Configuration spaces of tori

0.34 MB·English

by  Yoel Feler

CONFIGURATION SPACES OF TORI
6
0
YOELFELER
0
2
Abstract. The configuration spaces Cn(T2) = {Q ⊂ T2| #Q = n} and
n
a En(T2)={(q1,...,qn)∈(T2)n|qi6=qj ∀i6=j}ofatorusT2=C/{alattice}
J are complex manifolds. We prove that for n > 4 any holomorphic self-map
F of Cn(T2) either carries the whole of Cn(T2) into an orbit of the diag-
2 onal Aut(T2) action in Cn(T2) or is of the form F(Q) = T(Q)Q, where
1 T: Cn(T2) → Aut(T2) is a holomorphic map. We also prove that for n > 4
anyendomorphismofthetorusbraidgroupBn(T2)=π1(Cn(T2))withanon-
V] abelianimagepreservesthepuretorusbraidgroupPBn(T2)=π1(En(T2)).
C
.
h
t
a
m
[ Contents

1
1. Introduction 2
v
1.1. Configuration spaces 2
3
8 1.2. Plan of the proof of Theorem 1.1 3
2 1.3. Notation and definitions 4
1 2. Some algebraic properties of torus braid groups 5
0 2.1. Zariski presentation and homomorphism B →B (T2) 5
n n
6
2.2. Normal series in the torus pure braid groups P (T2) 5
0 n
/ 2.3. Proof of Theorem 1.4 7
h
3. Ordered configuration spaces 7
t
a 3.1. A Cartesian product structure in En(T2) 7
m 3.2. Holomorphic mappings En(T2)→T2\{0} 8
: 3.3. A simplicial structure on the set of differences 11
v
3.4. Regular mappings En(T2)→T2 12
i
X 4. Holomorphic mappings of configuration spaces 13
r 4.1. Strictly equivariant mappings 13
a
4.2. Proof of Theorem 1.1 14
4.3. Splitting of abelian maps 16
4.4. Holomorphic mappings T2 →Cn(T2) 17
5. Biregular automorphisms 17
6. Configuration spaces of universal families 18
7. Acknowledgements 21
References 21
2000 Mathematics Subject Classification. 32H02,32H25,32C18,32M05,14J50.
Key words and phrases. configurationspace,torusbraidgroup,holomorphicendomorphism.
ThismaterialisbaseduponPhDthesissupportedbyTechnion.
1
2 Y.FELER
1. Introduction
Inthispaperwestudycertainpropertiesofthetorusconfigurationspacesandtorus
braid groups.
1.1. Configuration spaces. The nth configuration space Cn = Cn(X) of a com-
plexspaceX consistsofallnpointsubsets(“configurations”)Q={q ,...,q }⊂X.
1 n
An automorphism T ∈ Aut X gives rise to the holomorphic endomorphism F of
T
Cn defined by F (Q) = TQ = {Tq ,...,Tq }. If Aut X is a complex Lie group,
T 1 n
one may take T = T(Q) depending analytically on a configuration Q ⊂ X and
define the corresponding holomorphic endomorphism F by F (Q) = T(Q)Q.
T T
Such a map F is called tame. On the other hand, choosing a base configura-
T
tion Q0 = {q0,...,q0} ⊂ X, one may define an endomorphism F , F (Q) =
1 n T,Q0 T,Q0
T(Q)Q0 = {T(Q)q0,...,T(Q)q0}, which certainly maps the whole configuration
1 n
space into one orbit (Aut X)Q0 of the diagonal Aut X action in Cn; endomor-
phisms that have the latter property are said to be orbit-like.
Ofcourse,configurationspaces ofa certainspecific space X may admit “sporadic”
holomorphicendomorphisms,whichareneither tamenororbit-like;butitis notto
be expected that there is a general construction of such endomorphisms.
LetusrestrictourselvestothesimplestinterestingcasewhenX isanon-hyperbolic
Riemann surface1, i.e. one of the following curves: complex projective line CP1;
complex affine line C; complex affine line with one puncture C∗ =C\{0}; a torus
T2 =C/{a lattice of rank 2}.
