From non-K\"ahlerian surfaces to Cremona group of P^2(C)

For any minimal compact complex surface S with n=b_2(S)>0 containing global spherical shells (GSS) we study the effectiveness of the 2n parameters given by the n blown up points. There exists a family of surfaces with GSS which contains as fibers S, some Inoue-Hirzebruch surface and non minimal surfaces, such that blown up points are generically effective parameters. These families are versal outside a non empty hypersurface T. We deduce that, for any configuration of rational curves, there is a non empty open set in the Oeljeklaus-Toma moduli space such that the corresponding surfaces are defined by a contracting germ in Cremona group, in particular admit a birational structure.


Introduction
Hopf surfaces are defined by contracting invertible germs F : (C 2 , 0) → (C 2 , 0). There are well-known normal forms F (z 1 , z 2 ) = (az 1 + tz m 2 , bz 2 ), 0 < |a| ≤ |b| < 1, (a − b m )t = 0, m ∈ N , which give effective parameters of the versal deformation and give charts with transition mappings in the group Aut(C 2 ) of polynomial automorphisms of C 2 , in particular in the Cremona group Bir(P 2 (C)) of birational mappings of P 2 (C). Hopf surfaces are particular cases of a larger family of compact complex surfaces in the VII 0 class of Kodaira, namely surfaces S containing global spherical shells (GSS). When b 2 (S) ≥ 1, these surfaces also called Kato surfaces admit neither affine nor projective stuctures [18,22,19]. Their explicit construction consists in the composition Π of n = b 2 (S) blowing-ups (depending on 2n parameters) followed by a special glueing by a germ of isomorphism σ (depending on an infinite number of parameters). These surfaces are not almost homogeneous [27] hence 0 ≤ dim H 0 (S, Θ) ≤ 1 and Chern classes of surfaces in class VII 0 satisfy the conditions b 2 (S) = c 2 (S) = −c 2 1 (S). By Riemann-Roch formula, we obtain the dimension of the base of the versal deformation of S, 2n ≤ dim H 1 (S, Θ) = 2b 2 (S) + dim H 0 (S, Θ) ≤ 2n + 1, where Θ is the sheaf of holomorphic vector fields. Some questions are raised (1) Are the 2n parameters of the blown up points effective parameters ?
(2) If there are non trivial global vector fields, there is at least one missing parameter. How to choose it ?
(3) Do compact surfaces with GSS admit a birational structure, i.e. is there an atlas with transition mappings in Cremona group Bir(P 2 (C)). More precisely is there in each conjugation class of contracting germs of the form Πσ (or of strict germs, following Favre terminology [14]) a birational representative ?
Known results: • If S is a Enoki surface (see [9]) known normal forms, namely F (z 1 , z 2 ) = t n z 1 z n 2 + n−1 i=0 a i t i+1 z i+1 2 , tz 2 , 0 < |t| < 1, are birational. The parameters t and a i , i = 0, . . . , n−1 are effective at a = (a 0 , . . . , a n−1 ) = 0 (i.e. if there are global vector fields or S is a Inoue surface) and give the versal deformation. If a = 0, there is no global vector fields, t and a i , i = 0, . . . , n − 1 but one a j = 0 give the versal deformation. The complex numbers a i are the coordinates of the blown up points O i in the successive exceptional curves C i .
• If S is a Inoue-Hirzebruch surface (see [6]) is the composition of blowing-ups hence is birational. Here p q r s ∈ Gl(2, Z) is the composition of matrices 1 1 0 1 or 0 1 1 1 with at least one of the second type. There is no parameters because these surfaces are logarithmically rigid.
• If S is of intermediate type (see definition in section 2), there are normal forms due to C.Favre [14] F (z 1 , z 2 ) = (λz 1 z s 2 + P (z 2 ), z k 2 ), λ ∈ C , s ∈ N , k ≥ 2, where P is a special polynomial. These normal forms are adapted to logarithmic deformations and show the existence of a foliation, however are not birational. In [26] K.Oeljeklaus and M.Toma explain how to recover second Betti number which is now hidden and give coarse moduli spaces of surfaces with fixed intersection matrix, • Some special cases of intermediate surfaces are obtained from Hénon mappings H or composition of Hénon mappings. More precisely, the germ of H at the fixed point at infinity is strict, hence yields a surface with a GSS [17,11]. These germs are birational.
Motivation: A.Teleman [28,29] proved that for b 2 (S) = 1, 2, any minimal surface in class VII + 0 contains a cycle of rational curves, therefore has a deformation into a surface with GSS. In order to prove that any surface in class VII + 0 contains a GSS, we should solve the following Problem: Let S → ∆ be a family of compact surfaces over the disc such that for every u ∈ ∆ , S u contains a GSS. Does S 0 contain a GSS ? In other words, are the surface with GSS closed in families ? To solve this problem we have to study families of surfaces in which curves do not fit into flat families, the volume of some curves in these families may be not uniformly bounded (see [13]) and configurations of curves change. Favre normal forms of polynomial germs associated to surfaces with GSS, cannot be used because the discriminant of the intersection form is fixed. Moreover, if using the algorithm in [26] we put F under the form Πσ, σ is not fixed in the logarithmic family, depends on the blown up points and degenerates when a generic blown up point approaches the intersection of two curves. Therefore this article and [8], section 5, focus on the problem of finding effective parameters and new normal forms of contracting germs in intermediate cases of surfaces with fixed σ, such that surfaces are minimal or not and intersection matrices are not fixed. Since usual holomorphic objects, curves or foliations, do not fit in global family, it turns out that birational structures could be the adapted notion. Clearly the problem of their unicity raises.
