Matumoto, Takao

Nakagawa, Noriaki

Osaka University Knowledge Archive

Complement to explicit description of Hopf surfaces and their automorphism groups

Osaka City University Repository

Matumoto, T. and Nakagawa, N.
Osaka J. Math.
48 (2011), 583–588
COMPLEMENT TO EXPLICIT DESCRIPTION
OF HOPF SURFACES
AND THEIR AUTOMORPHISM GROUPS
TAKAO MATUMOTO and NORIAKI NAKAGAWA
(Received August 3, 2009, revised February 8, 2010)
Abstract
In the previous paper we determined fAut(X) of each Hopf surface X D W=G
with W D C2   (0, 0) so that its holomorphic automorphism group is given by
Aut(X) D fAut(X)=G. We calculate the group of connected components 0(Aut(X))
by reviewing the classification.
A Hopf surface X is a compact complex surface whose universal covering space
is W D C2   (0, 0). So, X D W=G by denoting its covering transformation group by
G. A Hopf surface is called primary if its fundamental group G is isomorphic to the
group Z of integers, and secondary otherwise.
In Theorem 1 of [3] we determined fAut(X ) of each secondary Hopf surface X D
W=G so that its holomorphic automorphism group is given by Aut(X ) D fAut(X )=G,
where fAut(X ) is the normalizer of G in the holomorphic automorphism group of C2
fixing (0, 0). Moreover, in the following cases (2) and (3) it coincides with the nor-
malizer of G in GL(2, C). We calculate 0(Aut(X )) including primary Hopf surfaces
as a continuation by correcting some parts of [3].
Before stating Theorem 2 we review the classification of Hopf surfaces. Except
the special case (0) that G is not given by any subgroup of GL(2, C), we may assume
that G  GL(2, C) and G is an extension of H D fg 2 G I jdet gj D 1g by Z. Let a be
a primitive m-th root of 1, n D exp( i=n),  D 4,  D exp(2 i=5) and , ,  2 C
with 0 < jj, jj, j j < 1. Also, we define K D fg 2 G I det g D 1g  H .
The case (1) that G  GL(2, C) and G is abelian is divided into two cases:
(A) G is generated by g(z1, z2)D (z1,z2) and h(z1, z2)D (az1, anz2) with (m, n)D1 and
(B) G is generated by g(z1, z2) D (z1 C z2, z2) and h(z1, z2) D (az1, az2).
The case (2) that G is not abelian and decomposable, that is, isomorphic to ZH
is divided into six cases.
2000 Mathematics Subject Classification. 32J15.
584 T. MATUMOTO AND N. NAKAGAWA
(C1) G D h I i  H1 where H1 D haI i  B 02k (2lC1) and K D A2(2lC1) with (2k(2l C
1), m) D 1, 2l C 1  3 and k  3. Note that B 02k (2lC1) D
D
h0 D

s 0
0 s 1

, h D

0 d
d 0
E
such that s and d have finite orders 2l C 1 and 2k respectively. Note also that K D
A2(2lC1) is generated by  h0.
(C2) G D h I i  H2 where H2 D haI i  Bn and K D Bn with (m, 4n) D 1 and n  2.
Here Bn D
D
0 i
i 0

,

n 0
0  1n
E
is the binary dihedral group of order 4n.
(C3) G D h I i  H3 where H3 D haI i  C and K D C with (m, 6) D 1. Here C D
D
i 0
0  i

, (1=p2)


3

3
  
E
is the binary tetrahedral group of order 24.
(C4) G D h I i  H4 where H4 D haI i  C 083k and K D B2 with (m, 6) D 1 and k 
2. Here C 083k D
D
i 0
0  i

, (!=p2)


3

3
  
E
is a group of order 8  3k and ! is a
primitive 3k-th root of 1.
(C5) G D h I i  H5 where H5 D haI i  D and K D D with (m, 6) D 1. Here D D
D
 0
0  1

, (1=p2)


3

3
  
E
is the binary octahedral group of order 48.
(C6) G D h I i  H6 where H6 D haI i  E and K D E with (m, 30) D 1. Here
E D
D

3 0
0 2

,

0  1
1 0

, (1=p5)