V. Lin (see [1, 2, 3], etc.) proved that for n > 4 and X = C or X = CP1 every
holomorphic endomorphism F of Cn(X) is either tame or orbit-like. Moreover, F
is tame if and only if it is non-abelian, meaning that the image F∗(π1(Cn(X))) of
the induced endomorphism F∗ of the fundamental group π1(Cn(X)) (which is the
braidgroupB (X)of X)is a non-abeliangroup; otherwiseF is orbit-like. Similar
n
results were obtained by V. Zinde (see [4, 5, 6, 7, 8]) for X =C∗.
In this paper, we complete the story for all non-hyperbolic Riemann surfaces; one
of our main results is as follows.
Theorem 1.1. For n > 4, each holomorphic map F: Cn(T2) → Cn(T2) is either
tame or orbit-like. In particular, any automorphism of Cn(T2) is tame.
Moreover, F is tame if and only if the induced endomorphism F∗ of the funda-
mental group π1(Cn(T2)) is non-abelian, i.e. its image F∗(π1(Cn(T2))) is a non-
abelian group.
Thistheoremhastwoimmediatecorollaries. Thefirstshowsthatnon-abelianholo-
morphicendomorphismsofCn(T2)admitasimpleclassificationuptoaholomorphic
homotopy.2
Corollary 1.2. For n > 4, the set H(Cn(T2),Cn(T2)) of all holomorphic homo-
topy classes of non-abelian holomorphic endomorphisms of Cn(T2) is in a natural
one-to-one correspondence with the set H(Cn(T2),Aut T2) of all holomorphic ho-
motopy classes of holomorphic maps Cn(T2)→Aut T2.
1TheautomorphismgroupofahyperbolicRiemannsurfaceX isdiscrete;infact,a“generic”
hyperbolicRiemannsurfacedoesnotadmitanon-identicalautomorphism.
2Thatis,ahomotopywithinthespaceofholomorphicmappings.
CONFIGURATION SPACES OF TORI 3
Corollary 1.3. Let n > 4 and G= Aut Cn(T2). Then the orbits of the natural
G-action in Cn(T2) coincide with the orbits of the diagonal (Aut T2)-action in
Cn(T2).
Our second main theorem deals with the torus braid group B (T2) = π (Cn(T2))
n 1
andits purebraidsubgroupP (T2),whichisthe fundamentalgroupoftheordered
n
configuration space
En(T2)={q =(q ,...,q )∈(T2)n| q 6=q ∀i6=j}.
1 n i j
Theorem 1.4. a) Let n > 4. Then the pure braid group P (T2) ⊂ B (T2) is
n n
invariantunderanynon-abelianendomorphismϕofthewholebraidgroupB (T2),
n
that is, ϕ(P (T2))⊆P (T2).
n n
b) Let n > 4 and n > m. Then any homomorphism ϕ: B (T2) → B (T2) is
n m
abelian, i.e. ϕ(B (T2)) is an abelian group.
n
ForautomorphismsoftheclassicalArtinbraidgroupB (C),ananalogueofpart(a)
n
wasprovedbyArtinhimselfin1947(see[9]). V.Lin[10,2,11,12,3]generalizedthis
resultofArtintonon-abelianendomorphismsofB (C)andB (CP1),respectively;
n n
the case of B (C∗) was handled by V. Zinde [7, 8]. In 1992, N. Ivanov [13] proved
n
a similar result for automorphisms of the braid group B (X) of any finite type
n
Riemann surface but for X = CP1. Theorem 1.4(a) completes the story for non-
hyperbolic curves. Analogues of statement (b) were known for the braid groups
B (C), B (CP1) and B (C∗) (see papers by V. Lin and V. Zinde quoted above)
n n n
1.2. Plan of the proof of Theorem 1.1. First, due to Theorem 1.4(a), every
continuous non-abelian self-map F fits into a commutative diagram
En(T2) f- En(T2)
p p (1.1)
? ?
Cn(T2) - Cn(T2),
F
where p is the natural projection
En(T2)∋q =(q ,...,q )7→{q ,...,q }=Q∈Cn(T2). (1.2)
1 n 1 n
The map f is strictly equivariant with respect to the standard action of the sym-
metric group S(n) in En(T2), meaning that there is an automorphism α of S(n)
such that
F(σq)=α(σ)F(q) for all q ∈En(T2) and σ ∈S(n).
