A NNALES SCIENTIFIQUES DE L ’É.N.S.
B. A ZEVEDO S CÁRDUA
Transversely affine and transversely projective holomorphic foliations
Annales scientifiques de l’É.N.S. 4e série, tome 30, no2 (1997), p. 169-204
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4° serie, t. 30, 1997, p. 169 a 204.
TRANSVERSELY AFFINE AND TRANSVERSELY PROJECTIVE HOLOMORPHIC FOLIATIONS
BY B. AZEVEDO SCARDUA
ABSTRACT. - Let T be a codimension one holomorphic singular foliation on Mn. T is transversely affine respectively transversely projective if so it is its regular foliation. We consider foliations which are transversely affine or projective in M\A for some analytic codimension one invariant subset A C M. Examples are logarithmic and Riccati foliations on CP(2). In the projective case ther is a dual foliation ^"-L generically transverse to T-'.
T1' is a fibration if F is Riccati. We prove: 1. Let 7 be given on CP(2), transversely affine outside an algebraic invariant curve A. Suppose that T has reduced non-degenerate singularities in A. Then F is logarithmic. 2. Let y be given on CP(n), transversely projective non-affine, outside an invariant algebraic hypersurface A. Then J:l• extends to CP(n). If this extension has a meromorphic first integral, then F is Riccati rational pull-back.
Introduction
In this paper we consider holomorphic singular foliations of codimension one on a complex n-manifold M, n > 2. Let T be such a foliation and assume that the singular set of T, denoted s{^\ has codimension > 2. Define M' = M\s{y) and T ' = T J M ' the non singular associated foliation. Thus T ' can be defined by a covering of M' by open subsets Ui, i C J, and distinguished mappings fi:Ui —^ C, i.e. each fi is a holomorphic submersion and the leaves of ^/Ui are the connected components of the level surfaces /^(rr), x C C. Whenever U^Uj / (f) we have fi = fijofj for some local biholomorphism fir fj^i n Vj) C C ^ fi(Ui n Uj) C C. If Ui n Uj n £4 ^ (j> then we have in the common domain the cocycle condition fij o /^ = f^. The transversal structure of T in M is defined by the pseudogroup {fij}-i,j € I so that T has a "simple" transversal structure if this pseudogroup is "simple" for some choice. The correct meaning of the expression "simple" above is given by the notion of transversely homogeneous foliation (Chapter II §6) where the local biholomorphisms fij are restrictions of elements of a Lie group action on an homogeneous space. In the codimension one case the remarkable examples are derived from the following ones: transversely additive, affine and projective structures; where the submersions fi:Ui —> C are related by fi = fj + bij, fi = aijfj + bij and fi = a^jfj^ij., {dij^bij, Cij, dij e C); respectively, where in the affine case we require aij ^ 0 and in the projective case that aijdij — bijCij = 1. Of course the afffine case is a particular case of the projective case but we shall deal with the affine and the projective non-affine separately. We will investigate how often these structures appear. We remark that the existence of an affine resp. projective transverse structure implies that the non-singular ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE. - 0012-95 93/97/027$ 7.00/© Gauthier-Villars
associated foliation is given by a holomorphic resp. meromorphic submersion in any simply connected open set. This is a consequence of the well known notion of development of a transversely projective foliation (see [17] for instance). Using well known extension theorems for holomorphic or meromorphic functions through codimension > 2 analytic subsets (in our case s(^F)) we can obtain a holomorphic resp. meromorphic first integral for a transversely affine resp. projective foliation on a simply connected manifold and then conclude that there exists no transversely affine foliation on a compact simply connected manifold (for instance, the complex projective n-space CP(n)), and that the transversely projective foliations on CP(n) are the ones which have a rational first integral. Motivated by this we will consider foliations which are transversely affine or projective in M\S' for some analytic codimension one set S C M, invariant by the foliation T. Well known examples of these foliations are given by linear logarithmic and Riccati foliations on CP(n) and its pull-backs to spaces M (see Chapter I, §1, Example 1.3, and Chapter II,
§1, Example 1.1, for the definitions). These two families of examples play a fundamental role in our study being used as models.
In Chapter I we study transversely affine foliations proving the following (see Thm. 4.5):
THEOREM I. - Let T be a codimension one foliation on CP(n) which is transversely affine outside an algebraic codimension one invariant subset S C CP(n). Suppose that T has reduced non-degenerate singularities in S fsee Ch. I Section 2 for the definitions/
Then T is a logarithmic foliation.
For the proof of this theorem we need to study the holonomy of an irreducible component So of S. This goes as follows (see Theorem 4.1 and Proposition 5.1):
THEOREM II. - Let T be a foliation on M2 having A C M as an analytical connected invariant curve. Suppose
i) all singularities ofT in A are of1st-order
ii) the foliation T obtained by the resolution of the singularities ofT in A exhibits some linearizable non-resonant singularity. Then the following conditions are equivalent:
a) T is transversely affine in some neighborhood of A minus A and its local separatrices sep(A);
b) the holonomy group of the leaf A\s(^F) and of any projective line in the desingulariz.ation of T in A is a solvable group and we have the solvability compatibility between them (see Ch. I, Section 5 for definition). This is called the property (S) for the holonomy of A.
Using this theorem an the topological invariance of the projective holonomy, for stable deformations of germs of 1-forms having a generic first jet [15] we obtain the following theorem (see Proposition 5.2, Ch. I).
THEOREM III. - Let w = Adx + Bdy be a germ of holomorphic 1-form in the origin ofC2
having w^ generic as first y-jet, v > 2 and let w' = A'dx + B ' d y be a stable deformation of w. Suppose w has a multiform integrating factor of the form f = Hf^, fj G Vs, \j C C*.
Then w' has an integrating factor of the same type.
Chapter II is devoted to the study of foliations which are transversely projective outside an invariant codimension one analytic subset. We associate to such a projective non-affine structure for T in M, a dual codimension one foliation T1- on M which is transverse
4e SERIE - TOME 30 - 1997 - N° 2
to F almost everywhere. The duality between T and T1' is such that one determines the other. For example if T is a Riccati foliation T\ p(x)dy - (y^a^x) + yb(x) -h c(x))dx = 0 on C x C then the natural dual foliation J7-1- is the fibration x = Cte by vertical projective lines of C x C. The existence of such a dual fibration is persistent under rational pull-backs.
One central result proved in II §4 states that indeed this characterizes the existence of the pull-back from a Riccati foliation (see Theorem 4.1, Ch. II).
THEOREM IV. - Let T be a foliation on CP(n) which is transversely projective but not transversely affine, outside an invariant analytic subset S of codimension one. Then the dual foliation T1' on CP(n)\S extends to a foliation on CP(n) and if T1' has a meromorphic first integral then T is the rational pull-back of a Riccati foliation on CP(2).
We also study the cases where T1' has an affine transverse structure in CP(n)\S and the local case for T-'. The techniques introduced here are used to give different proofs of well known results about stability of logarithmic foliations on CP(n), n > 3 [3] and rational foliations on CP(n), n>_3 having first integrals of the form fp/ gq, (p^q) = 1 [18]. We also give a proof of a theorem due to A. Lins Neto and D. Cerveau on the existence of meromorphic first integrals for foliations on CP(n), n > 3, having a complete intersection Kupka component [11]. One important remark about the generality of the context is the following (see Theorem 6.1):
THEOREM V. - Let T be a holomorphic singular transversely homogeneous foliation of codimension one on M71. Then F is transversely projective foliation on At71.
These notes are derived from my doctoral thesis ([27]) held at IMPA in the year of 1994, under the advise of Prof. Cesar Camacho to whom I am very grateful and who suggested to me the subject. I am also in debt with A. Lins Neto, P. Sad and M.Brunella for many valuable conversations and suggestions during the preparation of my thesis and of this text.
