Spaces of geometrically generic configurations

0.4 MB·English

by  Yoel Feler

SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS
6
0
YOELFELER
0
2
Abstract. LetXdenoteeitherCPmorCm. Westudycertainanalyticprop-
n ertiesofthespaceEn(X,gp)oforderedgeometricallygenericn-pointconfigu-
Ja rationsinX. Thisspaceconsistsofallq=(q1,...,qn)∈Xnsuchthatnom+1
ofthepointsq1,...,qnbelongtoahyperplaneinX. Inparticular,weshowthat
4 fornbigenough anyholomorphicmapf:En(CPm,gp)→En(CPm,gp)com-
1 muting with the natural action of the symmetric group S(n) inEn(CPm,gp)
is of the form f(q) = τ(q)q = (τ(q)q1,...,τ(q)qn), q ∈ En(CPm,gp), where
] τ: En(CPm,gp)→PSL(m+1,C) is an S(n)-invariant holomorphic map. A
V
similarresultholdstrueformappingsoftheconfiguration spaceEn(Cm,gp).
C
.
h
t
a
m Contents
[ 1. Introduction 1

1 2. Some properties of spaces of generic configurations 4
v 2.1. Notation and definitions 4
2 2.2. Irreducibility of determinant polynomials 6
6 2.3. The direct decomposition of En(CPm,gp) 6
3
2.4. Determinant cross ratios 8
1
0 3. Holomorphic functions omitting the values 0 and 1 10
6 3.1. ABC lemma 10
0 3.2. Explicit description of all holomorphic functions En →C\{0,1}. 14
h/ 4. Simplicial complex of holomorphic functions En →C\{0,1} 16
t 4.1. Simplicial complex of holomorphic functions omitting two values 16
a
m 4.2. S(n) action in L△(En) 19
4.3. Maps of L△(En(X,gp)) induced by holomorphic endomorphisms 22
:
v 5. The proof of Theorem 1.4 29
i 5.1. Proof of Theorem 1.4 29
X
References 30
r
a
1. Introduction
In this paper we study certain analytic properties of the spaces of geometrically
generic point configurations in projective and affine spaces.
The most traditional non-ordered configuration space Cn = Cn(X) of a complex
space X consists of all n point subsets (“configurations”) Q = {q ,...,q } ⊂ X.
1 n
2000 Mathematics Subject Classification. 14J50,32H25,32H02,32M99.
Key words and phrases. configuration space, geometrically generic configurations, vector
braids,holomorphicendomorphism.
ThismaterialisbaseduponaPhDthesissupportedbyTechnion.
1
SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS 2
If X carries an additional geometric structure, it may be taken into account. Say
if X is either the projective space CPm or the affine space Cm and n > m then
the space Cn = Cn(X,gp) of geometrically generic configurations consists of all
n point subsets Q ⊂ X such that no hyperplane in X contains more than m
points of Q. The corresponding ordered configuration space En = En(X,gp) con-
sists of all q = (q ,...,q ) ∈ Xn such that the set Q = {q ,...,q } ⊂ X be-
1 n 1 n
longs to Cn(X,gp). The left action of the symmetric group S(n) on En defined by
σ(q1,...,qn)=(qσ−1(1),...,qσ−1(n))isfree;thecorrespondingorbitmapcanbeiden-
tified with the tautological covering map p: En ∋ (q ,...,q ) 7→ {q ,...,q } ∈ Cn,
1 n 1 n
which is an unbranched Galois covering with the Galois group S(n).
When m = 1, that is, X = CP1 or X = C1, the spaces of geometrically generic
configurationsinX coincidewiththeusualconfigurationspacesCn(X)andEn(X).
But for m > 1 the spaces Cn(X,gp) and En(X,gp) seem even more natural since
they both are Stein manifolds, unlike Cn(X) and En(X). In the same time, the
spacesCn(X,gp)andEn(X,gp)havemorecomplicatedtopology;forinstance,their
fundamental groups are highly nontrivial, whereas π (Cn(X)) = S(n) and En(X)
1
is simply connected.