Moreover,f is non-constant and holomorphic whenever F is so.
AtorusT2 =C/{a lattice of rank2}whichwedealwithisanadditivecomplexLie
group. To study non-constant strictly equivariant holomorphic endomorphisms f
of the space En =En(T2), we start with an explicit description of all non-constant
holomorphic maps λ: En → T2 \{0}. The set L of all such maps is finite and
separates points of a certain submanifold M ⊂ En of complex codimension 1. An
endomorphism f induces a self-map f∗ of L via f∗: L∋λ7→f∗λ=λ◦f ∈L.
The map f∗ carries important information about f. In order to investigate
behaviour of f∗ and then recover this information, we endow L with the following
simplicial structure: a subset ∆s = {λ ,...,λ } ⊆ L is said to be an s-simplex
0 s
whenever λ − λ ∈ L for all distinct i,j. The action of S(n) in En induces a
i j
simplicial S(n)-action in the complex L; the orbits of this action may be exhibited
4 Y.FELER
explicitly. On the other hand, the map f∗: L→L defined above is simplicial and
preserves dimension of simplices; since f is equivariant, f∗ is nicely related to the
S(n) action on L.
Studying all these things together, we eventually come to the conclusion that f
is tame, meaning that there exists an S(n) invariant holomorphic map
t: En(T2)→Aut T2 (1.3)
and a permutation σ ∈S(n) such that
f(q)=σt(q)q =(t(q)qσ−1(1),...,t(q)qσ−1(n)) ∀q =(q1,...,qn)∈En(T2). (1.4)
The latter formula implies that the original endomorphism F is tame.
For the complete proofs, see Section 2 (the proof of Theorem 1.4 and some related
algebraic results), Section 3 (a simplicial complex of holomorphic maps En(T2)→
T2\{0}) and Section 4 (the proof of Theorem 1.1).
1.3. Notation and definitions. For the reader’s convenience, we collected here
the main notation and definitions used throughout the paper.
Definition 1.5. The ordered and non-ordered configuration spaces of a torus T2
are defined as follows:
En(T2)={(q ,...,q )∈(T2)n| q 6=q ∀i6=j},
1 n i j
Cn(T2)={Q⊂T2| #Q=n}.
The projection
p: En(T2)→Cn(T2), p(q)=p(q ,...,q )={q ,...,q }=Q,
1 n 1 n
is an S(n) Galois covering and we have the exact sequence of the corresponding
fundamental groups
1→π (En(T2))−p→∗ π (Cn(T2))→S(n)→1. (1.5)
1 1
The fundamental group π (Cn(T2)) is called the torus braid group and is denoted
1
by B (T2). Its normal subgroup P (T2) =d=ef π (En(T2)) is called the pure torus
n n 1
braid group.
Definition 1.6. A group homomorphism ϕ: G → H is said to be abelian if its
imageϕ(G) isanabeliansubgroupofH. AcontinuousmapF: X →Y ofarcwise-
connected spaces is called abelian if the induced homomorphism F∗: π1(X) →
π (Y) of the fundamental groups is abelian.
1
For a complex space X, we denote by Aut X the group of all biholomorphic auto-
morphisms of X. For algebraic X, Aut X stands for the group of all biregular
reg
automorphisms.
The group Aut T2 = Aut T2 is a compact complex Lie group isomorphic
reg
to a semidirect product T2 ⋋ Z ; here k = 2 if the torus T2 has no complex
k
multiplications; otherwise, either k =4 or k =6.
Definition 1.7. A holomorphic endomorphism F of Cn(T2) is said to be tame if
there is a holomorphic map T: Cn(T2)→Aut T2 such that F(Q)=T(Q)Q for all
Q∈Cn(T2).
CONFIGURATION SPACES OF TORI 5
Definition 1.8. A holomorphic map F: Cn(T2) →Cn(T2) is said to be orbit-like
if its image F(Cn(T2)) is contained in one orbit of the diagonal Aut T2 action in
Cn(T2)).
2. Some algebraic properties of torus braid groups
The mainaim ofthis sectionis to proveTheorem1.4. To this end, we firstneed to
prove some auxiliary results, which also seem to be of independent interest.