I would like to thank Prof. E. Ghys for valuable discussions during the beggining of this work, for suggesting me the book of C. Godbillon "Feuilletages: Etudes Geometries I" and the references on real transversely affine foliations, which where very valuable, and for suggesting me the geometric approach I use here. I am grateful to D. Cerveau for pointing out the necessity of the use of Stein's Fatorization Theorem in Chapter II. Finally I want to thank the referee for his kind interest and careful reading of the original manuscript, which has helped me to improve the paper.
Chapter I
Transversely Affine Holomorphic Foliations 1. Transversely affine foliations and differential forms
Throughout this chapter I, except for explicit mention, the 1-form 0 will be assumed to have singular set s(Q.) of codimension bigger than one.
The problem of deciding wether there exist affine transverse structures for a given foliation is equivalent to a problem on differential forms as stated below (see [1] for the case of real non-singular foliations):
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PROPOSITION 1.1. - Let ^l be an integrable meromorphic 1-form which defines T outside the polar divisor (^)oo. The foliation T is transversely affine in M if and only if there exists a 1-form 77 in M satisfying: 77 is meromorphic, closed, d^l = 77 A 0, (77)00 = (^)oo and Resr] = —{order of'(0)oo|^) for each irreducible component L of (n)oo, and (77)00 has order one. Furthermore, two pairs (0, rf) and (0', y/) define the same affine structure for T in M if and only if there exists a meromorphic map g'.M —> C satisfying ^l' = g^l
and T}' = 77 + dg-.
Remark 1.1. - (a) For the case where M is open and ^ is holomorphic the form 77 is holomorphic. (b) The existence of a meromorphic 1-form 0 which defines T globally in M is always true if M is a complex projective space CP(n) or an algebraic non-singular projective variety (see [13] for instance), but is not really necessary (see Section 6 of Chapter I).
Proof of Proposition 1.1. - Let fl be a meromorphic 1-form which defines T in M and suppose {yi: Ui —> C} is a transversal affine structure for T in M. Since the submersions Vi define T locally, we can write 0|^ = gi dyi for some meromorphic Q{. In Ui nUj ^ (/) we have: (1) gi dyi == gj dyj\ (2) yi = a^ % + &^. From (2) we have dyi == a^ d%
and then from (1) we have a^^ = ^ so that d g i / g i = d g j / g j and this allows us to define 77 in M\5(^~) by 77!^ == d g i / g i . The 1-form 77 is closed, meromorphic and satisfies d^l = 77 At?. Since codimension (^(J')) > 1 we can extend by Hartogs' Extension Theorem (see [30]) the 1-form 77 meromorphic to M. We also have (77)00 = (^)oo of order one and Resj^ = - order of 0|^, for each component L of (0)oo: In fact, it is clear by the construction that (77)00 == (^)oo. Now given a point p C (^)oo say, p C £, L an irreducible component of (0)oo, choose a holomorphic function x: U —» C defined in p € U such that a^.O is holomorphic at p, where 71 = order of (0)oo along L. Then o^.O = g d y m SL small neighborhood of p so that by construction we have
» = x^.gdy and 77 = ^^ = -^ + ^ . Since g is holomorphic along L it follows that ResL 77 = —n. This proves the first part of the proposition.
Assume now that 0 and 77 are as in the statement. Since 77 is holomorphic and closed in M\(0)oo, there exists an open cover {Ui} of M\(0)oo and there are holomorphic functions hi G Hol(L^) such that 77) = dhi. We define gi = exp(^), gi G V(L^)* to obtain 77]^ = d g i / g i . Now, from condition dO = 77 A 0 we have d ( ^ ) = 0 , and then 0 = gi dyi for some holomorphic function yi e V(L^). This we can do in M\(n)oo. Now, given a point pi e (^)oo we can choose a local chart (rr,2/) G ^ such that ^ = (0,0), (0)oon?7= {^/ = 0} and r](x,y) = -^+^ where n = order of (n)oo and / € V(;7,)*.
Therefore we have 77 = ^^P = d^. 9i = /.2/~n. Thus the 1-form ^ is closed and holomorphic so that it can be writen ^ == dyi for some holomorphic yi. Thus we have covered M\s(^) with open sets Ui where we have the relations ^ = gi dyi, 77 = d9±. In each Ui C\Uj ^ (/) we have d9j- = 77 = dgl and ^ dyi = 0. = gj dyj. The first equality implies gj = a^.^ for some locally constant a^ and it follows from the second equality that dyi = a^ dyj and then yi = a^ yj + bij with bij locally constant in Ui H Uj. This shows that T is transversely affine in M.
4'^ S^RIE - TOME 30 - 1997 - N° 2
Now we prove the last part of the proposition. Let (0,^) be given and let g: M —^ C be a meromorphic function. We define 0' = gfl and 77' = T] + d9-. Using the same notation above we have r/|^ = r^ + ^ = ^ + ^ = ^ and ^ = ^|^ = {ggi)dy^
and this shows that: Q\ = a^ g'j and y[ = yi so that a^ = a^ and b^ = bij. Hence, the pairs (0,yy) and (^',T/) define the same transversal structure for T in M. Finally, suppose that (^,77) and (O',T/) define the same transversal structure for T in M. Since
^ and ^/ define ^7, we have 0' = ^0 for some g: M —» C meromorphic. Using the same notation above we write (locally) Q, = gidyi, Q,' = g^dyi, T] = d g i / g i and T]' = d g [ / g [ \ but g[ = ggi so y/ = 77 + d g / g completing the proof. D
Example 1.1. - Transversely affine foliations on simply-connected manifolds. Let M be simply-connected and let T be given by a holomorphic 1-form 0. The transversal affine structures for T are given by the holomorphic maps f:M —> C which are submersions outside s{y): In fact, it is a consequence of the well-known notion of development of a transversely homogeneous foliation (see [17] Prop. 3.3 pp.247-248), that the foliation exhibits a holomorphic first integral on M' = M\s{J^) (notice that M' is also simply connected). Hartogs' theorem [20] implies that this first integral extends holomorphically to M. In particular, the existence of an affine transverse structure on a punctured neighborhood of a singularity implies that this singularity has a (local) holomorphic first integral and is therefore of first order (see Section 2 for the definition).
Example 1.2. - Let $: N —» M be a holomorphic map transverse to the foliation T. If T is transversely affine then so it is the induced foliation ^*^7. This is easily verified by taking the local submersions which define the affine transverse strcture for T.
Example 1.3. - Logarithmic foliations on CP(n). The foliation T on CP(n) is called logarithmic if there is a rational map TT: CP(n) —> CP(m) such that 7 = TT*(L) where L is the linear logarithmic foliation on CP(m) given by 0 = nr=i x^ S^J=i ^j^ = 0 ln some affine chart (a:i,... ,Xm) e C^ <-^ CP(m). If we define the 1-form rj = ^^ d±L we can conclude from Proposition 1.1 that L is transversely affine in CP(m)\A where A C CP(m) is the algebraic invariant set given by IJJLi {^j = 0}, hence using Example 1.2 we conclude that T is transversely affine outside an algebraic invariant set D = TT-^A) C CP(n). Let 7r(^i,... ,Xn) = (fi(x^,... ,Xn),... , / m ( ^ i , . . . ,j^))_m affine charts; where the fi's are irreducible smooth polynomials; then D = |j • {fj =0}
and the hypersurfaces {fj = 0} are the compact leaves of T\ they have linearizable holonomy and any other leaf has trivial holonomy. For more information on logarithmic foliations the reader should consult [2].
