Quadratic forms and singularities of genus one or two

0.52 MB·English

by  Georges Dloussky

Quadratic forms and singularities of genus one or two
Georges Dloussky∗
8
0
0
2

n Abstract
a
J We study singularities obtained by the contraction of the maximal divisor in

7 compact (non-k˝ahlerian) surfaces which contain global spherical shells. These
singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein

] or Gorenstein. A family of polynomials depending on the configuration of the
V
curves computes the discriminants of the quadratic forms of these singularities.
C
We introduce a multiplicative branch topological invariant which determines the
.
h twistingofanon-vanishingholomorphic1-formonthecomplementofthesingular
t point.
a
m
[ Contents

2
v 0 Introduction 2
7
6 1 Preliminaries 2
4
1.1 Basic results on singularities . . . . . . . . . . . . . . . . . . . . . . 2
2
1.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0
7 1.3 Surfaces with global spherical shells . . . . . . . . . . . . . . . . . 4
0 1.4 Intersection matrix of the exceptional divisor . . . . . . . . . . . . 8
/
h
2 Normal singularities associated to surfaces with GSS 10
t
a 2.1 Genus of the singularities . . . . . . . . . . . . . . . . . . . . . . . 10
m
2.2 Q-Gorenstein and numerically Gorenstein singularities . . . . . . . 12
:
v
3 Discriminant of the singularities 13
i
X 3.1 A family P of polynomials . . . . . . . . . . . . . . . . . . . . . . . 13
r 3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
a
3.3 A multiplicative topological invariant associated to singularities . . 19
3.4 Twisted holomorphic 1-forms near the isolated singularity . . . . . 22
4 Proof of the main theorem 23
4.1 Expression of determinants by polynomials . . . . . . . . . . . . . 26
4.2 The reduction lemma. . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Relation between determinants and polynomials of P. . . . . . . . 31
∗Georges Dloussky, [email protected], , version December 2007. Keywords: singularity,
compact surface, complex geometry. AMS subject classification: 32 S 25, 32 S 45, 32 J 15
1
0 Introduction
We are interested in a large class of singularities which generalizecusps, obtained
by the contraction of all the rational curves in compact surfaces S wich contain
global spherical shells. Particular cases are Inoue-Hirzebruch surfaces with two
“dual”cyclesofrationalcurves. Thedualitycanbeexplainedbytheconstruction
of these surfaces by sequences of blowing-ups [3]. Several authors have studied
cusps [9, 11, 19, 20, 15]. In general, the maximal divisor is composed of a cycle
with branches. These (non-k¨ahlerian) surfaces contain exactly n=b (S) rational
2
curves. The intersection matrices M(S) have been completely classified [18, 1];
they are negative definite in all cases except when the maximal divisor is a cy-
cle D of n rational curves such that D2 = 0. In this article we study normal
singularities obtained by contraction of the exceptional divisor and the link be-
tweentheintersectionmatrixandglobaltopologicaloranalyticalpropertiesofthe
surface S. They are elliptic or of genus two in which case they are Gorenstein.
Using the existence of global section on S of −mK ⊗L for a suitable integer
S
m ≥ 1 and a flat line bundle L ∈ H1(S,C⋆), we show that these singularities
areQ-Gorenstein(resp. numericallyGorenstein)ifandonlyiftheglobalproperty
H0(S,−mK )6=0(resp. H0(S,−K ⊗L)6=0)holds. Themainpartofthisarticle
S S
isdevotedtothe studyofthediscriminantofthequadraticformassociatedtothe
singularity. In [1]the quadratic formhas been decomposedinto a sumof squares.