2.1. Zariski presentation and homomorphismB →B (T2). Wewillusethe
n n
following presentation of the torus braid group B (T2) found by O. Zariski [14]3.
n
Generators:
σ1,...,σn−1; a1,a2; (2.1)
relations:
σ σ =σ σ for |i−j|≥2, i,j =1,...,n−3; (2.2)
i j j i
σ σ σ =σ σ σ for i=1,...,n−2; (2.3)
i i+1 i i+1 i i+1
σ a =a σ for k =1,2 and i=2,...,n−1; (2.4)
i k k i
(σ−1a )2 =(a σ−1)2 for k =1,2; (2.5)
1 k k 1
σ1...σn−2σn2−1σn−2...σ1 =a1a−21a−11a2; (2.6)
a σ−1a−1σ a−1σ−1a σ =σ2. (2.7)
2 1 1 1 2 1 1 1 1
2.2. Normal series in the torus pure braid groups P (T2). Theexacthomo-
n
topy sequence (1.5) of the covering p: En(T2)→Cn(T2) may be written as
1→P (T2)−p→∗ B (T2)−µ→S(n)→1, (2.8)
n n
where the epimorphism µ is defined by
µ(σ )=(i,i+1) for i=1,...,n−1 and µ(a )=µ(a )=1. (2.9)
i 1 2
Let us take a point (c ,...,c )∈En(T2) and for each m =1,...,n−1 consider the
1 n
ordered configuration space En−m(T2 \{c ,...,c }) of the torus T2 punctured at
1 m
m points c ,...,c ; a point q ∈En−m(T2\{c ,...,c }) has n−m pairwise distinct
1 m 1 m
componentsq ,...,q ∈T2\{c ,...,c }. Eachoftheseconfigurationspacesisan
m+1 n 1 m
Eilenberg–MacLane K(π,1) space for its fundamental group π (see E. Fadell and
L. Neuwirth [16], Corollary2.2), which is the pure braid group(on n−m strands)
Pn−m;m of the punctured torus T2\{c1,...,cm}. The projections
t : En(T2)→T2, (q ,...,q )7→q , (2.10)
1 1 n 1
t : En−m(T2\{c ,...,c })→T2\{c ,...,c },
m+1 1 m 1 m
(2.11)
(q ,...,q )7→q , 1≤m≤n−2,
m+1 n m+1
definelocallytrivial,smoothfibrings4withfibresisomorphictoEn−1(T2\{c })and
1
En−m−1(T2\{c ,...,c }),respectively. Thefinalsegmentsoftheexacthomotopy
1 m+1
sequences of these fibrings look, respectively, as
1→Pn−1;1 →Pn(T2)→Z2 →1
3Another presentationwasfoundbyJ.Birman[15].
4See[16]forthecaseofarbitraryRiemannsurfaces.
6 Y.FELER
and
1→Pn−m−1;m+1 →Pn−m;m →Fm →1,
whereF standsforafreegroupofrankm. Thisleadsimmediatelytothefollowing
m
statement, which is an analogue of a well-known Markov theorem for Artin pure
braid groups, proved in [17].
Proposition 2.1. The subgroups Pn−s;s fit into the normal series:
{1}⊂P1;n−1 ⊂···⊂Pn−m−1;m+1 ⊂Pn−m;m ⊂···⊂Pn−1;1 ⊂Pn;0 =Pn(T2)
with the factors
P1;n−1/{1}=P1;n−1 =π1(T2\{a1,...,an−1})∼=Fn−1, ...,
Pn−m;m/Pn−m−1;m+1 ∼=Fm, ..., (2.12)
Pn−1;1/Pn−2;2 ∼=F2, Pn(T2)/Pn−1;1 ∼=Z2.
Inparticular,anynon-trivialsubgroupH ⊆P (T2)admitsanon-trivialhomomor-
n
phism to Z.
The last sentence of the proposition implies:
Proposition 2.2. A group G with finite abelianization G/G′ = G/[G,G] has no
non-trivial homomorphisms to the pure torus braid group P (T2).
n
Recall that the Artin braid group Bn =π1(Cn(C)) has generators σ1,...,σn−1 and
thesystemofdefiningrelations(2.2),(2.3). Thus,wehaveanaturalhomomorphism
i: B →B (T2)sendingallgeneratorsσ’s ofB tothe elementsofthe same name
n n n
in B (T2) (according to N. Ivanov [13], i is injective; however,we will not use this
n
fact).