4
   
2
  
3

2
  
3
   
4
E
is the binary icosahedral group of
order 120.
The case (3) that G is indecomposable, that is, not isomorphic to the product
Z  H : G is given as
G D G0 [ gG0, G0 D h 2 I i  H and g D  u
in the following cases from (D1) to (D6) and in the exceptional case (D7)
G D G0 [ gG0 [ g2G0, G0 D h 3 I i  H and g D  u,
where H is a finite cyclic group or one of H in the case (2) and u 2 GL(2, C).
(D1) The case H is abelian and K D
D
s 0
0 s 1
E
is of order mK  3: We can take
u D

0 1
1 0

and the case is divided into the following three cases.
(D1-1) H D haI i  K with K D
D
s 0
0 s 1
E
where s has the finite order mK  3 and
(m, mK ) D 1.
(D1-2) H D haI i 
D
b 0
0 b 1
E

D
c 0
0  c 1
E
where b and c have finite orders 2l C
1  1 and 2k with k  3 respectively. Moreover, we have (m, 2l C 1) D (m, 2) D 1,
mK D 2k 1(2l C 1) and s D bc2.
(D1-3) H D haI i
D
b 0
0 b 1
E

D
c 0
0  c
E
where b and c have finite orders 2lC1  3
and 2k with k  3 respectively. Moreover, (m, 2l C 1) D (m, 2) D 1, mK D 2(2l C 1)
and s D  b.
HOPF SURFACES AND THEIR AUTOMORPHISM GROUPS 585
(D2) The case H is abelian and K D fI g: H D haI i 
D
c 0
0  c
E
and u D

0 1
1 0

,
where c is a primitive 2k-th root of 1 with k  3 and (2, m) D 1.
(D3) H D H1 as in (C1) and u D

0 t 1
t 0

. We have two conjugate classes: t D 1 and
t D i . Their transformation groups are not isomorphic to each other.
(D4 and D5) H D haI i  Bn , K D Bn as in (C2) and u D

2n 0
0  12n

where (D4) is
the case n  3 and (D5) is the case n D 2.
(D6-1)1 H D haI i  C , K D C as in (C3) and u D

 0
0  1

.
(D6-2) H D haI i  C 083k with k  2, K D B2 as in (C4) and u D

 0
0  1

.
(D7) H D haI i  B2, K D B2 as in (D5) and u D (1=
p
2)


3

3
  

.
Since we did not give a proof of the classification for the indecomposable cases
(D1) to (D7), we will give an outline of proof. We may assume K ¤ f I g, since G is
abelian otherwise (cf. [1] Proposition 8).
We consider the case that H is abelian at first. Since H operates on S3 freely,
H is a cyclic group of order m H . We may assume that the generator is

d 0
0 dn

.
The matrix u is also determined as u D

0 t 1
t 0

in this case by [1] Lemma 6 and
Proposition 8. The condition n2  1 mod m H should be satisfied because the con-
jugation of the generator by u is contained in H . Since u does not commute with
the generator of H , we get the condition n ¥ 1 mod m H . Note that n2  1 mod pk
implies n  1 mod pk for odd prime p. In case p D 2, n2  1 mod 2k implies
n  1 mod 2k for k D 1, 2 and n  1 or n  1 C 2k 1 mod 2k for k  3. Let
m H D 2k pk11    p
kq
q be the prime decomposition.
(D1) Assume first that n  1 mod 2k . Then, we get the case (D1-1). When
k  3 and n   1C 2k 1 mod 2k , we get the case (D1-2). When n  1C 2k 1 mod 2k
with k  3, it is easy to see that K D fI g if and only if n  1 mod pk jj for every
odd prime p j . Since we treat the case K ¤ fI g in (D1), the case when n  1 C
2k 1 mod 2k with k  3 and n ¥ 1 mod pk jj for some odd prime p j is named (D1-3).
We see also that the matrix u above with any t 2 C is conjugate to u with t D 1 in
these three cases.
(D2) This is the remaining case when n  1C2k 1 mod 2k with k  3 and n  1
mod pk jj for every odd prime p j . Note that this case K D fI g is studied separately
in [1] p. 229 and u is determined as above in the indecomposable case. For the matrix
u we can take t D 1 in the same way as in the case (D1).
When K is not abelian, [1, 2] Lemma 7 and Lemma 70 imply that K and u are
uniquely determined as in the cases (D4) to (D7) including the case (D6-2).
1The case (D6-1) is denoted by (D6) in [3] and (D6-2) was missing there.
586 T. MATUMOTO AND N. NAKAGAWA
Finally when K is abelian and H is not abelian, we may assume that H D H1 as
in (C1). Since K D A2(2lC1), we are concerned only with Step 3 of [1] pp. 235–236.
Hence, the group G is conjugate in GL(2, C) to one of the non-isomorphic groups
hh0, h,  u1i and hh0, h,  u2i, where h0 D