The intersectionmatrix is completely determined by the sequence σ of (opposite)
self-intersectionsof the rationalcurveswhen takeninthe canonicalorder,i.e. the
order in which the curves are obtained in a repeted sequence of blowing-ups. Let
(Y,y) = (Y ,y) be the associated singularity obtained by the contraction of the
σ
rational curves. We introduce a family of polynomials P = P which have
σ A(σ)
integervaluesonintegers,dependingonthe configurationofthedualgraphofthe
singularity, such that the discriminant is the square of this polynomial. When we
fix the sequence σ we obtain an integer ∆ which is a multiplicative topological
σ
invarianti.e. satisfies∆ =∆ ∆ . We showthat∆ isequaltothe productof
σσ′ σ σ′ σ
the determinants of the branches. We apply this result to determine the twisting
integer of holomorphic 1-formsin a neighbourhoodof the singularity. We develop
here rather the algebraic point of view, however these singularities have deep re-
lations with properties of compactcomplex surfaces S containing globalspherical
shells, the classification of singular contracting germs of mappings and dynami-
cal systems: for instance, the integer ∆ is equal to the integer k = k(S) wich
σ
appears in the normal form of contracting germs F(z ,z ) = (λz zs+P(z ),zk)
1 2 1 2 2 2
which define S [2, 4, 5, 7].
I thank Karl Oeljeklaus for fruitful discussions on that subject.
1 Preliminaries
1.1 Basic results on singularities
Let D ,...,D be compact curves on a (not necessarilly compact) complex
0 n−1
surface X, and D =D +···+D the associated reduced divisor. We assume
0 n−1
that D is exceptional i.e. the intersection matrix M of D is negative definite. We
denotebyO thestructuralsheafofX,K =detT⋆X thecanonicalbundleand
X X
byΩ2 itssheafofsections. ItiswellknownbyGrauert’stheoremthatthereexists
X
a propermappingΠ:X →Y suchthateachconnectedcomponentof|D|=∪ D
i i
2
iscontractedontoapointy whichisanormalsingularityofY. For|D|connected,
denote by
r :H0(X,Ω2 )→H0(Y\{y},Ω2 )
X Y\{y}
the canonical morphism induced by Π. We define the geometric genus of the
singularity (Y,y) by
p =p (Y,y)=h0(Y,R1Π O ).
g g ∗ X
When Y is Stein, we have p =dimH0(Y\{y},Ω2 )/rH0(X,Ω2).
g Y\{y} X
A singularity (Y,y) is called rational (resp. elliptic) if p (Y,y) = 0 (resp.
g
p (Y,y) = 1). Therefore a singularity is rational if for every 2-form ω on Y\{y},
g
the 2-form Π⋆ω extends to a 2-form on X.
Proposition 1. 1 Let Π : X → Y be the proper morphism obtained by the con-
traction of an exceptional divisor:
1) p =h0(Y,R1Π O ) is independant of the choice of the desingularization.
g ∗ X
• If X is compact then p =χ(O )−χ(O )
g Y X
• If X is spc and Y is Stein then p =h1(X,O )
g X
2) The following sequence
0→H1(Y,O )→H1(X,O )→H0Y,R1Π O )→H2(Y,O )→H2(X,O )
Y X ⋆ X Y X
is exact.
We give now a criterion of rationality [21], p. 152:
Proposition 1. 2 Let Π : X → Y be the minimal resolution of the singularity
(Y,y) and denote by D the irreducible components of the exceptional divisor D.
i
If (Y,y) is rational, then:
i) the curves D are regular and rational
i
ii) for i6=j, D ∩D =∅ or D meets D tranversally. If D , D , D are distinct
i j i j i j k
irreducible components, D ∩D ∩D is empty
i j k
iii) D contains no cycle.
Definition 1. 3 A singularity (Y,y) is called Gorenstein if the dualizing sheaf
ω is trivial, i.e. there exists a small neighbourhood U of y and a non-vanishing
Y
holomorphic 2-form on U \{y}.