Example 1.4. — Bernoulli foliations on CP{n + 1). In CP(n +1) we consider affine coordinates ( r c i , . . . , Xn^ y) G C71^1 <—^ CP(n-l-l). Let 0 be the meromorphic 1-form given by 0(a:i,... ,Xn,y) = (11^=1 Pj^jWv - E^=i(n^ Pi^i^^^^j) - ybj(xj))dxj', where pj, bj, Cj are polynomials of one variable. We say that Q, defines a Bernoulli foliation of order k on CP(n + 1), if ^ satisfies the following integrability condition:
Ci{xi).bj[xj) = Cj{xj).bi{xi) V % , j . Under this hypothesis we define the 1-form T] := k^ + E^-iy -—'j—-L P j ^ j ) PJ-C1———--—"—dxj, and we obtain a transversal affine structure for y = ^'(0) outside of an algebraic invariant set F C CP(n 4- 1), which is a
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUP^RIEURE
finite union of hyperplanes CP(n) C CP(n + 1). If n = 1 we have ^(x,y) = p(x)dy - (yk c(x) - yb(x))dx which is the pull-back of the particular Riccati foliation (p(u)dv -{k- l)(c(u)v2 - vb(u))du by a map (u, v) = (x, ^-1). The point p^ G CP(2) given by x = 0, y = oo is a dicritical singularity of T (see definitions in §2). This dicritical singularity plays a fundamental role in the study of the structure of T and is the responsible for the non linearization of T. In fact in general T is not the pull-back of a linear logarithmic foliation because of the non-algebraic separatrices of poo.
Example 1.5 (see [I],[16]). - We will define a transversely affine foliation on a compact 3-manifold. This will be a non-singular foliation with dense leaves which are biholomorphic to C* x C* or cylinders C*/Z x C*. We begin with a general construction inspired in [1]
and [16]. Let M be a compact complex n- manifold. Let w be a closed 1-form on M and let /: M —^ M be a biholomorphism such that /*w = Aw for some A C C* with |A| ^ 1.
Define fl on M x C* by fl(x,t) = t.w(x). Then we have dfl = rj A ^ where rj(x,t) is defined by rf(x, t} = ^. We have drj = 0 and T] holomorphic, thus 0 defines a codimension one foliation F on M x C* which is transversely affine in the sense of Definition 1.1.
Now we consider the action <1>:Z x (M x C*) —> M x C*,n,(rc,t) i—> (^(x),^^).
This is a locally free action generated by the biholomorphism ( ^ : M x C * — ^ M x C * , y(x,t) == (f(x),\-n). We have ^* ^(x,t) = X-lt.\w(x) = ^(x,t) and ^rj = 7^.
Thus, the foliation T induces a codimension one foliation T on the quotient manifold V = (M x C*)/Z, this foliation inherits a transverse affine structure induced by the pair (^,77). For instance, we consider a variant of the Fumess example (see [1]): Consider the unimodular map U = ^ ^ j : C2 -^ C2; [/(a-, ?/) = (a; + ^ rr + 2?/). This map induces a biholomorphism /: M -> M, where M = C * / Z x C * / Z a n d where C* = C/Z has the coordinate obtained from the action Z x C —^ C(n,^) —^ ^ + n , and C*/Z is defined from the action Z x C* —^ C*, (n,t) -^ /^.t; where /^ G C*\5'1 is arbitrary.
The biholomorphism / is induced by F: C* x C* -^ C* x C*, F(z,w) = (zw.zw2).
We consider w = (1 + V5)dx - 2dy in C2. We have ?7*w = A.w where A = —— and (7 is Z x Z invariant (Z x Z acts on C x C by the natural product action) so that it induces a 1-form w in C* x C*, this last is also Z x Z invariant so that it induces a closed holomorphic 1-form w in the bitorus M. The 1-form w satisfies /*w = A.w. The foliation induced on V = (M x C*)/Z = ((C*/Z x C*/Z) x C*)/Z is transversely affine, has dense leaves and its leaves are biholomorphic t o C * / Z x C * o r C * x C * .
Example 1.6. - The Integration Lemma for closed rational 1-forms. Let T be a foliation on CP(n) which is given by a closed meromorphic 1-form, say, w. Then T has a transverse structure by translations in CP(n)\(w)oo where the polar divisor (w)oo is invariant and algebraic of codimension one. The Integration Lemma ([12]), states that if w = TT*W where TT: C^^O —^ CP(n) is the canonical projection then we have
w = Z^==i ^7^ + d[—^-r) for some Xj; e C, and some homogeneous polynomials /,, ^ in C^. We have Uy = order of (w)oo along the hypersurface {fj = 0) and (^)oo = U^i (A- = 0). so that (w)oo = 7r((w)oo) == U^=i 7r(/, = 0). As it is easy to see T may not be of a logarithmic or Bernoulli type. The reason is on the type of the singularities that may arise.
40 SERIE - TOME 30 - 1997 - N° 2
As a corollary of Example 1.1 we obtain:
PROPOSITION 1.2. - There is no transversely affine foliation on CP(n).
Proof. - In fact, CP(n) is simply-connected and, since it is compact, it admits no nonconstant holomorphic function. D
2. Resolution of singularities
Let F be a holomorphic singular codimension one foliation with isolated singularities on a compact two dimensional complex manifold M2. Let A C M be an analytic invariant curve. A theorem of Seidenberg [28] gives a resolution of the singular points of T on A.
THEOREM 2.1 [28]. - There is a finite sequence of blow-ups at the points of s{^F} such that their composition gives a proper holomorphic map TT: M —r M a complex compact 2-manifold M and a foliation .77* = TT*^" with isolated singularities such that:
i) 7^~l(s(y)) = U ==i -PJ ls a fi^6 connected union of complex projective lines with normal crossings and TT: M\ |j ^g Pj —> M\s(^F) is a biholomorhism (the union
D = TT-^A) = Tr-1^.?7)) U Tr^A^^)) = |j^=o p? is called the desingularizing divisor of s(^) D A), Po is the closure q/'7^-l(A\5(^)) on M);
ii) At any singularity p G U^o ^3 °f ^* tnere ls a local chart ( x ^ y ) such that x(p) = y(p) = Qandy is given by one of the P faff forms: (i)xdy—\ydx-\-h.o.t^ X ^ Q+
(non-degenerate linear part); (ii) ^p+l dy-^y(l-{-\xp)dx-^(h.o.t)dx, p > 1 (called saddle- node). In case (i) we say that p is resonant if A G Q-. Let p G s(J^), be a singular point of
^~, by the Separatrix Theorem [5] the foliation T admits at least one separatrix through p\
if the number of these separatrices is finite the singularity is called non-dicritical. This fact is equivalent to the fact that all the projective lines Pj belonging to Tr"1^) are tangent to JF*. The foliation .77* is called the resolution of the foliation T (for more information the reader should consult [4] or [24]). We remark that if a foliation T has only non-dicritical singularities in an invariant irreducible hypersurface A C M then it is well defined the analytic codimension one set sep(A) of the local separatrices of T through the points of A D •s-(^7), in a neighborhood of A in M.
DPJINITION 2.1. - A singularity 'p G ^'(^r) is said to be of first order when it is non-dicritical and there are no saddle-nodes in its resolution (see [6]).
We finish this section d e f i n i n g what we will consider as an extended affine structure.
DKIINITION 2.2. - Let T be given by ^2, and let A C M be an analytic invariant hypersurfaee, not containing dieritieal singularities of ^l. A 1-form 77 defined in a neighborhood of A is adapted to il ulon^ the hypersurface A if: (i) 77 is meromorphic, closed, dil = 77 A ^; (ii) the polar divisor (77)00 = A U sep (A) U (^)oo. has order one along A and (^)oo, and Res^ 77 = — (order of (^)^ along L) for each irreducible non-invariant component L of (^2)oo.
For example, if we consider ^2 == xdy — yk dx in affine coordinates in CP(2) then rj = k^ + ^ is an adapted form to ^2 along the algebraic leaf {y = 0} and also along the algebraic leaf {x = 0}. The same does not hold for the singular leaf Loo == CP(2)\C2, because Res^^ T] = —(k + 1) and (order of (0)oo along Loo) == k + 2.
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3. Extended affine structures
Our basic tools in the study of the holonomy of transversely affine foliations are the two following lemmas.