s 0
0 s 1

, h D

0 d
d 0

, u1 D

0 1
1 0

and
u2 D

0  i
i 0

. We get the case (D3) and no others. This is the end of the outline
of proof.
The case (D3) is subdivided and remarkable since the diffeomorphic classes can
be distinguished by 0(Aut(X )).
Theorem 2. The group of connected components 0(Aut(X )) of the holomorphic
automorphism group Aut(X ) D fAut(X )=G of each Hopf surface X D fC2   (0, 0)g=G
is described as follows.
(0) The case that G is not given by any subgroup of GL(2, C) (1(X )  Z  Zm):
0(Aut(X )) D 0.
(1) The case that G is contained in GL(2, C) and abelian (1(X )  Z  Zm):
0(Aut(X )) D 0,
except the case (A2) when  D  and n ¥ n2  1 mod m where
0(Aut(X ))  Z2.
(2) The case that G is decomposable and not abelian:
0(Aut(X )) D 0 in the cases (C5) and (C6).
0(Aut(X ))  Z2 in the cases (C1), (C2) with n  3, (C3) and (C4).
0(Aut(X ))  D3 in the case (C2) with n D 2, where Dk is the dihedral group of
order 2k.
(3) The case that G is indecomposable:
0(Aut(X )) D 0 in the cases (D1-1) with odd mK , (D1-2), (D1-3), (D2), (D3) with
t D i , (D4), (D5), (D6-1), (D6-2) and (D7).
0(Aut(X ))  Z2 in the cases (D1-1) with even mK and (D3) with t D 1.
Corollary. For any primary Hopf surface X we see that 0(Aut(X )) D 0.
It is quite interesting to see that any decomposable Hopf surface with non-trivial
0(Aut(X )) is a double or triple covering of the corresponding indecomposable Hopf
surface: (A2) ! (D1) or (D2), (C1) ! (D3) with t D i , (C2) with n  3 ! (D4),
(C2) with n D 2 ! (D5) or (D7), (C3) ! (D6-1) and (C4) ! (D6-2). In fact, the
covering transformation element u in these cases is non-trivial in 0(Aut(X )). Simi-
larly (D1-1) with mK D 2l is a double covering of (D1-1) with mK D 4l. But as is
well-known, the natural operation of
D
i 0
0  i
E.D
 1 0
0  1
E
 Z2 on the Hopf sur-
face in the case (C1) is not fixed point free, and the same for the case (D3) with t D 1.
HOPF SURFACES AND THEIR AUTOMORPHISM GROUPS 587
The proof of Theorem 2 follows easily from Theorem 1 of [3]. In the cases (0) and (1)
except (A2), we have 0(fAut(X )) D 0 and hence 0(Aut(X )) D 0. In the case (A2)
0(Aut(X )) D C I 

0 1
1 0

C I  Z2.
Now note that 0((C I G 0)=G) D 0((C I G 0)=(C I G)) D 0((C I G 0)=(C I 
G 00)) D G 0=G 00 for a normal subgroup G 00 of a finite subgroup G 0 of GL(2, C), provided
that C I  G 00 D C I  G and 0(G 0) D 0(C I  G 0).
Then, in the case (2) we get the result by noticing that B2n=Bn  Z2, D=B2 
S4=D2  D3, D=C  Z2 and C I C 083k D C
 I C . In fact, we take G 0 D B2(2lC1) and
G 00 D B2lC1 in the case (C1). Here, Bm D
D
0 i
i 0