Since there is only a finite number of linearly independant 2-forms in the comple-
ment ofthe exceptionaldivisorD modulo H0(X,Ω2), a 2-formextends meromor-
phically across D. Therefore we have (see [23])
Lemma 1. 4 Let Y be a Gorenstein normal surface and Π:X →Y be the min-
imal desingularization. Then there is a unique effective divisor ∆ on X supported
in D =Π−1(Sing(Y)) such that
ω ≃Π⋆ω ⊗ O (−∆)
X Y X
OX
3
1.2 Lattices
Here are recalled some well known facts about lattices (see [24]). We call lattice,
denoted by L,< . , . > , a free Z-module L, endowed with an integral non
degenerate symetric bilinear form
(cid:0) (cid:1)
< . , . > : L×L −→ Z
(x,y) 7−→ < x , y >.
If B ={e ,...,e } is a basis of L, the determinant of the matrix
1 n
< e , e > ,
i j 1≤i,j≤n
(cid:0) (cid:1)
is independent of the choice of the basis; this integer, denoted by d(L) is called
the discriminant of the lattice. A lattice is unimodular if d(L) = ±1. Let L∨ :=
Hom (L,Z) be the dual of L. The mapping
Z
φ: L −→ L∨
x 7−→ < . , x >
identifies L with a sublattice of L∨ of same rank, since d(L) 6= 0. Moreover, if
L :=L⊗ Q, it is possible to identify L∨ with the sub-Z-module
Q Z
x∈L | ∀y ∈L, < x , y >∈Z
Q
of L . So, we may wri(cid:8)te L⊂L∨ ⊂L , where L and L∨ h(cid:9)ave same rank.
Q Q
Lemma 1. 5 1) The index of L in L∨ is |d(L)|.
2) If M is a submodule of L of the same rank, then the index of M in L satifies
[L : M]2 = d(M) d(L)−1.
In particular d(M) and d(L) have same sign.
1.3 Surfaces with global spherical shells
We recall some properties of these surfaces which have been first introduced by
Ma. Kato [12] and we refer to [1] for details.
Definition 1. 6 Let S be a compact complex surface. We say that S contains a
global spherical shell, if there is a biholomorphic map ϕ : U → S from a neigh-
bourhood U ⊂C2\{0} of the sphere S3 into S such that S\ϕ(S3) is connected.
Hopf surfaces are the simplest examples of surfaces with GSS (see [1]), however
they contain no rational curves and elliptic curves have self-intersection equal to
0, hence no singularity can be obtained.
Let S be a minimal surface containing a GSS with n = b (S). It is known
2
that S contains n rational curves and to each curve it is possible to associate a
contracting germ of mapping F = Πσ = Π ···Π σ : (C2,0) → (C2,0) where
0 n−1
Π=Π ···Π :BΠ →B is a sequence ofn blowing-ups. If we wantto obtaina
0 n−1
minimal surface,the sequence ofblowing-upshas tobe donein the followingway:
• Π blows up the origin of the two dimensional unit ball B,
0
4
• Π blows up a point O ∈C =Π−1(0),...
1 0 0 0
• Π blows up a point O ∈C =Π−1(O ), for i=0,...,n−2, and
i+1 i i i i−1
• σ :B¯ →BΠ sends isomorphicallya neighbourhoodofB¯ onto a small ballin
BΠ in such a way that σ(0)∈C .
n−1
It is easy to see that the homologicalgroups satisfy
H (S,Z)≃Z, H (S,Z)≃Zn
1 2
In particular, b (S)=n.
2
The universal covering space (S˜,ω,S) of S contains only rational curves (C )
i i∈Z
with a canonical order relation, “the order of creation” ([1], p 29). Following [1],
we can associate to S the following invariants:
• The family of opposite self-intersection of curves of the universal covering
space of S, denoted by
a(S):=(a ) =(−C2)
i i∈Z i i∈Z
this family is periodic of period n,
•
j+n−1 n−1
σ (S):= a =− D2+2 ♯{rationalcurveswithnodes}
n i i
i=j i=0
X X
where j is any index, and the D are the rational curves of S. It can be seen
i
that 2n≤σ (S)≤3n ([1], p 43).
n
• The intersection matrix of the n rational curves of S,
M(S):=(D .D ).
i j
ImportantRemark: Theessentialfactusefultounderstandthedualgraph
or the intersection matrix is that
– if a =−D2 =2 then D meets D ,
i i i i+1
– if a =−D2 =3 then D meets D ,...,
i i i i+2
– if a =−D2 =k+2 then D meets D ,
i i i i+k+1
the indices being in Z/nZ, in particular D may meet itself: we obtain a
i
rational curve with double point.