Let Aut X be the subgroup of the holomorphic automorphism group AutX of
g
X consisting of all elements A ∈ AutX that respect the geometrical structure of
X. I. e., for X = CPm we have Aut X = PSL(m+1,C) = AutX, whereas for
g
X =Cm thegroupAut X =Aff(m,C)ofallaffinetransformationsofCm ismuch
g
smaller than the huge group Aut(Cm) of all biholomolomorphic (or polynomial)
automorphisms of Cm. Notice that in both cases Aut X is a complex Lie group.
g
The natural action of Aut X in X induces the diagonalAut X actions in Cn and
g g
En defined by
AQ=A{q ,...,q }=de=f {Aq ,...,Aq } ∀Q={q ,...,q }∈Cn
1 n 1 n 1 n
and
Aq =A(q ,...,q )=d=ef (Aq ,...,Aq ) ∀q =(q ,...,q )∈En.
1 n 1 n 1 n
Therefore, any automorphism T ∈ Aut X gives rise to the holomorphic endo-
g
morphism (i.e., holomorphic self-mapping) F of Cn defined by F (Q) = TQ =
T T
{Tq ,...,Tq }. This example may be generalized by making T depending analyti-
1 n
cally on a configuration Z ∈Cn; a rigorous definition looks as follows.
Definition 1.1. A holomorphic endomorphismF of Cn is saidto be tame if there
isaholomorphicmapT: Cn →Aut X suchthatF(Q)=F (Q)=de=f T(Q)Qforall
g T
Q∈Cn. Similarly, a holomorphicendomorphismf ofEn is called tame if there are
an S(n)-invariant holomorphic map τ: En →Aut X and a permutation σ ∈S(n)
g
such that f(q)=f (q)=de=f στ(q)q for all q ∈En.
τ,σ
When X = Cm, a holomorphic endomorphism F of Cn(Cm,gp) is said to be
quasitame if there is a holomorphic map T: Cn(Cm,gp) → PSL(m+1,C) such
that F(Q) = F (Q) =de=f T(Q)Q for all Q ∈ Cn(Cm,gp).1 Similarly, a holomorphic
T
endomorphism f of En = En(Cm,gp) is quasitame if there are an S(n)-invariant
holomorphic map τ: En(Cm,gp) → PSL(m+1,C) and a permutation σ ∈ S(n)
such that τ(q)q ,...,τ(q)q ∈ Cm and f(q) = στ(q)q for any q = (q ,...,q ) ∈
1 n 1 n
En(Cm,gp). (cid:13)
1Noticethatthelatterconditionimpliesthat T(Q)Q⊂Cm foranyQ∈Cn(Cm,gp).
SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS 3
The left S(n) action on En(X,gp) induces the left S(n) action on the set of all
mappings f: En(X,gp)→En(X,gp) defined by
σf =σ(f1,...,fn)=(fσ−1(1),...,fσ−1(n))
for f =(f ,...,f ): En(X,gp)→En(X,gp) and σ ∈S(n).
1 n
Definition 1.2. Acontinuousmapf: En(X,gp)→En(X,gp)issaidtobestrictly
equivariant if there exists an automorphism α of the group S(n) such that
f(σq)=α(σ)f(q) for all q ∈En(X) and σ ∈S(n).
(cid:13)
Remark 1.3. Any tame endomorphism f = f of En is strictly equivariant;
τ,σ
the corresponding automorphism α ∈ AutS(n) is just the inner automorphism
s7→σsσ−1. The same holds true for quasitame maps. (cid:13)
The following theorem contains the main results of this work.
Theorem 1.4. Let m>1, n≥m+3 and n6=2m+2.
a) Any strictly equivariant holomorphic map f: En(CPm,gp)→En(CPm,gp) is
tame.
b) Any strictly equivariant holomorphic map f: En(Cm,gp) → En(Cm,gp) is
quasitame.
Inmoredetail,form>1,n≥m+3,n6=2m+2andastrictlyequivariantholo-
morphicmapf: En(X,gp)→En(X,gp),thereexistanS(n)-invariantholomorphic
map τ: En(X,gp)→PSL(m+1,C) and a permutation σ ∈S(n) such that
f(q)=στ(q)q =(τ(q)qσ−1(1),...,τ(q)qσ−1(n)) (1.1)
for all q =(q ,...,q )∈En(X,gp).
1 n
Remark 1.5. For a holomorphic endomorphism f of En(X,gp) commuting with
the S(n)-action the representation (1.1) can be simplified as follows: there is an
S(n)-invariant holomorphic map τ: En(X,gp)→PSL(m+1,C) such that
f(q)=τ(q)q for all q ∈En(X,gp).