,

m 0
0  1m
E
and the fact

i 0
0  i


B2lC1 is essential. In the other cases we take G 0 D B2n and G 00 D Bn in the case (C2)
with n  3, G 0 D D and G 00 D B2 in the case (C2) with n D 2, G 0 D D and G 00 D C
in the cases (C3) and (C4), G 0 D G 00 D D in the case (C5) and G 0 D G 00 D E in the
case (C6).
The proof for the case (3) is a little more delicate. In the case (D1) we have
fAut(X ) D C I  BmK . Moreover, in the case (D1-1) if mK is odd then  I 2 C I  G
implies C I  G D C I  BmK , otherwise C I  G D C I  BmK =2. In the case (D1-2)

c 0
0  c 1

2 G implies

i 0
0  i

2 G and hence

c 0
0 c 1

,

0 i
i 0

2 C I  G. On the
other hand we can take bc D mK for mK D 2k 1(2l C 1). So, we have C I  G D
C I  BmK . In the cases (D1-3) and (D2)

c 0
0  c

2 C I G implies

i 0
0  i

2 C I G
and hence C I  G D C I  B2(2lC1), where l  1 in the case (D1-3) and l D 0 in the
case (D2). We used the correction fAut(X ) D C I  B2 below in the case (D2) with
t D 1. Note that

i 0
0  i

D

0  i
i 0

0  1
 1 0

is contained in C I G and C I G D
C I  B2(2lC1) in the case (D3) with t D i . In the case (D3) with t D 1 the same elem-
ent is not contained in C I G and hence C I G D C I  B2lC1. So, we get the result
in these cases. For the other cases we have only to note that C I  G D C I  B2n
in the case (D4), C I  G D C I  B4 in the case (D5), C I  G D C I  D in the
case (D6-1) and C I G D C I C in the case (D7). Also, it is not difficult to see that
fAut(X ) D C I D D C I G and 0(Aut(X )) D 0 in the case (D6-2) which was missing
in Theorem 1 of [3]. This completes the proof of Theorem 2.
Finally we note that there are six errors in [3] to be corrected. First ‘splits’ in
the 7-th line from the bottom of the p. 1 should read ‘decomposes’. Second shift the
condition from k  1 to k  2 in the case (C4); the action C 024 on S3 has fixed points.
Third in the case (D2) where K D fI g and H is abelian we should correct m D
2(2l C 1)  6 to m D 2k(2l C 1) with k  3, where m denotes the order of H . (In
the new description given in this paper m D 2l C 1 and the order of H is 2k(2l C 1).)
Fourth we should not restrict ourselves to t ¤ 1 in the case (D3); there are just two
conjugate classes as stated above. Fifth there is one more case (D6-2).
588 T. MATUMOTO AND N. NAKAGAWA
Sixth in the case (D2) of Theorem 1 we should correct the conclusion as follows:
(D2) fAut(X ) D C I
D
i 0
0  i

,

0 t 1
t 0
E
 C I  B2.
The proof which is located in the last five lines of p. 422 of [3] should read:
In the case (D2) we have NGL(2,C)(H ) D
n
a0 0
0 d 0

,

0 b0
c0 0

a0, b0, c0, d 0 2 C
o
.
The commutator of u and

a0 0
0 d 0

is
 d 0=a0 0
0 a0=d 0

and that of u and

0 b0
c0 0

is

c0t 2=b0 0
0 b0t2=c0

. Since they should be contained in K , we have a0 D d 0 and
c0 D b0t2. So, we get the result.
References
[1] M. Kato: Topology of Hopf surfaces, J. Math. Soc. Japan 27 (1975), 222–238.
[2] M. Kato: Erratum to: “Topology of Hopf surfaces”, J. Math. Soc. Japan 41 (1989), 173–174.
[3] T. Matumoto and N. Nakagawa: Explicit description of Hopf surfaces and their automorphism
groups, Osaka J. Math. 37 (2000), 417–424.
[4] N. Nakagawa: Moduli spaces for secondary Hopf surfaces, Hiroshima Math. J. 40 (2010),
37–45.
Takao Matumoto
16-7-1002 Osuga-cho
Minami-ku, Hiroshima 732–0821
Japan
e-mail: matumoto@math.sci.hiroshima-u.ac.jp
Noriaki Nakagawa
Tohnan High School
Minami-ku, Kyoto 601–8348
Japan
e-mail: wp-n-nkgw@s4.dion.ne.jp


http://ir.library.osaka-u.ac.jp/dspace/bitstream/11094/8669/1/ojm48_02_583.pdf

Complement to explicit description of Hopf surfaces and their automorphism groups

Abstract

Similar works

Full text

Available Versions

Osaka University Knowledge Archive

Osaka City University Repository