• n classes of contractingholomorphic germsof mappings F =Πσ :(C2,0)→
(C2,0) ([1], p 32).
Proposition 1. 7 LetS beasurfacecontainingaGSSwithb (S)=n,D ,...,D
2 0 n−1
the n rational curves and M(S) the intersection matrix.
1) If σ (S)=2n, then detM(S)=0.
n
2) If σ (S) > 2n, then ZD is a complete sublattice of H (S,Z) and its
n i 2
0≤i≤n−1
X
index satisfies
2
H (S,Z) : ZD = detM(S).
2 i
0≤i≤n−1
(cid:2) X (cid:3)
In particular, detM(S) is the square of an integer ≥1.
5
Proof: If σ (S)=2n, S is an Inoue surface; if σ (S)>2n, detM(S)6=0 so the
n n
sublattice is complete and the result is a mere consequence of lemma 5. (cid:3)
In order to give a precise description of the intersection matrix we need the
following definitions:
Definition 1. 8 Let 1≤p≤n. A p-uple σ =(a ,...,a ) of a(S) is called
i i+p−1
• a singular p-sequence of a(S) if
σ =(p+2,2,...,2).
p
It will be denoted by s . | {z }
p
• a regular p-sequence of a(S) if
σ =(2,2,...,2)
p
and σ has no common element with|a s{izngu}lar sequence. Such a p-uple will
be denoted by r .
p
For example s = (3),s = (4,2),s = (5,2,2),... are singular sequences, r =
1 2 3 3
(2,2,2)isaregularsequence. Itiseasytoseethatifwewanttohave,forexample,
acurvewithself-intersection-4,necessarily,thecurvewhichfollowsinthesequence
ofrepetedblowing-upsmusthaveself-intersection-2,soitiseasytosee([1],p39),
that a(S) admits a unique partition by N singular sequences and ρ ≤ N regular
sequences of maximal length. More precisely, since a(S) is periodic it is possible
to find a n-uple σ such that
σ =σ ···σ ,
p0 pρ+N−1
where σ is a regular or a singular p -sequence with
pi i
N+ρ−1
p =n
i
i=0
X
and if σ is regular it is between (mod. N +ρ) two singular sequences.
pi
Notation: We shall write
a(S)=(σ)=(σ ···σ ).
p0 pN+ρ−1
The sequenceσ is overlinedto indicatethatthesequenceσ isinfinitely repetedto
obtainthesequencea(S)=(a ) . Thesequencea(S)maybedefinedbyanother
i i∈Z
period. For example
a(S)=(σ ···σ σ ).
p1 pN+ρ−1 p0
If σ (S) = 2n, a(S) = (r ); if σ (S) = 3n, a(S) is only composed of singular
n n n
sequences and S is called a Inoue-Hirzebruch surface. Moreover if a(S) is com-
posed by the repetition of an even (resp. odd) number of sequences σ , we shall
pi
say that S is an even (resp. odd) Inoue-Hirzebruch surface. An even (resp. odd)
Inoue-Hirzebruch surface has exactly 2 cycles (resp. 1 cycle) of rational curves.
Another used terminology is respectively hyperbolic Inoue surface and half Inoue
surface.
6
WerecallthatforanyVII -classsurfacewithoutnonconstantmeromorphicfunc-
0
tions, the numerical characters of S are [10, I p755, II p683]
h0,1 =1, h1,0 =h2,0 =h0,2 =0, −c2 =c =b (S), b+ =0, b− =b (S)
1 2 2 2 2 2
We shall need in the sequel the explicit description of the dual graph which is
composed of a cycle and branches. A branchA determines and is determined by
s
a piece Γ of the cycle Γ.
s
Theorem 1. 9 ([1] thm 2.39) Let S be a minimal surface containing a GSS,
n=b (S), D ,...,D its n rational curves and D =D +···+D .