(cid:13)
Remark 1.6. The holomorphic endomorphisms of the spaces Cn(CP1) and Cn(C)
were completely described by V. Lin (see, for instance, [6], [8] and [11]). More-
over, due to the works of V. Zinde [17]-[21] and the author [3], such a description
is known now for holomorphic endomorphisms of traditional configuration spaces
of all non-hyperbolic algebraic curves Γ. In all quoted papers, the first important
step includes a purely algebraicinvestigationof the fundamental groups π (Cn(Γ))
1
and π (En(Γ)), which are the braid group and the pure braid group of Γ, respec-
1
tively. Namely, it is shown that for n > 4 any endomorphism of π (Cn(Γ)) with
1
a non-abelian image preserves the subgroup π (En(Γ)). This property ensures the
1
liftingofany“sufficientlynon-trivial”holomorphicself-mapF ofCn(Γ)toastrictly
equivariant holomorphic self-map f of En(Γ), which, in turn, can be studied via
certain analytic and combinatorial methods.
Myoriginaltargetinthepresentworkwastouseasimilarmachineryinorderto
study the holomorphic endomorphisms of the spaces Cn(CPm,gp) and Cn(Cm,gp)
SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS 4
for m > 1. I did not succeed however in investigating of the corresponding funda-
mental groups and therefore restricted myself to the study of strictly equivariant
holomorphicendomorphismsofthe correspondingorderedconfigurationspaces. (cid:13)
Remark 1.7. Theorem 1.4(b) is not complete, since at the moment I do not know
whether there are strictly equivariant holomorphic endomorphisms of En(Cm,gp)
that are quasitame but not tame.
Although I think that Theorem 1.4 holds true for n = 2m+2, in this case I
could not overcome some technical difficulties which arise in the proof. (cid:13)
The plan of the proof is as follows. Let X = Cm or CPm. To study a strictly
equivariant holomorphic self-map f of the space En =En(X,gp), we start with an
explicit description of all non-constant holomorphic functions λ: En →C\{0,1}.2
The set L of all such maps is finite and separates points of a certain rather big
submanifold M ⊂ En of complex codimension m(m + 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 be-
haviour of f∗ and then recover f, we endow L with the following simplicial struc-
ture. Asubset∆s ={λ ,...,λ }⊆Lissaidtobeans-simplexwheneverλ /λ ∈L
0 s i j
for all distinct i,j. The action of S(n) in En induces a simplicial S(n) action in
the complex L. The orbits of this actionmay be exhibited 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 find a holomorphic map
τ: En(X,gp)→PSL(m+1,C)
and a permutation σ ∈S(n) such that f(q)=στ(q)q.
Remark 1.8. The topology of the considered spaces is of great interest by itself.
A.I.BarvinokcalculatedthefirsthomologygroupoftheorderedspaceEn(C2,gp),
see [1]. V. Moulton, [13], found the generators and some generating relations of
the fundamental groups π (En(Cm,gp)) and π (En(CPm,gp)). T. Terasoma, [15],
1 1
proved that this set of generating relations is complete for the case m=2. (cid:13)
Acknowledgments: ThispaperisbaseduponmyPhDThesissupportedbyTech-
nion. I wish to thank V. Lin who introduced me to the problem and encouraged
me to workonit. I alsowish to expressmy thanks to M. EntovandA. Vershik for
stimulating discussions and to B. Shapiro for useful references.
2. Some properties of spaces of generic configurations
2.1. Notation and definitions. The spaces En of ordered geometrically generic
configurations in CPm or Cm have the following explicit algebraic description.
Any point q ∈(CPm)n may be represented as a ‘matrix’
q [z :···:z ]
1 1,1 1,m+1
q = .. = .. ∈(CPm)n, (2.1)
. .
 qn   [zn,1 :···:zn,m+1] 
   
2Comparewith[11],[18],[19]and[3].
SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS 5
where q = [z : ··· : z ] ∈ CPm, j = 1,...,n. For m+1 distinct indices
j j,1 j,m+1
i ,...,i ∈{1,...,n}, the determinant
1 m+1
z ... z z
i1,1 i1,m i1,m+1
d (q)=(cid:12) .. .. .. .. (cid:12) (2.2)
i1,...,im+1 (cid:12) . . . . (cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) zim+1,1 ... zim+1,m zim+1,m+1 (cid:12)(cid:12)
(cid:12) (cid:12)
isahomogeneouspolynomialo(cid:12)fdegreem+1inthehomogeneousco(cid:12)ordinates[z :
1,1
···:z ], ...,[z :···:z ]. The space En(CPm,gp) consists of all matrices
1,m+1 n,1 n,m+1
qoftheform(2.1)suchthatd (q)6=0foralldistincti ,...,i ∈{1,...,n}.
i1,...,im+1 1 m+1
Similarly, the space En(Cm,gp) consists of all matrices
q z ,...,z ,1
1 1,1 1,m
q = .. = .. ∈(Cm)n (2.3)
. .
 qn   zn,1,...,zn,m,1 
   
with all q =(z ,...,z )∈Cm such that
j j,1 j,m
z ... z 1
i1,1 i1,m
d (q)=(cid:12) .. .. .. .. (cid:12)6=0 (2.4)
i1,...,im+1 (cid:12) . . . . (cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) zim+1,1 ... zim+1,m 1 (cid:12)(cid:12)
(cid:12) (cid:12)
for all distinct i ,...,i ∈{1,..(cid:12).,n}. (cid:12)
1 m+1
We refer to the components q ,...,q of a point q = (q ,...,q ) in En(CPm,gp) or
1 n 1 n
in En(Cm,gp) as to vector coordinates of q.
Although we use the same notation for determinant polynomials in both cases,
projective and affine, every time we meet them it will be clear from the context
which one we mean.
Notation 2.1. Byamultiindexwemeananorderedseti=(i ,...,i )withdistinct
1 s
i ,...,i ∈{1,...,n}. Sometimes we forget the order and write i∈i and #i=s. If
1 s
s=1 and i∈{1,...,n}, we may write i=i instead of i=(i).
Fortmultiindicesi =(i1,...,i1),...,i =(it,...,it)suchthati1,...,i1,...,it,...,it
1 1 s t 1 s 1 s 1 s
are distinct, we set d =d .
i1,...,it i11,...,i1s,...,it1,...,its
Let i = (i ,...,i ) and i′ = (i ,...,i ); for any i ∈ {i ,...,i } and any k =
1 s 1 s−1 1 s
1,...,s−1, we denote by D the (s−1)×(s−1) minor of the s×s matrix
i;i,k
z ,...,z ,1
i1,1 i1,s−1
Z = .. 
i .
 zis,1,...,zis,s−1,1 
complementary to the elements zi,k; for i=is, we write δi′;k instead of Di;is,k.
Let Is denote the set of all multiindices i = (i ,...,i ) such that 1 ≤ i < ... <
1 s 1
i ≤ n. For i = (i ,...,i ) ∈ Is and j = (j ,...,j ) ∈ It, we define the multiindices
s 1 s 1 t
i∩j and i\j in the evident way; if i∩j=∅, we define the multiindex i∪j∈Is+t
by an appropriate reordering of the components i ,...,i ,j ,...,j .
1 s 1 t
For 1≤j ≤n, set Is ={i∈Is| j 6∈i}. (cid:13)
j
SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS 6
2.2. Irreducibility of determinant polynomials.
Lemma 2.2. Allminorsofthematrices (2.1)and(2.3)areirreduciblehomogeneous
polynomials in the entries z .
t,s
Proof. Itsufficestoprovethelemmaforthematrix(2.3). Theproofisbyinduction
in the order of a minor. Clearly,any minor of order 1 is an irreducible polynomial.
Suppose that for some k, 1 ≤ k < m+1, all minors of order ≤ k are irreducible.
