ=id\M-B. Therefore we have Δ = Δ°μ*°f* = 0.
On the other hand 7Γo(/Γ) = {l}, since F is connected. Hence we have the following
exact sequence:
648 T. MATSUNO
(1) 1-*
ξ*
We denote i*πι(F) by K, ξ*πi(M-B) by H and πι{E~S-μ~\B)) by G. The
short exact sequence means that:
G = Ky<\H (a semi-direct product).
Now let B = Bi U ••• UBι be the irreducible decomposition of B and otj be the
meridian of Bj (see Figure 3).
Figure 3
REMARK. Here we assume that B has a finite irreducible decomposition for
simplecity. But even if B has an infinite irreducible decomposition the following
argument is the same.
From a theorem of Van Kampen [4] (see also Namba [2] Cor. 1.2.8), we have
the following exact sequence:
where Q = <ai, ..., aΐ>πx{M~B) is the smallest normal subgroup ofπi(M — B) which
contains a\, •••, aι.
μ~ι(B) is a hypersurface of E, which has the irreducible decomposition
ξ*(aj) is a meridian of μ~ι(Bj), for μ : E^>M is a line bundle and so dμ :
TpM-+Tμ(P)M is surjective. Then, from the theorem of Van Kampen again, we
have the folloing exact sequence:
THEOREM OF ZARISKI-VAN KAMPEN TYPE 649
where <f*(ύfi), ..., ξ*(aι)^>G is the smallest normal subgroup of G which contains
We denote <ξ*(ch), ..., ξM>G by N, ξ*«ai, ..., aι>πl{M~B)) by Q and
KN by i?.
Then we can easily check that
(2) NΓ)H=Q and RΠNH=N.
Consider the natural exact sequence
l->/?/ΛT-> G/ΛΓ-> G/R-+1.
Note that, by (1) and (2),
Hence, we have the exact sequence
(3) 1->R/N
The homomorphism g : (NH)/N^G/N denned by
g: nh(modN) ^>h(nιodN)(n^N,
is well-defined and satisfies f°g= the identity. Hence the exact sequence (3) splits,
so
G/N = (R/N)><](NH/N) ( a semi-direct product).
We can easily check that
where ^a^qaq'^a^K, q^Q^>κ is the smallest normal subgroup of K which
contains {a~1qaq~1\a^K, q^Q}. Furthermore, note that if K and Q are respec-
tively generated by {#i, ..., a
n
) and {qi, ..., qt}, then
KΓ)N=<aJ1qkajqϊ1\l<j<n, l<k<t>κ.
We assume that <ύri, ..., aι^πi(M~B) has a finite presentation as follows :
<oti, ..., β 'z> 7 r i ( Λ f " 5 ) =</?i, ..., βt\Π = l, ..., D = l (some relations)>.
Since K=i*m(F) is isomorphic to the w-th free group </i , ..., 7^>, we have :
KnN = <γ71ζ*(βk)γjζ*(βk)-1\l<j<n, l<k<t>κ.
Thus,
κ/(κnN)=<γh ..., /nlrj^^
Since G/iV is isomorphic to πi(E-S), R/N is isomorphic to K/(KΠN) and
NH/N = H/(NΠH) = H/Q is isomorphic to ri(Λf), we have:
650 T. MATSUNO
πi(E-S) = (K/(KΠN))Xπi(M) (a semi-direct product).
Now useing φ and θ (defined in section 2), we have:
ξMΎjξM-ι = φ{θ{βk)){γj) (see Figure 4).
Hence,
This completes the proof of Main Theorem.
braid
4. Proof of Lemma 1. (Due to M.Sakuma): For a given point q£=M — B,
we can take a neiborhood U of q such that
(i) μ'ΛU) « UU Oi(μ : Uf^U) We write μ
homeomorphic
(ii) The following diagram is commutative.
μ\OSP1 (where P i : (p, z) ->/>)
Here we define a map Aί: Uf-*C as follows:
C.
Then we can write CΛ as follows :
We write Zj = hi(qi), then there exists a positive number ε > 0 such that
(1) Imhi C Int(Z)e(^ί)), where D
e
(zt) is an ε-disk whose center is Zi and /«ί
(D
ε
(zi)) is the interior of D
e
(zi).
(2) Dε(zι), ..., Dε(zn) are disjoint each other.
From Lemma 2 bellow, there exists a fiber preserving homeomorphism Φ such
THEOREM OF ZARISKI-VAN KAMPEN TYPE 651
that Φ(Ui)=Ux{x}.
u
So we can take local coordinates of μ :
Lemma 1.
— S — μ~ι(B)->M — B. This shows
Lemma 2. Let D be an ε-dίsk of C whose center is the origin. Let U be
the neίborhood of q as above. Let h : U-^Int(D) be a continuous map such that
h(q) = 0, where Int(D) is the interior of D. Put 0 = {(x, h(x))<Ξ U X Int(D)\x^