2 0 n−1 0 n−1
1) If σ (S)=2n, then D is a cycle and D2 =−2 for i=0,...,n−1.
n i
2) If 2n<σ (S)<3n, then there are ρ(S)≥1 branches and
n
ρ(S)
D = (A +Γ )
s s
s=1
X
where
i) A is a branch for s=1,...,ρ(S),
s
ii) Γ= ρ(S)Γ is a cycle,
s=1 s
iii) A and Γ are defined in the following way: For each sequence of integers
s P s
(a ,...,a )=(r s ···s 2a )
t+1 t+l+k1+···+kp+2 l k1 kp t+l+k1+···+kp+2
contained in a(S)=(σ ···σ ), where
0 N+ρ−1
• l ≥1 and r is a regular l-sequence,
l
• p≥1, i=1,...,p, k ≥1 and s , is a singular k -sequence,
i ki i
we have the following decomposition into branch A and corresponding piece
s
of cycle Γ (where p=p to simplify notations):
s s
Selfint(A ) = (2,...,2, k +2, 2,...,2,..., k +2, 2,...,2, 2)
s 2 p−1
 k1−1 k3−1 kp−1
 | {z } | {z } I|f {pz≡}1(mod 2)
Selfint(Γ ) = (2,...,2, k +2, 2,...,2,..., k +2, 2,...,2, k +2)
s 1 p−2 p
 | l{−z1 } |k2{−z1} |kp−{1z−1}
Selfint(A ) = (2,...,2, k +2, 2,...,2,..., k +2, 2,...,2, k +2)
s 2 p−2 p
 k1−1 k3−1 kp−1−1
 | {z } | {z } I|f {pz≡}0(mod 2)
Selfint(Γ ) = (2,...,2, k +2, 2,...,2,..., k +2, 2,...,2, 2)
s 1 p−1
iv) The top of the bran|chl{−zA1 }is its first|vke2{r−zt1ex} (or curve); the|rokop{t−z1of}A is the
s s
first vertex (or curve) of Γ where t=s+1 (mod ρ(S)).
t
7
3) If σ (S)=3n, D has no branch and
n
i) If a(S)=(s ···s ) then
k1 k2p
D =Γ+Γ′
where Γ and Γ′ are two cycles
Selfint(Γ) = (k +2,2,...,2, k +2,2,...,2,..., k +2,2,...,2)
1 3 2p−1
 k2−1 k4−1 k2p−1
 Selfint(Γ′) = (2,...,2|, k{2z+}2,2,...,2|,k4{z+2},...,2,...,2,k2p|+{2z) }
ii) If a(S)=(s ···s |k1{−z)1t}hen D is c|okn3{t−za1in}s only one cy|kc2lpe{−z1a−n1}d
k1 k2p+1
Selfint(D) = (k +2,2,...,2, k +2,2,...,2,..., k +2,
1 3 2p+1
k2−1 k4−1
| {z } | {z }
2,...,2,k +2,2,...,2,...,k +2,2,...,2)
2 2p
k1−1 k3−1 k2p+1−1
| {z } | {z } | {z }
1.4 Intersection matrix of the exceptional divisor
Let σ = σ ···σ where σ = r = (2,2,...,2) is a regular sequence of
0 N+ρ−1 i pi
length p or σ = s = (p + 2,2,...,2) is a singular sequence of length p ,
i i pi i i
i=0,...,N +ρ−1. We suppose that
• there are N singular sequences and ρ≤N regular sequences
• ifσ isregular,thenσ andσ aresingular,indicesbeinginZ/(N+ρ)Z.
i i−1 i+1
Let n= N+ρ−1p be the number of integers in the sequence σ.