LEMMA 3.1. - Let T be given by 0, let A C M be an analytic non-singular invariant hypersurface and let rj be an adapted form to 0 along A.
(1) Suppose Res^rj = a ^ {2,3,...}. Then given a regular point p e A^JF) there exists a local chart (x,y) G U such that p = (0,0), A D U = [y = 0}, ^ = g d y and rj = a^ + ^ where g is meromorphic in U. Furthermore if (x,y) G U is another such system with U H U / (f) and connected, then we have y = c.y for some c G C*.
(2) Suppose R€SA r] = k G {2, 3,...} and suppose that we have 0 = g dy, rj = k^ + dg- for some local chart (A, y) C U with U H A = {y = 0} and g meromorphic in U. Then given a regular point p G A\5(J') there exists a local chart (x, y) 6 U such that p = (0,0), A H U = {y = 0}, n = g d y and r] = k^ + d9.
~ y 9 ^
Furthermore if {x, y) G U is another such chart with U H U ^ (f) and connected, then we have ^~1 == ^(^/fc-l) for some homography h{z} = ——.
Remark 3.1. - We will show as a consequence of Lemma 3.2 that condition (2) is always satisfied if s^) D A contains some linearizable non-resonant singularity.
Proof of Lemma 3.1. - We will assume that M is 2-dimensional. The general case is proved in the same way. First we consider the case (1) where RQS^T] = a ^ { 2 , 3 , . . . } and make the following claim:
CLAIM 1. - Given a holomorphic function r(y) defined in a neighborhood of y = 0 e C with r(0) = 1, there exists a local holomorphic non-vanishing function u = u(y) defined in a neighborhood of y = 0 e C, such that ——^ = r(y).
Proof of Claim 1. - To prove the claim we consider the distinct cases a = 1 and a i { 1 , 2 , 3 , . . . , } .
Case 1. - a = 1: We define ^(y) = -^ - 1. Since ^(0) = 0 we have
^ ( y ) / y holomorphic in y = 0. So it is enough to define u(y) = exp ( f i(I) dy\
which is holomorphic, non-vanishing and satisfies ^(y) = (—r - l)/2/, which gives
/ \ \y/
r ( y } = u^
W u(y}+y.u'(y}'u^y^y.u'^yy
Case 2. - a ^ {1,2,3,...}: In this case we solve the problem formally and then we conclude that the solution converges. First we rewrite —u— = r as ^uy^- = -J—
0 u-\-y.u' {uyV r.y" •
We can write ^y = 1 + a^y + a^y2 + • • • in a convergent series. Thus, we have
^ = ^ - + ^ T + " - + ^ r + - - and since a i { 1 , 2 , 3 , . . . } we have the formal solution ^-r = ^ + ^ + • . . which gives u^ = ^.,^,^... ; this formal solution is convergent in a neighborhood of 0 e C. In fact, since 1 + a^y + a| + • • • is convergent in some neighborhood of the origin we have that lim sup^^ v/fofcj < +00 and then lim sup^_^ ^/|^^r • Ok\ < oo so that the series 1 + ^j • a^y+ ^j • a^y2 + • • • is convergent in some neighborhood of 0 e C. This proves Claim 1. D
4" SfiRIE - TOME 30 - 1997 - N° 2
Now given a local coordinate system (re, y ) G U with A n U = {y = 0} and 0 = g dy we have 77 = a^ + ^ + dL for some holomorphic local function r = r{y) with r(0) = 1.
We define y := u(y).y where u is given by the Claim 1 above. Then we have r{y)dy = ua{y)dy. Now, define g := ^'r(j/), so we have g d y = g d y and since u0'.^ = g.r we have ^ + f = ^ + ^ and then we^have ^ + ^ = ^ + ^ + ^ = 77(3:, ^/) which proves the first part of case (1). Now we make another claim:
CLAIM 2. - Let u = u(y) be a holomorphic local function defined near y = 0 € C with 'u(O) 7^ 0. Assume that we have r.u" = u + 2/.1A' for some r, a E C. Then n is locally constant provided that a ^ { 2 , 3 , . . . } . If a == k e { 2 , 3 , . . . } then we have
^fc-l = ^ 1 fc_i for some a € C.
Proof of Claim 2. - We write z^) = n(0) + ^'(O)?/ + • • • + ^'^^^ + • • • in convergent power series. Assume that a ^ { 2 , 3 , . . . , } . Derivating the expression r.i^ = u + y.^—
and using r.^Aa-l(0) / 0 we obtain by induction that IA^(O) = 0, Vfc > 1 and then u is constant. Suppose now a = k e { 2 , 3 , . . . , }. From r.z^ = u + y.^j we obtain ^^ = ?- and then . 1 ! = ^-r 4- o; for some constant a e C. The claim now follows easily.
This is enough to finish the proof of Case 1: In fact, given {x,y) G U, {x,y) e U such that ^ = g d y , T) = ad2/y 9 y i + d9-, 0 = g d y , rj = a^- + ^, and E7n£/ / </> then writting y = u.y we obtain r.ZA" = n + ^ for some r e C*. Using Claim 2 we conclude that:
a i { 2 , 3 , . . . } ^ y = c.y for some c C C* a = fc G { 2 , 3 , . . . } ^ ^•-1 = y^^rr for some A, a C C. This finishes the proof of Case 1.
Case 2. - ResA?7 = k e { 2 , 3 , . . . } .
Let (a:, 2/) G i7 be a local coordinate system such that 0 == g dy, A = {y = 0} and then T) = kdy -^-dg -\- d^- for some holomorphic r = r{y) with r(0) = 1 and U H I/ 7^ ^.
In 77 n £7 ^ (^ we have ^d^ = ^d^/ and ^ + ^ = ^ + ^ + ^. This gives
^ = c . ^ . for some constant c € C*. Therefore Res{y=o} (yT^y) = 0 and this allows us to use the same proof given for Claim 1 to show that there exists a new coordinate system {x,y) C U such that rj(x,y) = k^- + dR and Q = g d y . Since A is connected this implies that the first part of (2) is true. The last part follows from what we have observed above. D
LEMMA 3.2 (Extension Lemma). - Let T be given by 0 on M2 and let A C M2 be an analytic smooth invariant curve. Suppose:
(1) Given any singularity p € A D s{7} there is a local coordinate system ( x ^ y ) such that p = (0,0), A = {y = 0} and T is given by xdy - \y dx = 0, A € C*\Q+.
(2) One of the singularities, say po € A H s(y), is non-resonant (which means that we have X i Q in (1)J
(3) There exists a 1-form T] defined in some neighborhood of A minus A and its local separatrices satisfying: (i) 77 is meromorphic and closed; (ii) dQ, = y^AO; (iii) (^)oo = (^)oo has order one and Resi, r] = - order of (0)oo along L, for each non-invariant component L of (»)oo.
Then T] extends meromorphically to a neighborhood of A as an adapted form to 0 along the curve A.
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Proof. - Using Hartogs' Extension Theorem (see [30]) we conclude that it is enough to prove that T] extends as a meromorphic 1-form to a neighborhood of an arbitrary singularity p G s{^} n A. First we consider the case p = po given in (2). Choosing local coordinates {x, y) such that pa = (0,0), A = {y = 0}, H(x, y) = g(xdy - \y dx), A ^ Q we can write rj{x, y) = Ai^ + \2^ + ^ + df with / e V*({^/ ^ 0}). From condition dn = 7^ A ^ we conclude that df A (rcch/ - A^/ dx) = 0 and then a;/., -^ \y fy = 0. Using Laurent Series
^ = E A? ^y we obtain (% + Aj) . f,j = 0, V(%J) e Z2 and since A ^ Q we obtain ijez
fij = 0, V(%j) / (0,0), so / is constant and rj = Ai^ + X^ + ^. (It is now easy to check that we also have 1 + A = AiA + A2. This fact will be used fater).