Lemma 2.3. Let n > 4 and let ϕ: B (T2) → B (T2) be a homomorphism such
n m
that the composition
Φ=µ◦ϕ◦i: B −→i B (T2)−ϕ→B (T2)−µ→S(m) (2.13)
n n m
is an abelian homomorphism. Then ϕ is abelian. In particular, ϕ is abelian.
whenever µ◦ϕ is so.
Proof. LetΦ′: B′ →S(m)denotetherestrictionofΦtothecommutatorsubgroup
n
B′ of the groupB . Since Φ is abelian, Φ′ is trivialand henceϕ(i(B′))⊆Kerµ=
n n n
P (T2). By Gorin–Lin Theorem [18], for n > 4 the group B′ is perfect5, and
m n
Proposition 2.2 shows that ϕ(i(B′)) = 1. Hence ϕ◦i is abelian, and relations
n
σkσk+1σk = σk+1σkσk+1 imply ϕ(i(σ1)) = ... = ϕ(i(σn−1)). By definition, i sends
the generators σ1,...,σn−1 of Bn into the corresponding generators σ1,...,σn−1 of
B (T2); thus, the latter ones satisfy
n
ϕ(σ1)=...=ϕ(σn−1). (2.14)
Together with (2.4) this shows that
ϕ(a )ϕ(σ )=ϕ(σ )ϕ(a ) for all j,k. (2.15)
k j j k
By (2.14) and (2.6),
(ϕ(σ ))2(n−1) =ϕ(a )ϕ(a )−1ϕ(a )−1ϕ(a ), (2.16)
1 1 2 1 2
5Thatis,B′ coincideswithitscommutator subgroupB′′=[B′,B′].
n n n n
CONFIGURATION SPACES OF TORI 7
and by (2.15) and (2.7),
ϕ(a )ϕ(a )−1ϕ(a )−1ϕ(a )=ϕ(σ )2. (2.17)
2 1 2 1 1
Relations (2.15) and (2.17) imply
ϕ(σ )2 =ϕ(a )−1ϕ(a )ϕ(a )ϕ(a )−1. (2.18)
1 2 1 2 1
Multiplying (2.16) and (2.18) we obtain
1=(ϕ(σ ))2(n−1)(ϕ(σ ))2 =(ϕ(σ ))2n.
1 1 1
However, by E. Fadell and L. Neuwirth [16], Theorem 8, the torus braid group
B (T2) has no elements of a finite order, this implies ϕ(σ )=1. Thus
m 1
ϕ(a )−1ϕ(a )ϕ(a )ϕ(a )−1 =1,
2 1 2 1
and ϕ is an abelian homomorphism. (cid:3)
2.3. Proof of Theorem 1.4. Letn>4andletϕbeanon-abelianendomorphism
ofB (T2). ByLemma2.3,thehomomorphismΦ=µ◦ϕ◦i:B →S(n)in(2.13)is
n n
non-abelian. Thus,byV.Lin’stheorem(see[2],Section4;acompleteproofmaybe
foundin[13,11,12]),ΦcoincideswiththestandardepimorphismB →S(n)upto
n
anautomorphismofS(n). Itfollowsthatthehomomorphismµ◦ϕ: B (T2)→S(n)
n
is surjective. N. Ivanov (see [13], Theorem 1) proved that for n > 4, any non-
abelian homomorphism B (T2) → S(n) whose image is a primitive permutation
n
groupon n letters coincides with the standardepimorphism µ: B (T2)→S(n) up
n
to an automorphism of S(n). Thus, Ker(µ◦ϕ) = P (T2) = Kerµ, which implies
n
ϕ(P (T2))⊆Kerµ=P (T2) thus proving part (a) of the theorem.6
n n
To prove part (b), we use another Lin’s theorem (see quotations above), which
says that for n > max(m,4) any homomorphism B → S(m) is abelian; it follows
n
that the homomorphismΦ=µ◦ϕ◦i: B →S(m) in (2.13) is abelian. By Lemma
n
2.3, ϕ is abelian. (cid:3)
We skip the proofs of the next two results, which will not be used in what follows.