By the Lagrange decomposition formula, any minor M of order k+1 is a linear
function of the entries z ,...,z of its first column with coefficients that are
i1,j1 ik+1,j1
certainminorsoforderk,i. e.,polynomialsofallentriesofM butz ,...,z .
i1,j1 ik+1,j1
By the induction hypothesis, all the latter polynomials are irreducible; moreover,
they depend on different sets of variables and hence cannot be proportional (with
constantcoefficients)toasinglepolynomial. ThisimpliesthatM isirreducible. (cid:3)
Lemma 2.3. Let i ∈ {1,...,n}, i = (i ,...,i ) ∈ Im+1 and L ⊂ Cmn be a linear
0 0 1 m i0
subspace defined by the relations z =...=z .
i1,2 im,2
a) For i ∈ Im+1, the restriction d| of d to L is irreducible if and only if
i0 i L i
#(i∩i )<m. Moreover,if #(i∩i )=m, then d | =±D ·(z −z ), where
0 0 i L i;i,2 i,2 i1,2
i∈i\i .
0
b) For i∈Im+1, degd | =m.
i0 i L
Proof. Assume that #(i∩i ) = m. It follows that #(i\i ) = 1. Set (i) = i\i .
0 0 0
By the Lagrange determinant decomposition formula, we can show that d | =
i L
±D ·(z −z ). That is, d| is reducible and degd| =m.
i;i,2 i,2 i1,2 i L i L
Now assume that #(i∩i )<m. Let i,j ∈i\i be two distinct indices. It is clear
0 0
thatifweprovethatd | isirreducibleanddegd | =m,thesame
i,j,i2,...,im L i,j,i2,...,im L
statements for d| hold true. So set j=(i,j,i ,...,i ). Obviously,
i L 2 m
d | =(z −z )·D −(z −z )·D
j L i2,2 i,2 j;i,2 i2,2 j,2 j;j,2
and degd | =m. It follows that d | is a linear function of the variables z ,z
j L j L i,2 j,2
andz withcoefficientsD ,D andD −D . ByLemma2.2,D ,D
i2,2 j;i,2 j;j,2 j;i,2 j;i,2 j;i,2 j;j,2
are irreducible. Therefore the polynomials D ,D and D −D are pair-
j;i,2 j;j,2 j;i,2 j;i,2
wiseco-prime. The latterimplies thatd | is irreducible. This completes the proof
j L
of the lemma. (cid:3)
2.3. ThedirectdecompositionofEn(CPm,gp). HereweshowthatEn(CPm,gp)
admits a naturalrepresentationas a Cartesianproductof its subspace of codimen-
sion m(m+2) and the group PSL(m+1,C).
Definition 2.4. Set
v =[1:0:···:0], v =[0:1:0:···:0],...,v =[0:···:0:1]
1 2 m+1
(2.5)
and w =[1:···:1].
The subspace M ⊂En(CPm,gp) defined by
m,n
M ={q =(q ,...,q )∈En(CPm,gp)| q =v ∀i=1,...,m+1, q =w}
m,n 1 n i i m+2
is called the reduced space of geometrically generic ordered configurations. (cid:13)
Lemma 2.5. Let n ≥ m+3. For every q ∈ En(CPm,gp), there is a unique γ(q) ∈
PSL(m+1,C) such that γ(q)q ∈M . The map
m,n
γ: En(CPm,gp)∋q 7→γ(q)∈PSL(m+1,C)
SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS 7
is holomorphic.
[z :···:z ]
1,1 1,m+1
Proof. 3 For q =  ..  ∈ En(CPm,gp) and i ∈ {1,...,m+1},
.
 [zn,1 :···:zn,m+1] 
 
the matrices
z ,..., z
1,1 1,m+1
.
 . 
.
z ,..., z
1,1 1,m+1  zi−1,1 ,..., zi−1,m+1 
A(q)= ...  , Ai(q)= zm+2,1 ,..., zm+2,m+1 
 zm+1,1 ,..., zm+1,m+1   zi+1,1 ,..., zi+1,m+1 
   .. 
 . 
 
 z ,..., z 
 m+1,1 m+1,m+1 
are defined up to multiplication of their rows by non-zero complex numbers. The
determinants detA(q) and all D = detA are homogeneous polynomials non-
i i
vanishingonEn(CPm,gp);therefore,thedeterminantoftheadjunctmatrixX(q)=
Aadj(q)(whoseelementsx arethealgebraicco-factorsoftheelementsz ofA(q))
i,j j,i
is equal to (detA(q))m and hence does not vanish on En(CPm,gp). It follows that
foreveryq ∈En(CPm,gp)thematrixT(q)=(x /D )m+1 isnon-degenerate. Itis
i,j j i,j=1
easy to verify that T(q) determines a well-defined automorphism γ(q) of CPm and
γ(q)q ∈M .