Therefore 77 extends meromorphically to a neighborhood of po having poles of order one. Now, this implies that rj extends meromorphically to all A.\{s(^) n A) having order one polar divisor (Hartogs' Extension Theorem).
Now we fix an arbitrary singularity p € A D 5(^~) p ^ p^ and choose (x, y) as in (1).
Again we have T] = Ai^ +\^ + ^ +df and xf^+Xyfy = 0 and then (i+\j) • /„ = 0, V(%j) e Z2. Since (77)00 has order one along A\(A 0 ^(.F)) we have / holomorphic along A and then f^ = 0, V ( z , j ) e Z x Z-. Now, fixed j e Z+ since A ^ Q+ we have /„ = 0, V z € Z _ . Thus we have fij = 0, V ( z , j ) ^ Z2_, so / is holomorphic and this proves the Lemma 3.2. D
We finish this section with a lemma that we will use to linearize some singularities in the proof of the main theorems. Consider ^ a germ of singular foliation on (C2,0) with non-degenerate linear part and in the Siegel domain', i.e., xdy-Xydx-^-h.o.t., A C C*\R_.
We can assume that {y = 0} is a separatrix of T_.
LEMMA 3.3. - Let ^ be as above and let p G (C^O^O. Suppose that there exists a local transversal section E, E n [y = 0} = {p} and coordinate system y G S, y(p) = 0, such that the holonomy of the local separatrix {y = 0} is given by
/^)fc-l = i^^i f^ some k G { 2 , 3 , . . .},/z, a e C, ^-1 / 1. Then E can be made linear in some system of coordinates Z = T(y) for some homography T.
Proof. - It is enough to show that h: (S,p) ^ can be made linear in some coordinate system z C S, z(p) = 0 [24]. Let H: Pi(C) -^ Pi(C) be the homography H(z) = /^.
Since /^-1 ^ 1 there exists an other homography T such that if Z = T(y) e (C°b) then I?(Z) = /^Z. By the hypothesis we have ///(//)A-1 = f f ( / /A - l) and therefore /.(Z^-i = ^-1 . Z^-i so that /i(Z) = //. . Z. D
4. Statement and proof of the main results
THEOREM 4.1. - Let T be a foliation on M2 having A C M as cm analytical connected invariant curve, and given by a meromorphic I-form il. Suppose:
(i) all singularities^ of T in A are of 1st-order:
(ii) the foliation T obtained as the resolution of the singularities of F in A, has one lineariz.able non-resonant singularity.
Then the following conditions are equivalent:
(a) F is transversely affine in some neighborhood of A minus A and its local separatrices.
(b) The form 0 admits an adapted form along A.
4"' ShRIH - TOME 30 - 1997 - N° 2
Moreover, if one of these conditions holds then the holonomy group of A and of any component Pj of the desingularization divisor D of s{^F) D A is either linearizable or is a finite covering of a group of homographies. In the Imeariz.able case there exists a closed meromorphic 1-form wj defined in a neighborhood Uj of Pj, with (wj)oo == Pj U sep (Pj), such that 7\^j, is given by Wj outside the polar divisor (w,)oo.
We remark that the hypothesis i) above comes from the difficult exhibited by our approach in dealing with the dicritical case (the complementar of the divisor D is not necessarily a Stein manifold so that Levi's Extension Theorem does not apply), and from the fact that we do not know wether an affine transverse structure defined in the complementar of the separatrices of a germ of saddle-node extends to these separatrices in the sense of section 3 above. With respect to hypothesis ii) above, it seems that actually it is possible to construct examples of germs of resonant non degenerate singularities, which admit affine transverse structures on the complementar of the two local separatrices, but do not exhibit a Liouvillian first integrals i.e., extended affine structures. The construction of these examples is based on the techniques of [25].
Proof. - The implication (b) => (a) is a straighforward consequence of Proposition 1.1.
Now we prove that (a) =^ (b). Let TT: M —^ M, T = Tr*^) be the resolution of the singular points of T in A and let D = TT'^A) = N . Pj, the desingulariz.ing divisor of s(^F) n A given by Theorem 2.1. Let q^ e P^ be a linearizable non-resonant singularity of T. The foliation T is transversely affine in ^\[(Uj Pj) u ^^(^(A))] for some neighborhood V of D in M. Therefore there exists a pair (0, rj) with 0 = TT*(O), rj = TT*(^), and rj meromorphic closed in ^\[(Uj Fj) u ^{^PW)} satisfying the conditions stated in Proposition 1.1. Our objective is to show that rj extends meromorphically to V. Using Lemma 3.2 we show that rj extends meromorphically to P^ minus the other singular points of J^ in Pj^. But this extension already allows us to calculate the holonomy of the leaf P^
of T. According to Lemma 3.1 this holonomy is either linearizable or is a finite covering of a group of homographies and in particular given any singular point q'- G Pj^ H s{J^) there is a local coordinate y e S in any local transversal E, such that the holonomy of the separatrix P^ in this singularity is of one of the following forms: (A) h{y) = a.y, a € C*; (B) h^yY = j^r a € C*, & € C (in this case we have Resp^rj = k + 1).
In case (B) we have two possibilities:
(1) If a^ ^- 1: In this case the homography [z \-> f—) can be made linear in some local coordinate y G S which is obtained by an homography from y and therefore we can assume that h(y) = p^y as in (A) (see Lemma 3.3). Therefore in this case and in case (A) the singularity is linearizable and we use Lemma 3.2 to extend rj to the singularity q1- .
(2) ak = 1: In this case we have that q^ is a singularity of the form u\ = g{xdy - Xydx + h.o.t.), X = -^ e Q-, (m,n) = 1, and we assume that it is nonlinearizable with (y = 0) C P^. The local holonomy h of q, in P^ satisfies
^{vY = i^ f c . This implies that mk = In for some I C N. Furthermore we can assume that a = 7^ and so this holonomy is conjugated to the holonomy of (y = 0) of the germ of foliation ujk,i = I x dy+ k y(l + ^xkyl)dx = 0. Thus by [25] the foliation T is conjugated near q^ to the germ of foliation ujk.i- Thus there are local coordinates (x^^ya) centered at
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q^ € P^, such that for some meromorphic function (^ ^|rr = 9a^k,i (we observe that
^i = gk^i dyk,i where y^i = ^^ol^-i and g^ = -^^^r^2). We define
%o,a := (k + 1)^- + (< + 1)^- + d^. The form 77 extends to q^ as y^a because both define affine transverse structures outside the axis (x = 0),(^/ = 0) and have the same residue (fc + 1) along the axis {y = 0).
Thus we have showed that the form rj extends meromorphically to all P^ and given any projective Pj with Pj H P^ ^ cj), then rj extends meromorphically to Pj minus the other singularities of T in Pj. Using these arguments we can show that rj extends meromorphically to all the divisor D = |j Pj and the local separatrices 7^-l(sep(A)).
This shows that (a) =^ (b). The last part of Theorem 4.1 is a consequence of the following remarks:
(I): Let (a;, y) G U be a local chart around any linearizable singularity of T in Pj, say, a singular point p E Pj; H 5(^), such that we have 0(a;, ?/) = g(xdy — \y dx), X e C\Q4_.