Theorem 2.4. For n ≥ 5, the abelianization B′(T2)/[B′(T2),B′(T2)] of the
n n n
commutator subgroup B′(T2) of the torus braid group B (T2) is isomorphic to
n n
Z =Z/nZ.
n
Theorem 2.5. Let 3≤m≤n≤4 and let ϕ: B (T2)→B (T2) be a non-abelian
n m
homomorphism such that µ◦ϕ is non-abelian. Then ϕ(P (T2))⊆P (T2).
n m
3. Ordered configuration spaces
Here we establish some analytic properties of ordered configuration spaces.
3.1. A Cartesian product structure in En(T2). The torus T2 acts in En(T2)
by translations: q = (q ,...,q ) 7→ (q +t,...,q +t), t ∈ T2. Any orbit of this
1 n 1 n
action is isomorphic to T2 and intersects the submanifold
M˜0 ={q =(q1,...,qn−1,0)∈En(T2)}∼=En−1(T2\{0})
6Infact,weprovedaslightlystrongerpropertyϕ−1(Pn(T2))=Pn(T2).
8 Y.FELER
at a single point. This gives the Cartesian decomposition En(T2) = T2 × M˜ ,
0
defined by the following maps
En(T2)∋(q1,...,qn)7→(qn,(q1−qn,...,qn−1−qn,0))∈T2×M˜0,
T2×M˜0 ∋(t,(q1,...,qn−1,0))7→(q1+t,...,qn−1+t,t)∈En(T2).
It is easily seen that M˜ is a non-singular irreducible affine algebraic variety; in
0
particular, it is a Stein manifold, whereas En(T2) is not so.
3.2. Holomorphic mappings En(T2) → T2 \{0}. We continue to execute our
plan sketched in Section 1.2 with the following
Theorem 3.1. Any non-constantholomorphicmapf :En(T2)→T2\{0}isofthe
form
f(q ,q ,...,q )=m(q −q ) with some i6=j,
1 2 n i j
where m: T2 →T2 is either the identity or a complex multiplication.7
Notation 3.2. We denote by M the finite cyclic group of automorphisms of T2
consisting of ±identity and all complex multiplications in T2 (if they exist). M is
isomorphic either to Z or to Z or to Z . Let M consists of all m ∈ M with
2 4 6 +
0≤Argm<π (i.e. M contains 1, 2 or 3 elements).
+
To prove the theorem we need some preparation. The proof of the following well-
known statement is due to H. Huber [19], §6, Satz 2; it may also be found in Sh.
Kobayashi’sbook [20], Chapter VI, Sec. 2, remarks after Corollary 2.6.
Claim A.Let M andN be compactRiemann surfacesand A⊂M andB ⊂N be
finitesets. SupposethatthedomainN\B ishyperbolic,meaningthatitsuniversal
coveringisisomorphictotheunitdisc. ThenanyholomorphicmapM\A→N\B
extends to a holomorphic map M →N.
The following facts are also very well known.
Claim B. Any non-constant holomorphic self-map µ: T2 → T2 is regular, has no
critical points and hence is an unbranched covering of some finite degree k. If
µ(0)=0 then µ is also a regularendomorphismof the compactalgebraicgroupT2
anditskernelKerµisasubgroupoforderk. Everyholomorphicself-mapofT2isof
the formq 7→ P kmm(q−a),andanyautomorphismis ofthe formq 7→m(q−a),
m∈M
where m∈M, km ∈Z, and a∈T2.
Explanation. The graph G = {(p,q) ∈ T2 ×T2| q = µ(p)} of µ is an analytic
µ
subset of the projective variety T2 × T2. By general Chow’s theorem (see [21]
or [22], Chapter V, Theorem D7), G is a (non-singular) projective variety. Of
µ
course, the projections α: G ∋ (p,q) 7→ p ∈ T2 and β: G ∋ (p,q) 7→ q ∈ T2 are
µ µ
regular. Hence, the inverse map α−1: T2 ∋ p 7→ (p,µ(p)) ∈ G is regular, and the
µ
composition µ=β◦α−1 is regular, as well.