m,n
An automorphism of CPm is uniquely determined by its values at any m+2
pointsingeometricallygenericposition. Itfollowsthatforanyq ∈En(CPm,gp)the
constructed above element γ(q)∈PSL(m+1,C) is the only element of PSL(m+
1,C) that carries q to a point of M . Moreover, γ(q) holomorphically depends
m,n
on a point q. (cid:3)
Corollary 2.6. The mutually inverse maps
A: En(CPm,gp)∋q 7→A(q)=(γ(q),γ(q)q)∈PSL(m+1,C)×M
m,n
and
B: PSL(m+1,C)×M ∋(T,q˜)7→B(T,q˜)=T−1q˜∈En(CPm,gp)
m,n
induceanaturalbiholomorphicisomorphismEn(CPm,gp)∼=PSL(m+1,C)×Mm,n.
Remark 2.7. The above corollary implies that En(CPm,gp) and Cn(CPm,gp)
are irreducible non-singular complex affine algebraic varieties and, in particular,
Stein manifolds. Indeed, in the above direct decomposition of En(CPm,gp) both
PSL(m+1,C) and M are such varieties and hence En(CPm,gp) is also of the
m,n
same nature. Since the group S(n) is finite and its action on En(CPm,gp) is free,
the same is true for the quotient Cn(CPm,gp)=En(CPm,gp)/S(n).
The same properties hold true for En(Cm,gp) and Cn(Cm,gp).
Notice that for n > 1 the “standard” n-point configuration spaces En(X) and
Cn(X)ofa complexmanifoldX ofdimensionm>1cannotbe Steinmanifolds. (cid:13)
3For m = 1 the statement of Lemma is a common knowledge; the case m = 2 is treated in
[2], Chap. V, Sec. 109, Theorem 36. For m> 2, I could not find an appropriate reference and
sketchedtheproofhere.
SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS 8
2.4. Determinant cross ratios. Here we construct certain non-constant holo-
morphic functions En →C\{0,1},which are called “determinant cross ratios” (in
fact, later we shall show that there are no other functions with these properties).
Let us recall that, according to Notation 2.1, for distinct i ,...,i ,j,k and
1 m−1
i=(i ,...,i ) the notation d means the determinant d .
1 m−1 i,j,k i1,...,im−1,j,k
Definition 2.8. Let X be either CPm or Cm and let n≥ m+3. For any m+3-
dimensional multiindex I = (i ,...,i ) with distinct components i ∈ {1,...,n},
1 m+3 t
set i = (i ,...,i ), j = i , k = i , l = i , s = i ; the non-constant
1 m−1 m m+1 m+2 m+3
rational function
d (q) d (q)
e (q)=e (q)= i,j,k : i,k,s , q ∈(CPm)n, (2.6)
I i;j,k,l,s
d (q) d (q)
i,j,l i,l,s
is called a determinant cross ratio, or, in brief, a DCR. This function is regular on
the algebraic manifold En(X,gp)⊂(CPm)n.
The unordered set of indices {I} = {i ,...,i ,j,k,l,s} is called the support
1 m−1
of the function µ = e = e and is denoted by suppµ; its unordered subset
I i;j,k,l,s
{i}={i ,...,i } is called the essential support of µ and is denoted by supp µ
1 m−1 ess
(we often write I instead of {I} and i instead of {i}). In fact, the function e (q)=
I
e (q ,...,q ) depends only on the vector variables q with t∈I. (cid:13)
I 1 n t
Remark 2.9. Notice that two determinant cross ratios, say e = e and
I i;j,k,l,s
eI′ =ei′;j′,k′,l′,s′,coincideifandonlyif{i}={i′}and(j′,k′,l′,s′)isobtainedfrom
(j,k,l,s)byaKleinianpermutationoffourletters. Thesetofalldeterminantcross
ratios is denoted by DCR(En)=DCR(En(X,gp)). (cid:13)
The S(n) action in En induces an S(n) action on functions defined by (σλ)(q) =
λ(σ−1q)=λ(q ,...,q ) (λisafunctiononEn,q =(q ,...,q )∈En;noticethat
σ(1) σ(n) 1 n
σλ mayalso be written as(σ−1)∗λ=λ◦(σ−1), where σ andσ−1 areconsideredas
theself-mappingsofEn). Thisactioncarriesholomorphicfunctionstoholomorphic
functions.