i=0 i
ExamplPes 1. 10 For 0≤N ≤3 we have the following possible sequences:
• If N =0, σ =r ,
n
• If N =1, σ =s or σ =s r , p+m=n,
n p m
• If N =2, σ =s s , σ =s s r , σ =s r s , σ =s r s r ,
p0 p1 p0 p1 m0 p0 m0 p1 p0 m0 p1 m1
• If N =3, σ =s s s
p0 p1 p2
σ =s r s s , σ =s s r s , σ =s s s r ,
p0 m0 p1 p2 p0 p1 m0 p2 p0 p1 p2 m0
σ =s r s r s , σ =s s r s r , σ =s r s s r ,
p0 m0 p1 m1 p2 p0 p1 m0 p2 m1 p0 m0 p1 p2 m1
σ =s r s r s r .
p0 m0 p1 m1 p2 m2
To a sequence σ we associate a symmetric matrix of type (n,n), M(σ) = (m )
ij
“written on a torus”, with indices in Z/nZ defined in the following way: if σ =
σ ···σ =(a ,...,a )
0 N+ρ−1 0 n−1
a if a 6=n+1
i) m = i i
ii n−1 if a =n+1
i
(cid:26)
ii) For 0≤i<j ≤n−1,
−2 if j =i+m −1 and i=j+m −1 mod n
ii jj
m =m = −1 if j =i+m −1 or else i=j+m −1 mod n
ij ji  ii jj
0 in all other cases


8
Theorem 1. 11 (D1,N1) 1) Let S be a minimal complex compact surface con-
tainingaGSSwithn=b (S)>0,thenS containsnrationalcurvesD ,...,D
2 0 n−1
and there exists σ such that the intersection matrix M(S) of the rational curves
in S satisfy
M(S)=−M(σ).
Moreover the curve D is non-singular if and only if a 6=n+1.
i i
Conversely, for anyσ thereexistsasurfaceS containingaGSSsuchthat M(S)=
−M(σ).
2) For any σ 6=r , M(σ) is positive definite.
n
Examples 1. 12 1) For σ = r , M(σ) is not positive definite. The dual graph
n
of the curves has n vertices
(cid:117)(cid:49)
(cid:117)(cid:49)
(cid:117)(cid:49)
(cid:117)(cid:49) (cid:117)(cid:49)
(cid:117)(cid:49) (cid:117)(cid:49)
(cid:117)(cid:49) (cid:117)(cid:49)
(cid:117)(cid:49) (cid:117)(cid:49)
(cid:117)(cid:49)
This configuration of curves appear on Enoki surfaces [6, 17, 1].
2) If σ = s ···s we obtain respectively one or two cycles if N is odd (resp.
p0 pN−1
even). The singularities are cusps and surfaces are odd (resp. even) Inoue-
Hirzebruch surfaces [10, 17, 1]. When there are two cycles, one of the two cycles
determines the other. For example, if σ = s s s s , we obtain a cycle with
p0 p1 p2 p3
p +p curves and another with p +p curves.