To see that µ has no critical points, suppose to the contrary that the set of all
critical values C contains m ≥ 1 points. The restriction of µ to the complement
of µ−1(C) defines an unbranched covering T2 \µ−1(C) → T2 \C of some finite
7Ofcourse,thelattermayhappenonlyifourtorusadmitssuchamultiplication(byacomplex
multiplicationwemeananautomorphismofthetorusT2inducedbymultiplicationonthecomplex
linebyacomplexnumber6=1,−1).
CONFIGURATION SPACES OF TORI 9
degreek. TheRiemann–HurwitzformulaforEulercharacteristicsstatesthatχ(T2\
µ−1(C))=kχ(T2\C),i.e. #(µ−1(C))=km. Since#(µ−1(c))≤k−1forallc∈C,
we have km=#(µ−1(C))≤(k−1)m, which is impossible.
Assuming that µ(0)=0, define a map ν: T2×T2 →T2 by
ν(p,q)=µ(p+q)−µ(p)−µ(q).
Any loop in T2×T2 can be deformed to the subset (T2×{0})∪({0}×T2), where
ν = 0. Thus, ν induces the trivial homomorphism of the fundamental groups and
lifts to a holomorphic map ν˜: T2 ×T2 → C, which, by the maximum principle,
must be constant. It follows that ν = const = 0 and µ is a holomorphic group
endomorphismofT2. (Another argumentmaybe foundin [23], Chapter3,Section
3.1.)
The generalform of automorphisms and endomorphisms of T2 may be found in
[24], Chapter V, Section V.4.7, and [25], Chapter 1, Section 5, respectively.
(cid:3)
Lemma 3.3. For any a,b∈T2 everynon-constantholomorphic mapλ: T2\{a}→
T2\{b} extends to a biregular automorphism of T2 sending a to b.
Proof. ByClaimsAandB,λextendstoanunbranchedholomorphiccoveringmap
λ˜: T2 → T2 of some finite degree k. Clearly λ˜−1(a) = {b}; hence, k = 1 and λ˜ is
an automorphism. (cid:3)
Definition3.4. Aconfigurationa=(a ,...,a )∈Em(T2)issaidtobeexceptional
1 m
ifthere existi6=j anda non-constantholomorphicself-mapλ: T2 →T2 suchthat
λ(a )=λ(a ) and λ−1(λ(a ))⊆{a ,...,a }.
i j i 1 m
Lemma 3.5. a) The set A of all exceptional configurations a = (a ,...,a ) ∈
1 m
Em(T2) is contained in a subvariety M ⊂Em(T2) of codimension 1.
b) For any non-exceptional configuration (a ,...,a )∈Em(T2), every non-cons-
1 m
tant holomorphic map λ: T2\{a ,...,a }→T2\{0} extends to a biregular auto-
1 m
morphism of T2, sending a certain a to 0.
i
Proof. a) Let N denote the union of all finite subgroups of order ≤ m in T2; this
set is finite. Set M ={(a ,...,a )∈Em(T2)|a −a ∈N for some i6=j}; then M
1 m j i
is a subvariety in Em(T2) of codimension 1. We shall show that A⊆M.
Let a = (a ,...,a ) ∈ A and let i,j, and λ be as in Definition 3.4. Set
1 m
µ(t) = λ(t+a )−λ(a ), t ∈ T2. Then µ(0) = 0 and, by Claim B, µ is a group
i i
homomorphism with finite kernel #Kerµ. If t ∈ Kerµ, then λ(t+a ) = λ(a ),
i i
t + a ∈ λ−1(λ(a )) ⊆ {a ,...,a } and t ∈ {a − a ,...,0,...,a − a }; that
i i 1 m 1 i m i
is, Kerµ ⊆ {a − a ,...,0,...,a − a }. In particular, #Kerµ ≤ m and hence
1 i m i
Kerµ⊆N. Since µ(a −a )=0, we have a −a ∈N and a∈M.
j i j i
b) Let a=(a ,...,a )∈/ A. By Claims A and B, any non-constant holomorphic
1 m
map λ: T2 \{a ,...,a } → T2 \{0} extends to a finite unbranched holomorphic
1 m
covering map λ˜: T2 → T2. Clearly λ˜−1(0) ⊆ {a ,...,a }; in particular, λ˜(a ) = 0
1 m i
for a certain i. Since a ∈/ A, we have λ˜(a ) 6= 0 for all j 6= i; this means that
j
λ˜−1(0)={a } and degλ˜ =1. (cid:3)
i
Proof of Theorem 3.1 First, we notice that any holomorphic map T2 → T2 \{0}
is constant, since it lifts to a holomorphic map C → D = {ζ ∈ C| |ζ| < 1} of the
universal coverings,which is constant due to Liouville’s theorem.