Main results: A marked surface (S, C 0 ) is a surface S with a fixed rational curve C 0 . In section 2, we define large families Φ J,σ : S J,σ → B J of marked surfaces with GSS with fixed second Betti number n = b 2 (S) which use the same n charts of blowing-ups identified by a subset J ⊂ {0, . . . , n − 1}. The base admits a stratification by strata over which the intersection matrix of the n rational curves is fixed. With these fixed charts, we construct explicit global sections of the direct image sheaf of the vertical vector fields R 1 Φ J,σ, Θ over B J , which express the dependence on the parameters of the blown up points: [θ i ] are the infinitesimal deformations along the rational curves and [µ i ], i = 0, . . . , n − 1 the infinitesimal deformations transversaly to the rational curves. Surfaces with non trivial global vector fields exist over an analytic set of codimension at least 2. The choice of a rational curve C 0 (the first one in the construction) fixes the conjugacy class of a contracting germ. Using a result by A.Teleman [30], we obtain in section 3, Theorem 1. 1 Let (S, C 0 ) be a minimal marked surface containing a GSS of intermediate type, with n = b 2 (S). Let J = I ∞ (C 0 ) be the indices of the blown up points at infinity and let Φ J,σ : S J,σ → B J be the family of surfaces with GSS associated to J and σ. Then, there exists a non empty hypersurface ii) T J,σ intersect each stratum B J,M such that the corresponding surfaces admit twisted vector fields and Z ∩ B J,M ⊂ T J,σ , Corollary 1. 2 Any marked surface (S, C 0 ) belongs to a large family Φ J,σ : S J,σ → B J and there is a non empty hypersurface T J,σ such that over B J \ T J,σ this family is versal.
This answers to the first question and the result is the best possible because T J,σ is never empty. This article stems from discussions with Karl Oeljeklaus and Matei Toma at the university of Osnabrück about the case b 2 (S) = 2, I thank them for their relevant remarks. I thank Andrei Teleman for fruitful discussions in particular to have pointed out that thanks to his results [30] the cocycles θ i and µ i cannot be everywhere independent.
2 Surfaces with Global Spherical Shells 2.1 Basic constructions Definition 2. 4 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 neighbourhood U ⊂ C 2 \ {0} of the sphere S 3 into S such that S \ ϕ(S 3 ) is connected.
Hopf surfaces are the simplest examples of surfaces with GSS.
Let S be a surface containing a GSS with n = b 2 (S). It is known that S contains n rational curves and to each curve it is possible to associate a contracting germ of mapping F = Πσ = Π 0 · · · Π n−1 σ : (C 2 , 0) → (C 2 , 0) where Π = Π 0 · · · Π n−1 : B Π → B is a sequence of n blowing-ups and σ is a germ of isomorphism (see [5]). The surface is obtained by gluing two open shells as explained by the following picture • W 0 is the ball of radius 1 + blown up at the origin, The pseudoconcave boundary of U i is patched with the pseudoconvex boundary of U i+1 by Π i , for i = 0, . . . , n − 2 and the pseudoconcave boundary of U n−1 is patched with the pseudoconvex boundary of U 0 by σΠ 0 , where σ : is biholomorphic on its image, satisfying σ(0) = O n−1 .
If we want to obtain a minimal surface, the sequence of blowing-ups has to be made in the following way: • Π 0 blows up the origin of the two dimensional unit ball B, . . , n − 2, and • σ :B → B Π sends isomorphically a neighbourhood ofB onto a small ball in B Π in such a way that σ(0) ∈ C n−1 .
Each W i is covered by two charts with coordinates ( In these charts the exceptional curves has always the equations v i = 0 and v i = 0. A blown up point O i ∈ C i will be called generic if it is not at the intersection of two curves. The data (S, C) of a surface S and of a rational curve in S will be called a marked surface.
If we assume that S is minimal and that we are in the intermediate case, there is at least one blowing-up at a generic point, and one at the intersection of two curves (hence n ≥ 2). If there is only one tree i.e. one regular sequence and if we choose C 0 as being the curve which induces the root of the tree, we suppose that • Π 1 is a generic blowing-up, • Π n−1 blows-up the intersection of C n−2 with another rational curve and • σ(0) is one of the two intersection points of C n−1 with the previous curves.
The Enoki covering is obtained as in the following picture: of a surface with one tree If all, but one, blowing-ups are generic, then l = n − 1 • Π l (u l , v l ) = (v l + a l−1 , u l v l ) is also generic, but O l is the origin of the chart (u l , v l ), the intersection of two curves.
The general case of ρ ≥ 1 trees is obtained by joining ρ sequences similar to the previous one, i.e., where n 1 + · · · + n ρ = n.