Lemma 2.10. Foranyσ ∈S(n)andanyλ∈DCR(En)thefunctionσλalsobelongs
to DCR(En). Moreover,the S(n) action is transitive on the set DCR(En).
Proof. Let a = (a ,...,a) be a multiindex. For any σ ∈ S(n), let σ(a) denote
1 l
the multiindex (σ(a ),...,σ(a )). For any σ ∈ S(n), we have σd (q) = d (σ−1q) =
1 l a a
d (q); thus,
σ(a)
d (σ−1q) d (σ−1q)
σe (q)=e (σ−1q)= i,j,k : i,k,s
i;j,k,r,s i;j,k,r,s d (σ−1q) d (σ−1q)
i,j,r i,r,s
d (q) d (q)
σ(i),σ(j),σ(k) σ(i),σ(k),σ(s)
= : =e (q).
d (q) d (q) σ(i);σ(j),σ(k),σ(r),σ(s)
σ(i),σ(j),σ(r) σ(i),σ(r),σ(s)
LeteI,eI′ ∈DCR(En). SinceeachoftheorderedsetsI andI′ consistsofm+3≤n
distinct elements of the set {1,...,n}, there is a permutation σ ∈ S(n) such that
σI =I′ and hence σeI =eI′. (cid:3)
Lemma 2.11. Determinant cross ratios are invariants of the PSL(m+1,C) action
on the configuration space En(CPm,gp).
SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS 9
Proof. It is easy to observe that the following elementary operators do not change
any determinant cross ratio:
[z :···:z ]7→[a z :···:a z ] for a ·...·a 6=0;
1 m+1 1 1 m+1 m+1 1 m+1
[z :···:z :···:z :···:z ]7→[z :···:z :···:z :···:z ] ;
1 i j m+1 1 j i m+1
[z :···:z :···:z :···:z ]7→[z :···:z +z :···:z :···:z ] .
1 i j m+1 1 i j j m+1
AnyelementofPSL(m+1,C)canbedecomposedintoasequenceoftheelementary
operators. This proves the lemma. (cid:3)
Lemma 2.12. Any function λ∈DCR(En) omits the values 0 and 1.
Proof. By Lemma 2.11, any cross ratio λ is PSL(m+1,C) invariant. By Lemma
2.5, any orbit of PSL(m+1,C) action in En intersects the subspace M ; hence,
m,n
it sufices to show that λ does not accept the values 0 and 1 on M .
m,n
First, let I =(1,....,m+3), i=(1,...,m−1) and λ=e (q); then
I
d (q) d (q)
i,m,m+1 i,m+1,m+3
λ(q)=e (q)= : .
I
d (q) d (q)
i,m,m+2 i,m+2,m+3
For q =(q ,...,q )∈M with q =[z :···:z ], we have
1 n m,n t t,1 t,m+1
d (q)=1, d (q)=1,
i,m,m+1 i,m,m+2
d (q)=−z , d (q)=z −z ,
i,m+1,m+3 m+3,m i,m+2,m+3 m+3,m+1 m+3,m
and hence
z
m+3,m+1
e (q)=1− .
I
z
m+3,m
Since d (q) 6= 0, d (q) 6= 0 and d (q) 6= 0, we have
i,m,m+3 i,m+1,m+3 i,m+2,m+3
z ,z 6= 0 and z 6= z . Thus, e (q) 6= 0,1 on M
m+3,m m+3,m+1 m+3,m m+3,m+1 I m,n
and, consequently, on the whole of En.
By Lemma 2.10, S(n) acts transitively on the set DCR(En), which implies that
any λ∈DCR(En) does not accept the values 0 and 1. (cid:3)
Notation 2.13. Fors∈{1,...,m},setm(sˆ)=(1,...,sˆ,...,m). Fors=m,wewrite
sometimes m instead of m(mˆ).