Let (^, 0) C ?A{p} be any regular point of T and choose V a small neighborhood of (g, 0) such that V r\ {x = 0} = (f) and V n Pj is simply-connected. Let x = x - q, y = yx~x, g = gxl+x. Then ( x ^ y ) define new coordinates in V such that Pj nV = {y = 0} and 0 = g dy. Now, since the singularity is linearizable it follows (as remarked in the proof of Lemma 3.2) that we have rj{x, y) = a^ + b^ + ^, where l + A = a - A + 6 . Therefore w e h a v e ^ ^ ) = a ^ + ( l + A ) ^ - a A ^ + ^ = ad^ + ^ ^ = a f + f i n y . (II): For the linearizable case we assume that a ^ { 2 , 3 , . . . } . Notice that since the holonomy is linearizable all the singularities are linearizable ([24] and [25]). Using now Lemma 3.1 we can conclude that there exists a family of local charts (xa^Va) ^ Ua with the (7c/s covering a neighborhood of Pj, such that: (1) Pj D Ua = {y^ == 0}, Va;
(2) If y\ is regular it is given by dya = 0, and if T\ is singular it is given by Xc. dy^ - A, ^ dx^ = 0, A, G C\Q; (3) If U^HU^ ^ and (t/, U U^) H s(^) = (f>
then we have y^ = c^.% for some Ca/3 G C* and if Ua H s(^) / ^ then [/^ D s(^') = (^
and we have yaX^^ = Ca^-y^ for some Ca/3 G C*.
(Ill) Finally, we proceed as in [13] and [6]. We define local meromorphic forms w^ in the U^s by: w^x^y^} := d^- if T ^ is regular; w^x^y^} := ^ - \^ if^|^
is singular. Using condition (3) above we have Wa = w^ in each Ua n U^ -^ ^ and thul we have defined a closed meromorphic 1-form wj, (which defines J^) in a neighborhood of Pj, having order one polar divisor (wj)oo = Pj U sep(Pj). D
Remark 4.1. Generalised Levi's Extension Theorem.
Let M be a compact complex manifold (of dimesion > 2), and let A C M be an analytic subset of codimension one, such that M\A is a Stein manifold. Then any meromorphic differential q-form defined in a neighborhood of A extends meromorphically to M.
This is a consequence of Levi's extension Theorem [30] (see [6] Lemma 5 Section 3).
In particular if A C CP(n) is an algebraic hypersurface then CP(n)\A is a Stein manifold [30] and any meromorphic differential q-form uj defined in a neighborhood of A extends meromorphically to GP(n).
Remark 4.2. - Let T be a foliation on CP(2). The foliation T has degree n if and only if in affine coordinates (re, y) G C2 ^ CP(2), T is given by 0 = P dy — Q dx = 0
4e SfiRIE - TOME 30 - 1997 - N° 2
where P = Y^^Pj + x.g, Q = Y^^Qj + y.g where Pp Qj are homogeneous polynomials of degree j and g is an homogeneous polynomial of degree n+1. Geometrically the degree of T is the number of tangencies of its leaves with a generic projective line CP(1) C CP(2) (see [22]). If ^ is like above and the line L^ = CP(2)\C2 is not invariant then (0)oo = ^oo has order = deg.7" + 2. The Poincare Problem for foliations on CP(2) is to bound the degree of a projective foliation T in terms of the degree of an algebraic solution S C CP(2) of T (see [22] and [9]). In the non-dicritical case it is proved that deg.?7 < deg 5 + 2 [9].
The next theorem proves that we have an equality in the "Poincare Problem" for a foliation under our assumptions. We refer to [9], [13] and [22] for any further information on this subject.
THEOREM 4.2. - Let T be a foliation on CP(2) and let A be a smooth algebraic curve invariant by T. Suppose: (i) all singularities ofF in A are of first order; (ii) the foliation T obtained by the resolution of s(J^) DA has at least one linearizable non-resonant singularity;
(iii) T is transversely a/fine in some neighborhood of A minus A and its local separatrices.
Then: (a) T has a finite number of algebraic leaves; let Sep (^F) denote this set: (b) deg T + 2 = degree of Sep (JF).
Proof of Theorem 4.2. - The proof is based on the Index Theorem [5] and the Residue Theorem. Let TT: M —> CP(2), T = Tr*^), be the desingularization of the singularities of T in A. Let 7^~l(s(y)) = N Pj = D denote the divisor D of the desingularization and let A denote the curve TT'^A^A ft 5(^))). It follows from Theorem 4.1 that T can be given by a meromorphic 1-form ft, which admits an adapted form along A, say ^, defined in all CP(2). We have (^)oo = Sep(^) which proves (a).
By the Integration Lemma (see Example 1.6) we have r] = ^ Aj^- — n.^ in some affine chart ( x ^ y ) G C2 ^-» CP(2), where fj and g are polynomials transverse to CP(2)\C2, Sep(^) = |J, (/, = 0), (0)oo = (g = 0), g has degree one, n - order of (0)oo = deg J^.The Residue Theorem shows that (1) ^. Xj deg ^ = n == deg ^+2.
Let Ai = A = (/i - 0) and^A, = (/, = 0 ) , V j ^ 1. and A, = TT-^A.VA, H 5(^))).
Now we fix a singularity p G Aj; DD, say, p G A^ DP^. We know that the residue Xj and the index ind (p; P^) are related by the formula: Aj = 1 + (1 - a^).ind(p; P^) where a^ = Resp^(7r*^). (In fact as in the first part of the proof of Lemma 3.2 we have that 1 + ind(p; P^) = ind(p; P^).a^ 4- Res.rj). Hence we have ind(p; P^) = —^r^ for each singularity p G P^ H A^. Let w(Pi,) denote the weight of the projective P^ in the desingularization process, that is, the number of times that we have blowed-up points over Py plus one; we have —w(P^) := first Chern class of Py in M. Using Camacho-Sad Index Theorem [5] we obtain -w(P^) = ^ ^-nnp i^^P^)- Now we have
E = E + E + E,-
pGs((^)nP^ p6s(;r)np^ p€s(.p-)np^ pePi/ns(:r) pep.nlj^, ^p^- ,^
j^i
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Therefore we obtain
-w(P,)=^-#(P,nA,).^—
1^i
av~
1-^P.nA)^^- ^ #(P,nP.).^-
1av 1 P^np^^> ^ - 1
A^
and then
(2) w(P,).(a, - 1) = ^ #(P, H A,)(A, - 1) j¥i
+ #(P, n A).(Ai - l) + ^ #(P. n P^).(a^ - l).
^7^^
Now we sum over all P^ obtaining
^ w(P,).(a, - 1) = ^(^ #(P, n A,))(A, - 1)
v j^l v
+ ( A i - l ) . ^ # ( P , n A ) + ^ #(P,nP^).(a^-l).
^ ^,,vp,^v
We observe that:
(a) ^ #(P, n Ai))(Ai - 1) = #(s(.F) n A)(Ai - 1)
( b ) E(E^ P ' /nA ^)(^•- l )
j¥i '/
= ^(A, - l).degA,.deg/i - deg A. [^ A,. deg/, - ^;deg/,]
i^i .ȴ! j^i
= deg/i.[deg^-+ 2 - Ai.degA - ^deg/j].
j^
(c) ^ #(P, n P^).(a^ - 1)
^,i/
^t^i/
= ^ #(P, n P^).(a, - l)
^^^
^,1^= ^ ( a , - l ) # ( P , n P ^ ) + ^ (a,-l).#(P,nP^).
^'T^A1 v^^
Pi^nA^0 Pi/nA==<?1)
= ^ w(P,).(a, - 1) + ^ (w(P,) - l).(a, - 1) P,,nA=^ p^nA^^
= ^ w ( P , ) . ( a , - l ) - ^ ( a , - l ) .
v P^nA^^
Now, using (1), (2), (a), (b) and (c) we obtain (*) 0 = deg/i.[degJ^+ 2 - Ai.deg/i - E^ideg/,] + (Ai - 1Y#W) H A ) - Ep,nA^(^ - 1)' Now W^g t^ Index Theorem for the curve A we obtain: (deg/i)2 - ^OfT) n A) = ^ ^ ind(p^,A), where P^ H A = {p^} and ind(p^A) = -^. Thus we have Ep^A^^ - 1) = 46 SERIE - TOME 30 - 1997 - N° 2
(AI - l).[(deg/i)2 - #(^(^7) n A)]. Using this last equation and (*) we obtain 0 = deg/i.[deg^+2-E,>ideg/,] and then deg^+ 2 = E,->i deg/, = degSep(^). D In the following theorem we make hypothesis on all the singularities of T lying over algebraic leaves.