=2 ½
We may suppose, up to a conjugation of F by a linear map, that it means that the strict transform of the curve σ −1 (C n−1 ) intersects C 0 at the infinite point of the chart (u, v), i.e. the origin of (u , v ). This condition is convenient for computations. When n = 2, we denote by U 01 = U 0 ∩ Π 1 (U 1 ) ⊂ U 0 and U 10 = U 1 ∩ σΠ 0 (U 0 ) ⊂ U 1 the two connected components of the intersection U 0 ∩ U 1 of the images in S of U 0 and U 1 , denoted in the same way.
. We refer to [5] for the description of configurations of curves. We index the curves (C i ) i∈Z in the universal covering space following the canonical order (see [5]). Let a(S) = (a i ) i∈Z be the family of positive integers defined by a i = −C 2 i . By [5] p104, this family is periodic of period n and for any index i ∈ Z we define a positive integer independant of i, The family (a i ) i∈Z splits into sequences s p = (p + 2, 2, . . . , 2) and r m = (2, . . . , 2) of length respectively p and m, where p ≥ 1 and m ≥ 1. We call s p (resp. r m ), p ≥ 1 (m ≥ 1) the singular (resp. regular) sequence of length p (resp. m). We have ρ := {trees} = {regular sequences}.

Large families of marked surfaces
With the previous notations, we consider global families of minimal compact surfaces with the same charts, parameterized by the coordinates of the blown up points on the successive exceptional curves obtained in the construction of the surfaces and such that any marked surface with GSS (S, C 0 ) belongs to at least one of these families. More precisely, let F (z) = Π 0 · · · Π n−1 σ(z) be a germ associated to any marked surface (S, C 0 ) with tr(S) = 0. In order to fix the notations we suppose that C 0 = Π −1 0 (0) meets two other curves (see the picture after definition 2.5), hence σ(0) is the intersection of C n−1 with another curve. We suppose that We denote by I ∞ (C 0 ) ⊂ {0, . . . , n − 1} the subset of indices which correspond to blown up points at infinity, that is to say, Each generic blow-up may be deformed moving the blown up point (a i−1 , 0). If we do not want to change the configuration we take for all κ = 0, . . . , ρ − 1 (with n 0 = 0), The mapping σ is supposed to be fixed. We obtain a large family of compact surfaces which contains S such that all the surfaces S a have the same intersection matrix and n 1 + · · · + n ρ = n.
If for an index κ, a n1+···+nκ = 0, there is a jump in the configuration of the curves. For instance, if for all κ, κ = 0, . . . , ρ − 1 a n1+···+nκ = · · · = a n1+···+nκ+lκ−1 = 0 we obtain a Inoue-Hirzebruch surface. To be more precise the base C l0 × · · · × C lκ × · · · × C lρ−1 splits into locally closed submanifolds called strata • On second hand, towards Enoki surfaces. If for all indices such that O i is at the intersection of two rational curves, in particular for i ∈ J, the blown up point O i is moved to O i = (a i , 0) with a i = 0, all the blown up points become generic, the trace of the contracting germ is different from 0. We obtain also all the intermediate configurations.