U}czUxInΐ(D). Then there exists a homeomorphism Ψ : UxD^UxD such
that
(i) Ψ(U)=Ux{0}.
(ii) Ψ is fiber preserving, (i.e. the folloing diagram is commutative.)
u
(iii) Ψ\uχdD\ UXdD^UXdD is the identity map.
Proof of Lemma 2. First we define a homeomorphism H
x
 :
point x^ U as follows :
(i) H
x
(h(x))=0.
(ii) Hχ\dD =id\dD.
(iii) H
x
 is extended to D with radial extention (see Figure 5).
for each
D
Figure 5
Second we define a homeomorphism Ψ : U X £>—> U X D as follows :
Ψ(x,z)=Ψ(x,Hx(z)\
Ψ satisfies the above conditions. (q.e.d)
652 T. MATSUNO
5. Case of Trivial Line Bundle. In Main Theorem, we assumed the exis-
tence of a continuous section ξ such that ζ(M)ΠS = φ. In the case of the trivial
line bundle we can prove the following proposition :
Proposition 1. Let M be a connected complex manifold and μ : E-+M be
a trivial line bundle on M(i, e, E=Mx C and μ(p, z)=p for every point (p, z)
G J ί X C). Let /,..., f
n
 be holomorphic functions on M and S be the hypersurface
of E defined by
S = {(/>, z)tΞ
Then there is a continuous section ξ of μ: E^>M such that ξ(M)ΠS = φ.
Proof. We define a continuous function h : Af—• C by
h(p) = \
We define a section ξ: M—+E by
One can easily see that this section ξ of μ satisfies ξ(M)Γ\S=φ. In fact, if there
is a point p^M such that $(p)^.S, then
Since h(p)> 1
1
 hip) {h\p)}2 '" {hip)}"'
Hence
IΛ(/OI
hip) + {hip)}2+
Since {hip)}k>hip)ik=1,2,...),
Ί < \fιiP)\
=
 \MP)\+-+\MP)\
l ( ) l l ( ) l
A contradiction. (q.e.d)
Let //: C^+ 1->Cm be the trivial line bundle on Cm defined by
Let 5 be the hypersurface of Cm+1 defined by
THEOREM OF ZARISKI-VAN KAMPEN TYPE 653
= {(Zi, ..., Zm,
where z=(zi, ..., z
m
) and fi(z), ..., f
n
(z) are polynomials.
By Corollary to Main Theorem and Proposition 1, we have
Theorem 1. Let S be the hypersurface of Cm+1 defined by (1). Then,
Furthermore, let (Xo: Xι: ••• : Xm+i) be homogeneous coordinatss of Pm+1
such that CXi/XΌ, - , X«+i/XΌ) = Ui, ..., zm+i)^ Cm+1 and S be the closure of 5 in
Pm+1. Then we have the following theorem of Zariski:
Theorem 2(Zariski [5]).
Suppose that poo=(0 : ••• : 0 : 1) is not contained in S. Then
πι(Pm+ι-S)
Proof. Let //• be the hypersurface ofPm+1 defined by //ββ = {-XΌ=0}, (z.e.
hyperplane at infinity) and ύf be a meridian of //« in Pm+1 — S — H^ (see Figure 6).
m+1
\
B
Figure 6
From the theorem of Van Kampen [4], we have the following exact sequence:
1—^<<2>7Γl(cn+1"s)^7Γi(Cm+1 — S)^>πi(Pm+ι — S)—>1 (exact).
We can take a as {γnγn-i' 'ϊiY1 in Cm+1 — S (see Figure 7).
Thus,
This shows Theorem 2.
654 T. MATSUNO
REMARK. A similar theorem to Theorem 1 holds for μ: Bm(ε)x
B\ε')-^Bm(ε), where Bm(ε) is a m-dimensional complex ball; Bm(ε) = {(zi, ...,
£OT)eCm | |£i|2H h|^m|2<β2}. In this case, the existence of continuous section with
a similar conditions to Theorem 1 is obvious.
6. Calculations of Fundamental Groups of Finite Branched Coverings
EXAMPLE 1.
Let X be the surface in C3 defined by
X={λ, x, y)(ΞC3\y2=x(χ-l)(χ-λ)}.
X has two isolated singular points at (0,0,0) and (1,1,0). Hence X is normal. Let
π : X—+C2 be the projection map defined by
π(λ, x, y) = (λ, x).
Then π is a double branched covering of C2. The branch loucus 5 of π is a
curve in C2 and is written as :
S = {(λ, x)^C2\x(χ-l)(χ-λ) = 0}.
According to Theorem 1, we can calculate 7Γi( C2— S). Let μ : C2—• C be the trivial
line bundle on C defined by
μ(λ, x) = λ.
The branch locus B of μ is {0, l } c C and πi(C — B) is isomorphic to the free
group </?i, /?2>, where β\ and 02 are its generators and can be considered as the
THEOREM OF ZARISKI-VAN KAMPEN TYPE 655
A"
/
X=λ