THEOREM 4.3. - Let y, A be as in Theorem 4.2. Suppose: ( i ) all singularities ofT lying over algebraic leaves of F, are non-degenerate of the form xdy — Xydx + h.o.t. = 0, A G C\Q+; (ii) at least one of the singularities ofT in A is linearizable non-resonant;
(in) T is transversely affine in some neighborhood of A minus A and its local separatrices.
Then T is a logarithmic foliation and deg.77 + 2 = degSep (^r).
Proof. - As in the proof of Theorem 4.2, given any affine chart { x ^ y ) G C2 <-^
CP(2) such that the line CP(2)\C2 is not invariant and given a polynomial 1-form 0 = P dy — Q dx which defines T in C2, we can find a meromorphic 1-form rj defined in a neighborhood of A in CP(2) and adapted to 0 along this curve. Since CP(2)\A is a Stein manifold, rj extends meromorphically to all CP(2) (Remark 4.1). As in the proof of Theorem 4.2 we have rj = E^f^ where ^P^) n c2 = U(/j = °) and
^ Xj ' deg(fj) = degJ^ + 2 as a consequence of the Residue Theorem. Now according to Theorem 4.2 we have Sdeg/j = degJ^ + 2 and then E/^ - 1) • ^Sfj = 0 and this shows that A^ ^ { 2 , 3 , . . . } for some jo. Using now Theorem 4.1 we conclude that the algebraic leaf A^ = (/^ = 0 ) of T has a linearizable holonomy in the same way that in the proof of Theorem 4.1.
Therefore, (since the singularities of T on A are already reduced) according to Theorem 4.1 and to Remark 4.1, T is defined in CP(2) by a closed meromorphic 1-form w having order one polar divisor (w)oo == Sep^). By the Integration Lemma w is a logarithmic 1-form. D
Remark 4.3. - We remark that Theorem 4.3 still holds (and with the same proof) if we replace condition i) by: (i')all the singularities ofT lying on some algebraic leafofT are of first order and exhibit local meromorphic integrating factors (that is, the foliation is given by a closed meromorphic local 1-form in a neighborhood of a singularity): In fact using the abelian holonomy of a leaf A^\5(.7~) as in the proof above we can glue the local closed meromorphic 1-forms given by the local integrating factors around the singularities, in order to obtain a closed meromorphic 1-form uj which describes the foliation T in a neighborhood of the algebraic curve A^ (see [13] or [6] for a similar procedure). Thus we obtain:
THEOREM 4.3'. - Let T, A be as in Theorem 4.2. Suppose: ( i ) all singularities ofT lying over algebraic leaves of T, are of first order and admit local meromorphic integrating factors; (ii) at least one of the singularities of^F in A is linearizable non-resonant; ( H i ) 7 is transversely affine in some neighborhood of A minus A and its local separatrices.
Then T is given by a closed rational 1-form uj on CP(2) and deg T -h 2 == deg Sep (^).
Finally, we remark that in the next results we do not require that T exhibits a linearizable singularity in its desingularization. However we suppose that T is transversely affine in all CP(n) minus the algebraic invariant set S of codimension one.
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THEOREM 4.4. - Let T be a codimension one foliation on CP(n) which is transversely affine outside an algebraic codimension one invariant set S C CP(n). Suppose that T has only 1st-order singularities in some component So of S. Then deg.77 + 2 = deg S.
THEOREM 4.5. - Let y, S be as in Theorem 4.4 above. Suppose that T has only non- degenerate singularities in S. Then T is given by a closed rational 1-form on CP(2) and deg.77 + 2 == deg6'. The foliation T is a logarithmic foliation on CP(n) provided that it exhibits only non-resonant singularities on S.
We would like to call the reader's attention to the fact that both Theorems 4.4 and 4.5 above are stated for codimension one foliations on CP(n). We recall that according to [6] a codimension one foliation T on CP(n) is said to have only non-dicritical singularites in some algebraic codimension one invariant set S C CP(n) if there exists a linearly embedded E = CP(2) ^ CP(n) in general position with respect to T (see [6] for a definition), such that the induced foliation ^* = ^cpcz} ^ the ^l1151011
%:E -> CP(n)) has codimension > 2 singular set in CP(2) and has only non-dicritical singularities in 5* = ^{S) C CP(2). Proceeding the same way we say that T has only 1st (non-resonant) singularities in S if.77* has only 1st (non-resonant) singularities in 5*.
Proof of Theorem 4.4. - We can assume that n = 2: In fact if T is a codimension one foliation on CP(n) then given a generic linearly embedded CP(2) ^ CP(n) the induced foliation T* = f\^p^ has the same degree that 7. Moreover the singular set of ^*
consists of the intersection s(^) H CP(2) and of the tangencies of T with CP(2). The tangencies of T with CP(2) originate singularities wich have a local holomorphic first integral (in fact if p e CP(n)\s{^) then T has a local holomorphic first integral at p) and thus these are non-dicritical singularities. This shows that .F* has only non-dicritical singularities in S D CP(2). Thus we assume n = 2. Let 0 = P dy - Q dx be a polynomial 1-form which defines T in affine coordinates (x,y) e C2 as in the proof of Theorem 4.2, with S transverse to the line CP(2)\C2. Write S H C2 = \J^ {fj = 0) fj irreducible polynomial relatively prime with /, for i -^ j. Since F is transversely affine in CP(2)\S' we have a 1-form rj defined in CP(2)\S', closed and meromorphic with polar divisor (^)oo = (^)oo = (CP(2)\C2) and satisfying the conditions stated in Proposition 1.1. By the Integration Lemma we have rj = ^. Xj^ + ^ for some holomorphic F: C2^ -> C*.
By the Residue Theorem we have (*) ^. \j deg fj = deg^-h 2. Now we remark that the arguments used in the proof of Theorem 4.2 can be repeated in this case using equation (*) above even in the non-linearizable case (notice that we suppose the singularities to be of l^-order). Thus, we leave the rest of the proof to the reader. D
Proof of Theorem 4.5. - According to [6] if a codimension one foliation T on CP(n) is such that ^\^p^ is (given by a closed rational 1-form) a logarithmic foliation for some linearly embedded CP(2) <—^ CP(n), in general position with respect to T, then T is (given by a closed rational 1-form) a logarithmic foliation on CP(n). Therefore we will assume, as in the proof of Theorem 4.4, that n = 2. Let 0 = P dy - Qdx,
fn=Y,\jc^•-\-dj^be^sm the proof of Theorem 4.3 above. Since ^ Xj deg fj = deg JF+ 2 and ^degfj = degJ^ + 2 we have J^(Xj - l)deg^ = 0 and then there exists A,, i {2,3,...}. Now we put ^ = F.fl and rf = SA^A = rf - df. Then, according to
4e SfiRIE - TOME 30 - 1997 - N° 2
Proposition 1.1, the pair (^/,?/) defines the same affine structure for T in CP(2)\5' and in this case 77' is meromorphic in CP(2).
CLAIM. - For each regular point p € Aj^\s{y) there exists a local chart (re, y) G U such that p = (0,0), A^ n U = [y = 0}, ^ = F.^A/ and T/ = A^ • ^ + ^. Furthermore if ( x , y ) e U is another such chart with x{p) == ^(p) = 0, U H (7 7^ (f) then we have
?/ = c.^/ for some c € C*.
This claim is proved as Lemma 3.1 (1) because Ajp ^ {2,3,...}. Using the claim we prove that the holonomy of the algebraic leaf (/^ == 0) = A^ is linearizable in the sense of Theorem 4.1. Proceeding as in Theorem 4.3 we prove that T is a logarithmic foliation. D
The same way we prove Theorem 4.3' we can prove:
THEOREM 4.5'. - Let T, S be as in Theorem 4.4. Suppose that all singularities ofT lying over S are of first order and admit local meromorphic integrating factors. Then T is given by a closed rational 1-form uj on CP(2) and deg.77 + 2 = degSep (.F).