Proposition 2. 6
There is a monomial holomorphic function t : C Card J → C depending on the variables a j , j ∈ J such that over B J := {|t(a)| < 1} ⊂ C n , the family Φ J,σ : S J,σ → B J may be extended and for every a ∈ B J , t(a) = tr (S a ).
We have and with our convention on σ, The general case is obtained by the composition and other F k have similar expressions with σ = Id and m k ≥ 1, i.e.
and we extend the family thanks to proposition 2.6. We obtain larger strata of minimal surfaces, from dimension l + 1 to dimension n.
2) Example with 6 curves: If we start with the sequence Towards Enoki surfaces, we move each non generic point into generic one: • (42 2 3 22) with a 5 ∈ C , • (2222 3 2) with a 1 ∈ C , a 4 ∈ C , • (3 22 3 22) with a 1 = 0, a 2 ∈ C , a 5 ∈ C , • (42 2222) with a 4 ∈ C , a 5 ∈ C , • (222222) with a 1 ∈ C , a 4 ∈ C , a 5 ∈ C Proposition 2. 8 For any J = ∅, any invertible germ σ : Proof: 1) Let N = Card J ≥ 1, we may suppose that the numbering is chosen so that 0 ∈ J, and for all the other indices k 2) We suppose that G is a germ associated to a Inoue surface therefore there is an invertible germ ϕ : is the canonical germ associated to Inoue surfaces and n = N i=1 l i + N . Recall that a germ associated to a Inoue surface admits an invariant curve Γ, i.e. there is a unique germ of curve such that F |Γ : Γ → Γ is a contraction and the curve Γ induces the elliptic curve of the associated surface S(G) S(F ) [5]. The curves Γ and σ −1 (C n−1 ) are transversal, therefore replacing if necessary G by a conjugate, we may suppose that with A 10 = 0, the left member of the equality ϕ • G = F • ϕ is, which is impossible since there is no linear term in the left member.
Remain non minimal surfaces: we still extend the previous family on a small neighbourhood B J of B J , moving the blown up point transversally to the exceptional curves there are some questions: • Are the parameters a i , b i , i = 0, . . . , n − 1, effective ?
• Which parameter to add when h 1 (S a,b , Θ a,b ) = 2b 2 (S a,b ) + 1 in order to obtain a complete family ?
• If we choose σ = Id or more generally an invertible polynomial mapping, we obtain a birational polynomial germs. Does this families contain all the isomorphy classes of surfaces with fixed intersection matrix M ?

Minimal and non minimal deformations
Proof: The fundamental remark is that (a, b) ∈ H i if and only if in the construction of the surface S (a,b) there is a sequence of indices i, i + j 1 , . . . , i + j p = i mod n such that the curve C i+j k is blown up by C i+j k+1 . If this sequence of blow-ups ends before reaching the index i, say at i + j q , C i + C i+j1 + · · · + C i+jq will be a simply exceptional divisor. Therefore, the total transform of C i has to check We have . . . and this is the equation of a smooth hypersurface. The third assertion follows readily from these equations.

Infinitesimal deformations of the families S J,σ
We define the following cocycles which are the infinitesimal deformations of the families S J,σ → B J : • For i = 0, . . . , n−1, the cocycles θ i called the "tangent cocycles" move the blown up points O i along the curve C i and vanish only (at order two) at the point "at infinity" • For i = 0, . . . , n − 1, the cocycle µ i called the "tranversal cocycles" move O i transversaly to C i On a stratum where there are global twisted vector fields we need another infinitesimal deformation (see [8] for an explicit construction).
More precisely, Since θ i just moves the blown up point O i along the curve C i , all surfaces in these deformations are minimal. We introduce now n other cocycles which move the blown up point O i transversaly to the exceptional curves C i . They yield non minimal surfaces, for instance blown up Hopf surfaces but also surfaces with GSS blown up k times, 1 ≤ k ≤ n.
For any J ⊂ {0, . . . , n − 1}, the family S J,σ → B J is globally endowed with a family of Enoki coverings. Using the family of Enoki coverings, all the cocycles θ i , µ i are globally defined over S M,σ and give global sections