1 3 0 2
(cid:128)(cid:117)(cid:100)
-2 (cid:49)
(cid:117)(cid:49)
(cid:117)(cid:83)(cid:119)(cid:128)(cid:76)(cid:49)(cid:68) (cid:84) (cid:117)(cid:49) (cid:117)(cid:49)
(cid:49)
(cid:128)(cid:117)(cid:100)
(cid:100)
(cid:84)
(cid:84) (cid:84)
(cid:117)(cid:49) (cid:117)(cid:83)(cid:119)(cid:128)(cid:76)(cid:49)(cid:68) (cid:117)(cid:83)(cid:119)(cid:128)(cid:76)(cid:49)(cid:68) (cid:117)(cid:83)(cid:119)(cid:128)(cid:76)(cid:49)(cid:68)
(cid:40) (cid:110) (cid:100)
(cid:117)(cid:49)
(cid:117)(cid:49) (cid:117)(cid:49)
(cid:117)(cid:49)
(cid:117)(cid:49)
(cid:128)(cid:117)(cid:100) (cid:128)(cid:117)(cid:100)
(cid:110) (cid:40)
3)Theintermediatecase[17,1,4]. Therearebranchesandthenumberofbranches
is equal to the number of regular sequences in σ. For example, if σ = r s the
p0 p1
dual graph is
9
(cid:84)non singular rational curve of self-intersection (cid:117)(cid:110)
(cid:14)
(cid:117)(cid:49)
non singular rational curve of self-intersection (cid:117)(cid:49)
(cid:117)(cid:49) (cid:117)(cid:49)
(cid:117)(cid:49)
(cid:117)(cid:49) rational curve with double point
(cid:117)(cid:49) (cid:9)
(cid:117)(cid:49) (cid:84) (cid:117)(cid:119)(cid:128)(cid:100)(cid:76)(cid:49)(cid:68) (cid:117)(cid:9)(cid:83)(cid:128)(cid:100) (cid:117)(cid:49) (cid:117)(cid:49) (cid:117)(cid:49) (cid:117)(cid:49)
(cid:117)(cid:49) (cid:117)(cid:49) For(cid:128)(cid:40)(cid:41)(cid:100)
(cid:117)(cid:49)
For(cid:128)(cid:40)(cid:29)(cid:49)
2 Normal singularities associated to surfaces with
GSS
2.1 Genus of the singularities
IfS isaInoue-Hirzebruchsurfaceweobtainbycontractionofacycle,asingularity
calledacusp. TheyappearalsointhecompactificationofHilbertmodularsurfaces
[9]. WeareinterestedhereinthegeneralsituationofanysurfacecontainingaGSS.
Proposition 2. 13 LetS beacompact complex surfaceofclass VII without non
0
constant meromorphic. It is supposed that n := b (S) > 0, the maximal divisor
2
D is not trivial and the intersection matrix M(S) is negative definite. Denote by
Π : S → S¯ the contraction of the curves onto isolated singular points. Then the
following properties are equivalent:
i) D contains a cycle of rational curves;
ii) H1(S¯,OS¯)=0.
Proof: i) ⇒ ii) By Proposition 1, the sequence
(∗) 0→H1(S¯,OS¯)→H1(S,OS)→H0(S¯,R1Π⋆OS)→H2(S¯,OS¯)→0.
is exact. If D contains a cycle then h0(S¯,R1Π O ) ≥ 1. We suppose that
∗ S
h1(S¯,OS¯) = 1 and we shall derive a contradiction. With these assumptions,
h0(S¯,ωS¯) = h2(S¯,OS¯) = 1 and h0(S¯,R1Π⋆OS) = 1 since S¯ has no non con-
stant meromorphic functions. Denote by x , i=0,...,p the singular points of S¯,
i
Γ = Π−1(x ) and p (S,x ) the geometric genus of (S,x ). Then p (S,x ) =
i i g i i g i
h0(S¯,R1Π O ) = 1, therefore there are rational singular points and one elliptic
⋆ S
singularpoint. Moreoverthese singularitiesare GorensteinbecausePh0(S¯,ωS¯)=1
and a non-trivial section cannot vanish because there is no more curves. Hence
there are rational double points with trivial canonical divisor and one minimal
elliptic singularity, (S,x ) with canonical divisor Γ which is a cusp. Since there
0 0
is a global meromorphic 2-form on S, n = −K2 = −Γ2. By [17], S is an odd
S 0
Inoue-Hirzebruch surface (i.e. with one cycle); but such a surface has no canoni-
cal divisor (see for example [3])...a contradiction.
ii) ⇒ i) By the exact sequence (∗), h0(S¯,R1Π O )≤2 without any assumption
∗ S
and 1 ≤ h0(S¯,R1Π O ) by ii). Therefore there is a singular point, say (S,x )
∗ S 0
such that p (S,x ) ≥ 1. If Γ would be simply connected, then taking a 3-cover
g 0 0
space S′ of S we would obtain 3 copies of Γ hence h0(S¯′,R1Π O )≥3 which is
0 ∗ S′
10