x=0
λ
0 1/2 1
Figure 8
meridians of {0} and {l}, respectively. We may take Qo=-^~ as a reference point of
πi(C-B). In this case the standerd fiber F of μ\
c
-s-μ-\B) is C~{3-points}. We
define /i, 72 and 73 as the meridians of l-y, l)»(~o~, ~o~) and (~y, 0 j , respectively.
The image of 0i and 02 by θ : πi(C — B)—^B3 are tff and tfi, respectively. Then we
have
πi(C2-S) = <γh 72,
= </i, 72,
By using the Reidemeister- Schreier method (c.f. Rolfsen [3] P.315-P.316), we
can calculate ri(RegX), where RegX is the set of regular points of X. Since πi(C2
— B) is generated by three elements and since π is a double branched covering, we
take the 3-th free group F3 and the 5-th free group F5. As in Figure 9, we take their
generators {71, 72, 73} and {bi, bi, bz, b4, fa}, respectively, where^""1(7i)==:{^i, #2},
π~
1(γ2):={yi, 3 }^, π~\γz)={zu z2), and
, ; = 1, 2, 3, A? = l , 2>
=7372, 7i/2=727i>
bι=yixϊ
l
 xϊ
ι
— Z\X\λ
Then we transfer the relation of m(C2 — S)7273721n1 = l in the words of F 5 .
656 T. MATSUNO
z2| y^ζ^iί J\^)yi lZi
I
I 7C : doul: double cyclic covering
Figure 9
In a similar way, we transfer the relation 71/2/f V^1 — 1 in the words of F 5 :
We also transfer the relations /i = l, /l = l, 73 = 1 since the ramification index
of each irreducible component of S is equal to 2 :
X\X2 = 0 5 = 1
2—636465 — 1 .
Putting a\ — b\ and cii —bi, we have
m(Reg X) = <ah a2\aϊ=l,
= (Z/2Z)*(Z/2ZXϊτee product).
EXAMPLE 2.
Let X be the hypersurfaces of C4 defined by
X={(x, y, z, w)£ΞC4\wn=z2-χy2}(n>2).
The singular locus of X is the line {(x, y, z, w)^X\y = z=w = 0}. Hence X is
normal. Let π : X^>C3 be the projection map defined by :
π(x, y, z, w) = (x, y, z).
Then π is a cyclic branched covering of C3. The branch locus 5 of π is a
surface in C3 and is written as :
THEOREM OF ZARISKI-VAN KAMPEN TYPE 657
S = {(x, y, z)^C3\z2-χy2=0} (the Cartan umbrella).
According to Theorem 1, we can calculate πi(C3 — S). The result is
πi(C3-S) = <γ>(the free group).
From the Reidemeister-Schreier method again, we have:
πi(RegX) = {ie. RegX is simply connected).
EXAMPLE 3.
Let X be the hypersurface of Cm+2 defined by
u ..., z
m
)=0},
where g is a polynomial which is not constant. The singular locus of X is at most
(m — l)-dimensional. Hence X is normal. Let π : X—> Cm+1 be the projection map
defined by:
7Γ\Zi, ..., Zm + l, Zm+2) — \Zl, ..., Zm + l).
Then 71 is a branched covering of Cm+1. The branch locus 5 of n is a hypersurface
in Cm+1 and is written as :
By Theorem 1, πi(Cm+1-S) can be written as:
From the Reidemeister-Schreier method again, we have:
{Dor
m(RegX) = Z/qZ(3q(ΞZ) or
Z
(Le. πi(RegX) is isomorphic to a cyclic group).
References
[l] T. Matsuno : Normal singularities and fundamental groups, Osaka Univ. master theses (in
Japanese).
[2] M. Namba : Branched coverings and algebraic functions, Pitman Res. Notes Math. Ser., Vol.161,
Longman Sci. Tech., Harlow,1987.
[3] D. Rolfsen : Knots and links, Publish or Perish, Berkeley, 1976.
[4] van Kampen: On the fundamental group of an algebraic curves, Amer.J.Math. 55 (1933),
255-260.
[5] O. Zariski: Collected Papers Vol 3, MIT Press, Cambridge, 1973.
658 T. MATSUNO
Department of Mathematics
Faculty of Science
Osaka University
Toyonaka, Osaka 560, Japan


Osaka University Knowledge Archive

http://ir.library.osaka-u.ac.jp/dspace/bitstream/11094/7960/1/ojm32_03_08.pdf

On a theorem of Zariski-van Kampen type and its applications

Abstract

Similar works

Full text

Available Versions

Institutional Repositories DataBase (IRDB)

Osaka City University Repository

Osaka University Knowledge Archive