Splitting of the space of infinitesimal deformations
We divide minimal deformation in two types of deformations: logarithmic deformations for which the intersection matrix of the maximal divisor D does not change, in particular the surfaces remain minimal, and deformations in which the cycle may be smoothed at some singular points or disappear and surfaces may become non minimal.
Proof: Consider the exact sequence on S Supp(J D ) = D, and N Di the normal bundle of D i . The long exact sequence of cohomology gives We compute now H 1 (D, J D ): the restriction of ( ) to U gives

Infinitesimal non logarithmic deformations
We We suppose that there exists a linear relation We choose the curve C 0 such that O 0 is a generic point but O n−1 is the intersection of two curves. Hence D 0 the curve in S induced by C 0 is the root of a tree. We shall use this fact later. We have the following linear system where X i is a vector field over U i , i = 0, . . . , n − 1: We notice that by Hartogs theorem, X i extends to W i , hence X i is tangent to C i for i = 0, . . . , n − 1; moreover Therefore the i-th equation at O i gives β i = 0.
Remark 3. 12 If we replace the vector field ∂ ∂vi by any non vanishing transversal vector field, the proof works as well.
Now, we show that if O i is the intersection point of two curves, then α i = 0. In fact, there are two cases: (2) The vector field X k+1 is tangent to C k , therefore X k+2 is tangent to (the strict transform of) C k .
and α k+2 = 0; by induction we prove α k+1 = α k+2 = · · · = 0 till the moment O k+l is not the point C k+l ∩ C k but the point C k+l ∩ C k+l−1 . However if it happens it means that we are in the first case.
We have obtained The sequence of blowing-ups splits into subsequences where κ = 0, . . . , ρ − 1. The indices wich correspond to points O i at the intersection of two curves are i = n 1 + · · · + n κ + l k , . . . , n 1 + · · · + n κ + n κ+1 − 1, therefore for κ = 0, . . . , ρ − 1, The equations (E1) become It should be noticed that a block may be reduced to one line, if l κ = n κ+1 − 1, i.e. if there is in the block only one blowing-up at the intersection of two curves.
For κ = 0, . . . , ρ − 1, the vector fields X n1+···+nκ+lκ , . . . , X n1+···+nκ+1−1 glue together into a vector field that we shall still denote X n1+···+nκ+lκ . Hence setting we reduce the system to When ρ = 1, i.e. when there is only one tree, the linear system reduces to  Proof: In the explicit construction of Inoue-Hirzebruch surfaces [6], there is no generic blown up points and h 1 (S, Θ) = 2n, hence we have an explicit base of H 1 (S, Θ) and explicit universal deformation. It is easy to see that any singular point of a cycle may be smoothed for even as well for odd Inoue-Hirzebruch surface.
For moduli space of Oeljeklaus-Toma see [26]. Definition 3. 17 Let X be a complex manifold of dimension m. We shall say that X admits a birational structure if there is an atlas U = (U i ) i∈I with charts ϕ i : is the restriction of a birational mapping of P m (C). Proof: We take σ = Id, then the gluing map σ • Π is birational, then apply Corollary 3.16.

Infinitesimal logarithmic deformations
We have seen that a relation is only possible among infinitesimal logarithmic deformation. In fact it can contain neither θ i when the curve C i meets two other curves. In order to avoid an overflow of notations, we give a complete proof for surfaces with only one tree and we postpone it to the appendix 5.2. The idea of the computation is to work in the first infinitesimal neighbourhood of the maximal divisor. Vanishing of other coefficients should imply to work (if possible) in the successive infinitesimal neighbourhoods. • O k is a generic point but C k meets two other curves.
In particular, if the unique regular sequences r m are reduced to one curve (i.e. m = 1), is an independant family of H 1 (S, Θ) (resp. of H 1 (S, Θ(−LogD))) and a base if there is no non trivial global vector fields. Remark 3. 20 By induction it is possible to show that for any k < r + s − (p + q), a similar Cramer system may by defined and that α k = 0. However, it is not possible to achieve the proof in this way because when k = r + s − (p + q) a new unknown appears. This difficulty is explained by the fact that in general there is a relation among the [θ i ]'s or a class vanishes.