10 Y.FELER
The proof of the theorem is by induction on n, starting with n = 2. For a ∈ T2,
denote by λ = λ(·,a) the restriction of λ to the fibre p−1(a) = T2 \{a} of the
a
projection p: E2(T2)∋(q ,q )7→q ∈T2.
1 2 2
The set S = {a ∈ T2| λ is a constant map} is finite. Because, otherwise, by
a
the uniqueness theorem, λ =const for all a ∈ T2, λ =λ(q ,q ) does not depend
a 1 2
on q and may be considered as a holomorphic map T2 →T2\{0}, which must be
1
constant; however,this implies λ=const, contradicting our assumption.
By Lemma 3.3, for any a ∈/ S the map λ : T2 \{a} → T2 \ {0} extends to
a
a biholomorphic automorphism of T2 sending a to 0. This means that λ is of
a
the form λ (t) = m(t−a), with some m = m ∈ M. Thus, for all (q ,q ) in the
a a 1 2
connected, everywhere dense set E2(T2)\p−1(S) we have
λ(q ,q )=m(q −q ) (3.1)
1 2 1 2
with a certain m = m ∈ M. Since M is finite, the element m = m ∈ M on the
q2 q2
right hand side of (3.1) cannot depend on q and the latter formula holds true for
2
the whole of E2(T2), which completes the proof for n=2.
Assume that the theorem is already proved for some n = m−1 ≥ 2. For a =
(a ,...,a )∈Em−1(T2), denote by λ =λ(·,a ,...,a ) the restriction of λ to the
2 m a 2 m
fibre p−1(a) = T2 \{a ,...,a } of the projection p: Em(T2) ∋ (q ,q ,...,q ) 7→
2 m 1 2 m
(q ,...,q )∈Em−1(T2).
2 m
ItisclearthatS =de=f {a∈Em−1(T2)|λ is a constant map}isananalyticsubset
a
of Em−1(T2), and either (i) S =Em−1(T2) or (ii) dimCS ≤m−2.
In case (i), λ=λ(q ,...,q ) does not depend on q and may be considered as a
1 m 1
holomorphic map Em−1(T2) → T2 \{0}; by the induction hypothesis, λ is of the
desired form.
Let us consider case (ii). By Lemma 3.5(a), the set A of all exceptional config-
urations is contained in a subvariety M ⊂ Em−1(T2) of dimension m − 2. Let
a ∈ Em−1(T2) \ (S ∪ M). Then λ : T2 \ {a ,...,a } is a non-constant map.
a 2 m
By Lemma 3.5(b), λ extends to a biholomorphic map λ˜ : T2 → T2. Therefore,
a a
λ˜ (t) = m(t−a ), with some m = m ∈M and i = i . Thus, for all (q ,...,q ) in
a i a a 1 m
the connected, everywhere dense set Em(T2)\p−1(S∪M) we have
λ(q ,...,q )=m(q −q ) (3.2)
1 m 1 i
with certain m = m ∈ M and i = i . Since M is finite, m and i do not depend
q q
on q, and (3.2) holds true on the whole of Em(T2), which completes the step of
induction thus proving the theorem. (cid:3)
Definition 3.6. For any m∈M and i6=j ∈{1,...,n}, the map
+
em;i,j: En(T2)→T2\{0},
(3.3)
em;i,j(q)=m(qi−qj) for q =(q1,...,qn)∈En(T2),
is called a difference. For a difference µ = em;i,j, the unordered pair of variables
{q ,q }iscalledthesupportofµandtheautomorphismm∈M iscalledthemarker
i j +
of µ. We denote them by suppµ and m respectively. It follows from Theorem 3.1
µ
that any non-constant holomorphic map µ : En(T2) → T2 \{0} admits a unique
representation in the form of a difference, that is, µ = em;i,j for some uniquely
defined m∈M and i,j ∈{1,...,n}.
+