We have alsbo the following lemma:
Lemma 2.14. a) The map P: M →(Cn−m−2)m defined by
m,n
p (q), ..., p (q)
1,m+3 1,n
q 7→P(q)= ... ... ...  ,
p (q), ..., p (q)
m,m+3 m,n
 
with p (q) = e (q) for s = 1,...,m and t = m+3,...,n, is a holo-
s,t m(sˆ);s,m+1,m+2,t
morphic embedding.
b) M is a hyperbolic space.
m,n
Proof. a)Forq =(q ,...,q )∈M withq =[z :···:z ]wehavez =
1 n m,n i i,1 i,m+1 t,m+1
d (q)6=0 and z =±d (q)6=0. Furthermore,
m,m,t t,s m(sˆ),m+1,t
bd (q)=(−1)m−s, d (q)=(−1)m−s,
m(sˆ),s,m+1 m(sˆ),s,m+2
d (q)=−(−1)m−sz , d (q)=(−1)m−s(z −z );
m(sˆ),m+1,t t,m m(sˆ),m+2,t t,m+1 t,m
SPACES OF GEOMETRICALLY GENERIC CONFIGURATIONS 10
thus,
dm(sˆ),s,m+1(q) dm(sˆ),m+1,t(q) zt,m+1
p (q)=e (q)= : =1− .
s,t m(sˆ);s,m+1,m+2,t d (q) d (q) z
m(sˆ),s,m+2 m(sˆ),m+2,t t,s
If q′ = (q′,...,q′) ∈ M with q′ = [z′ : ··· : z′ ] and p (q) = p (q′) for
1 n m,n i i,1 i,m+1 s,t s,t
s= 1,...,m, then z :z =z′ :z′ for all s =1,...,m and hence q =q′.
t,m+1 t,s t,m+1 t,s t t
Thus, P(q)=P(q′) implies q =q′ and P is injective.
To see that P is an embedding, it suffices to observe that from the above calcu-
lation follows that at any point the Jacobi matrix of P is of maximal rank.
b) Since every determinant cross ratio omits values 0 and 1,
P(M )⊂(C\{0,1})m(n−m−2),
m,n
it follows that M is a Kobayashi’s hyperbolic complex manifold. (cid:3)
m,n
Lemma 2.14 and Corollary 2.6 imply that the determinant cross ratios generate
the whole algebra A = C[En(CPm,gp)]PSL(m+1,C) = C[M ] of invariants of the
m,n
PSL(m+1,C) action on the algebraic variety En(CPm,gp). We shall see that the
set DCR of all these generators coincides with the set L(En) of all non-constant
holomorphic functions En →C\{0,1} and that a strictly equivariant holomorphic
map f: En(CPm,gp)→En(CPm,gp) induces a selfmap f∗ of L(En); thereby, such
a map f induces an endomorphism of the algebra A. Together with Corollary 2.6,
this provides an important information about f and eventually leads to the proof
of Theorem 1.4.
3. Holomorphic functions omitting the values 0 and 1
In this section we describe explicitly all holomorphic functions µ: En →C\{0,1}.
3.1. ABC lemma. The following lemma plays a crucial part in an explicit de-
scription of all holomorphic functions En(Cm,gp)→C\{0,1}. Let us recall that,
according to Notation 2.1, for distinct i ,...,i ,j,k and i = (i ,...,i ) the
1 m−1 1 m−1
notation d means the determinant d .
i,j,k i1,...,im−1,j,k
Lemma 3.1. Let A,B,C ∈C[Cmn]=C[z ,...,z ] be pairwiseco-prime polyno-
1,1 n,m
mialsonCmnnon-vanishingontheconfigurationspaceEn(Cm,gp)⊂Cmn. Assume
that at least one of them is non-constant and A+B+C = 0. Then there exist a
multiindex i = (i ,...,i ), indices j,k,l,s, and a complex number α 6= 0 such
1 m−1
that all i ,...,i ,j,k,l,s are distinct and A=αd d , B =αd d and
1 m−1 i,j,k i,l,s i,j,l i,s,k
C =αd d .
i,j,s i,k,l
Proof. The polynomials A,B,C do not vanish on
En(Cm,gp)=Cmn\ {q∈Cmn| d (q)=0}.
i
[i
It follows from Lemma 2.2 that
A=α dai, B =β dbi, C =γ dci,
i i i
i∈YIm+1 i∈YIm+1 i∈YIm+1
where α,β,γ ∈ C\{0} and a ,b,c ∈ Z . The polynomials are homogeneous;
i i i +
thus, the equality A + B + C = 0 implies that degA = degB = degC; i.e.,