The hypersurface of non versality
The constructed families are generically versal. We show in this section that the locus of nonversality is non empty hypersurface. When b 2 (S) = 2 it is the ramification set of the canonical mapping from each stratum to Oeljeklaus-Toma moduli space. It is conjectured that it is a general phenomenon.
Remark that a numerically Q-anticanonical divisor D −K on a surface S is a solution of a linear system whose matrix is the intersection matrix M = M (S) of S. Therefore the index is the least integer m such that mD −K is a divisor and this integer is fixed on any logarithmic family Φ J,M,σ : S J,M,σ → B J,M .
, z k 2 ) be a Favre contracting germ associated to a surface of intermediate type S. Let µ = index(S) ∈ N and κ ∈ C such that where α is the vanishing order of θ along C n−1 and A(0) = 0. Since det DF (z) = λkz s+k−1 2 , comparison of lower degree terms gives Lemma 4. 23 Let S be a minimal complex surface, µ the index of S and κ such that H 0 (S, K ⊗−µ ⊗ L κ ) = 0. Then a section of K ⊗−µ ⊗ L κ vanishes on all the rational curves in S.
Proof: Let D i , i = 0, . . . , n − 1 be the n rational curves in S and suppose that We have k i ≥ 0 for all i = 0, . . . , n − 1; if one coefficient vanishes, say k 0 = 0, on one hand, since the maximal divisor is connected, and on second hand, by adjunction formula we obtain a contradiction. such that H 0 (S a , K −µ Sa ⊗ L κ(a) ) = 0. 2) If the surfaces admit twisted vector fields there exists a unique surjective holomorphic function such that the marked surface (S a , C 0,a ) is defined by a germ of the form 3 However the second possibility is excluded by Grauert semi-continuity theorem because on a whole stratum we would have H 0 (S a , K −µ ⊗ L α ) = 0 which is impossible because the twisting parameter is not constant. Therefore the slice has an extension. If {κ = α} ∩ (B J,M \ B J,M ) = ∅, the line bundle K −µ ⊗ L α has a section over S J,σ |{κ=α} hence the zero locus which is the union of all the rational curves by [10] would be is a flat family of divisors; however it is impossible because the configuration changes contradicting flatness (it can be seen that the curve whose self-intersection decreases has a volume which tends to infinity (see [13])). Therefore

Intermediate surfaces
We consider intermediate surfaces S, since the problem of normal forms is solved for the other cases. There are two curves: one rational curve with a double point D 2 1 = −1 with one tree where k = k(S) = 2, s = 1. Invariant vector fields θ exist if and only if λ = 1 in which case A germ of isomorphism ϕ which conjugates G a and G a leaves the line {z 2 = 0} invariant, therefore ϕ has the form ϕ(z) = (ϕ 1 (z), Bz 2 (1 + θ(z)). A simple computation shows that if G a and G a are conjugated then a 1 = ±a 1 .
Besides if we want to determine the twisting parameter κ such that H 0 (S, K −1 ⊗ L κ ) = 0, we have to solve the equation µ(G a (z)) = κ det DG a (z)µ(z).
Therefore all surfaces with global vector fields are obtained when ξ moves in C, and X acts by translation. In particular when b 2 (S) = 2, all surfaces are obtained by simple birational mappings obtained by composition of blowing-ups and an affine map at a suitable place. We extend the family to Enoki surfaces. The family of marked surfaces Φ J,σ : S J,σ → B J is defined by the family of polynomial germs G a (z 1 , z 2 ) = z 2 , z 2 2 (z 1 + a 1 ) + a 0 z 2 , a = (a 0 , a 1 ) We have tr (S) = tr DG a (0) = a 0 therefore |a 0 | < 1. The open set B J = ∆ a0 × C a1 has the following strata We have Since by Hartogs theorem the vector fields X 0 and X 1 extend on the whole blown up ball, they are tangent to the exceptional curves and we set By straightforward computations similar to those in the appendix we derive that Case J = {0, 1} With σ(z) = (z 1 + a 1 , z 2 ), the family of marked surfaces Φ J,σ : S J,σ → B J is associated to the family of polynomial germs det DG a (z 1 , z 2 ) = z 2 2 (z 1 + a 1 ), tr (S a ) = tr DG a (0) = a 0 a 1 , with |a 0 a 1 | < 1.
There are two lines of intermediate surfaces which meet at the point (0, 0) which parametrize the Inoue-Hirzebruch surface with two singular rational curves.
The involution of the Inoue-Hirzebruch surface which swaps the two cycles induces on the base of the versal family swapping of the two lines of intermediate surfaces.
We have obtained Theorem 4. 29 Let F = Πσ : (C 2 , 0) → (C 2 , 0) be any holomorphic germ, where Π = Π 0 Π 1 are blowing-ups and σ is any germ of isomorphism. Then F is conjugated to a birational map obtained by the composition of two blowing-ups and an affine map at a suitable place. If moreover, S is of intermediate type and there is no non-trivial invariant vector field, F is conjugated to the composition of two blowing-ups of the previous types.

On logarithmic deformations of surfaces with GSS, by Laurent Bruasse
The results contained in this section is a not yet published part of the thesis [3]. Notations are those of [4] and [10].
Let F be a reduced foliation on a compact complex surface S. We denote by T F (resp. N F ) the tangent (resp. normal) line bundle to F. Let p be a singular point of the foliation; in a neighbourhood of p endowed with a coordinate system (z, w) in which p = (0, 0), F is defined by a holomorphic vector field Let J(z, w) be the jacobian matrix of the mapping (A, B). Baum-Bott [1] and Brunella [4] have introduced the following two indices: where Res (0,0) is the residue at (0, 0) (see [16] p649). We denote by S(F) the singular set of F, it is a finite set of points, and let By [10], if S is a minimal compact complex surface with GSS, then Det F = n, T r( F) = 2n − σ n (S).
Proposition 5. 32 Let S be a minimal surface containing a GSS with n = b 2 (S) ≥ 1 and tr(S) = 0. If F is a reduced foliation on S, then If there is no non-trivial global vector fields this integer is precisely the number of generic blowingups.
We conclude by Riemann-Roch theorem that which is the annouced result.
We have a canonical injection 0 . The aim of the following proposition is to compare logarithmic deformations and deformations which respect the foliation: Since θ is tangent to D, ω i (θ) vanishes on D, therefore the morphism is well defined on U i . Moreover, by definition, the normal bundle N F is defined by the cocycle (g ij ) ij = (ω i /ω j ) ij ∈ H 1 (U, O ), therefore j is well defined on S and its kernel is clearly Im i. It remains to check that j is surjective: outside D it is obvious since the foliation has singular points only at the intersection of two curves and we have the exact sequence and U an open neighbourhood of x on which f is defined.
• If x is not at the intersection of two curves, let (z, w) be a coordinate system in which D = {z = 0} and F defined by ω = a(z, w)dz +zb(z, w)dw. Since f vanishes on D, f = zg. Let θ = zα(z, w) ∂ ∂z + β(z, w) ∂ ∂w be a logarithmic vector field. We have to find α and β such that f (z, w) = zg(z, w) = ω(θ) = z(aα + bβ) i.e. g ∈ (a, b). The are solutions because x is not a singular point of the foliation hence a is invertible at x.
We have to solve g = aα + bβ. By [20] p171 (see also [10] p1528), the order of θ is one, hence a or b is invertible and g ∈ (a, b).
Let S be a minimal compact complex surface containing a GSS with n = b 2 (S) ≥ 1 and tr(S) = 0. If S is not a Inoue-Hirzebruch surface then S admits a unique holomorphic foliation In particular any logarithmic deformation keeps the foliation.

Infinitesimal logarithmic deformations: the hard part
We give in this section the proof of proposition 3.19. Since σ(0) = O n−1 is the intersection of two transversal rational curves which are contracted by F , there is a conjugation by a linear map ϕ (in particular birational) such that ϕ −1 F ϕ = F = Π σ , satisfies It means that σ −1 (C n−1 ) is tangent to z 2 = 0 and the other curve is tangent to z 1 = 0, therefore their strict transforms meet the exceptional curve C 0 respectively at {u = v = 0} and {u = v = 0}. Therefore in the following computations we shall suppose that the condition (S) is satisfied. Let Π = Π l • Π l+1 • · · · • Π n−1 be the composition of blowing-ups at the intersection of two curves and of Π l , then it is the composition of mappings (u, v) → (uv, v) or (u , v ) → (v , u v ), and of Π l (u , v ) = (v + a l−1 , u v ), hence Π (x, y) = (x p y q + a l−1 , x r y s ) where p q r s is the composition of matrices 1 1 0 1 or 0 1 1 1 , the last one being of the second type, therefore det p q r s = ±1.