5. Solvable holonomy groups and transversely affine foliations
A subgroup G C Bih(C.O) is solvable if the group of commutators [G^G] is an abelian group. In particular any abelian subgroup G C Bih(C,0) is a solvable group. A less trivial example of solvable groups is given by the subgroups G C Vf-k where H^ = {g G Bih(C.O)/^) = ——==;A,a e C}, k e N. A theorem of
' yl+a-s^
Cerveau-Moussu ([14]) states that except for some exceptional cases these are the only non-commutative solvable groups. Let T be a foliation on M2 and let A C M2 be an analytical invariant curve. Under generic hypothesis on s{^) D A, T is transversely affine in some neighborhood of A minus A and its local separatrices if and only if the holonomy of A is a solvable group in a strong way which we define below:
DEFINITION 5.1. - Assume that ^(JF) n A is non-dicritical. We say that the holonomy of A has the property (<S) if:
(i) the holonomy group Gi of each component Pi of the divisor obtained in the desingularization T of T on A; is either an abelian analytically normalizable group (that is, the group embedds in the flow of a holomorphic vector field on (C, 0)), or a solvable normalizable group Gi ^-> H^ as above.
(ii) We have the following compatibility condition: Given any comer {q} = P^DPj, such that F has a holomorphic first integral in a neighborhood of g, say rc9^ with Pi = {x = 0) and Pj = (y = 0); then, if the holonomy group Gj of Pj is nonabelian Gj C H^, we have p\{kjq) in N. In the case both groups are nonabelian, if we take normalizing coordinates z and w such that the holonomy groups of Gi, and G. are of the form z i—^ , xz and
' ° J Vi+a^z
w \—> Aw respectively then (via the Dulac correspondence which is defined by the Vl+awfc•7
local first integral) we have z^ = aw ^ for some homography x i—^ T^jx '
PROPOSITION 5.1. - Let y, M and A be as in Theorem 4.1. Assume that each component Dj of the desingularUing divisor D of s^} H A exhibits some non-resonant linearizable singularity. Then the following conditions are equivalent:
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(i) T is transversely affine in some neighborhood of A minus A and its local separatrices.
(ii) The holonomy of A has the property (S).
In particular if M\A is a Stein manifold with M compact then any local separatrix of T through some singular point in s(T^ D A is the germ of a global analytic separatrix of 7 in M, provided that (i) or (ii) holds.
The proof of the Proposition 5.1 is based on the refered characterization of Cerveau- Moussu (see [14]) and in the following lemma whose proof is a straighforward calculation left to the reader.
LEMMA 5.1. - Let G C Bih(C^O) be a subgroup such that:
(i) There exists a holomorphic coordinate y 6 (C,0), y(0) = 0 such that each element g C G is of the form g(y) = -^== ; a,g G C,\g G C*, where k G { 1 , 2 , . . . } is independent of g; (ii) G contains a non-periodic linearizable element, say, go € G, ^o(^) =
\o ' z + h.o.t.^ \^ 7^ l,Vn G N*. Then there exists a holomorphic coordinate z G (C,0), z(ff) = 0, such that go(^) = Ao.^, and each g G G is of the form g(z) = ^ , g'^ —; indeed
•\yi-\-bg .z this holds for any holomorphic coordinate z which linearises go.
Proof of Proposition 5.1. - According to (the proof of) Lemma 3.1, (i) =^ (ii), except for the compatibility condition (ii). This condition is easily proved using the local expression 0 = g(pxdy + qydx)^rj = a^ + b^ + d9-, in suitable coordinates around the comer q, which admits a local holomorphic first integral xqyp (see the proof of Lemma 3.1). Now we proceed to prove that (ii) => (i): Let Gi denote the holonomy group of a component Pi of the divisor D.
1st case. - Gi is a commutative group. In this case since G{ contains a non-periodic linearizable element, Gi is linearizable in some coordinate system and therefore T is given by a closed meromorphic 1-form Wi defined in a neighborhood of Pi in M, with (wi)oo = Pi U sep (Pi) (see the last part of the proof of Theorem 4.1).
^na case. - G is a solvable non-commutative group. In this case since Gi contains a linearizable non-periodic element Gi is holomorphically conjugated to a subgroup of H^;
for some unique ki G { 1 , 2 , . . . } [14].
CLAIM 1. - There exists a collection of charts (xa^ya) G Ua, 01. G A, such that:
(i) U^^ Ua = V\sep (Pi), V = some neighborhood of Pi in M; (ii) U^ nP, = [y^ = 0}
and Ua n s(!F) = </>, Va G A; (iii) ^|^ is given by eh/a = 0; (iv) If Ua H (7^ / (/) then
^ = /w(2/JO for some homography /^^^ G Hi.
Proof of Claim 1. - The claim is proved using the embedding Gi (—^ Hfc,, Lemma 5.1 and a procedure similar to that used in [6]. D
Now, for each a G A there exists a holomorphic function g^ G V((7a) such that n(xa,ya) = Qa dy^ in U^. We therefore define the local model ^a(^a^a) = ( ^ + 1 ) ^ + ^ in [/,.
CLAIM 2. - In each Ua n U/s ^ (^ we have 770, = y^.
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Proof of Claim 1. - In fact in Ua <~\ U/s we have 0 = ga dya = g/s dy^ and
yka =
i^^'
so that^ = A • 7^
and that^^'
+1 =^•'rf^
and thus(^ + 1)^+^ = (^+1)^ + <^'3. D
It follows from the claim that there exists a meromorphic 1-form ^ in V\sep (Pi) with, (^)oo = (-P? U (^)oo) n (y\sep(P^)) wich defines a transverse affine structure to T in V\(Pi U sep (Pz)). This form rji extends to the singularities s{J^) Fl Pi as in Lemma 3.1 and part (2) of case (B) in the proof of Theorem 4.1. Now, using condition (iii) in Definition 5.1 we can glue rji to the analogous ones constructed in a neighborhood of the J9^s and obtain rj in a neighborhood of D in M. This form blows down and extends (by Hartogs5 Theorem) to a closed meromorphic 1-form rj in a neighborhood of A = 7r(D) as required to define an affine transverse structure to T in this neighborhood minus A U sep(A) as stated. D We recall that according to [15] a germ w == A dx + B dy has its z^-jet Wy called generic if w^(x^y) = a^{x^y)dx + b^(x^y)dy where dy, b^ are homogeneous polynomials of degree v having Py^{x^y) = xa^{x^y) + yby{x^y} as the tangent cone, satisfying:
(i) Wy is non-dicritic, i.e., P^+i ^ 0
(ii) the residues Xj; = ^ J j^— where the 7/s are generators ofI^C2-^?^! = 0)), are non-real and Py^{x,y} = c. II^i1 (v ~ ^a;). t i ^ t j \ / i ^ j , c ^ C so that w^ = c. n^"i (^ — ^a;) ' S^i ^^Z^^ • I11 particular a; is desingularized with one blow-up.
One basic tool here is the following consideration: Let a, /? be germs of holomorphic 1-forms in (C^O). The 1-form f3 is a stable deformation of a if there exists a family t i-^ Of, continuous in ^ G [0,1] such that 0,0 = a, ai = /3 and {o^} is topologically trivial in the sense that there exists a continuous family of germs of homeomorphisms { / ^ ( C ^ O ) —^ (C^O)} such that ho = Id and /^ is a topological equivalence between 0.1 and Oo, for all t. According to [15] any stable deformation of a germ of holomorphic 1-form w = Adx + B dy in (C^O) having ^-jet, w^ generic ^ > 2, has projective holonomy topologically conjugated to the projective holonomy of w.
The main result of this section is the following proposition:
PROPOSITION 5.2. - Let w = A dx + B dy be a germ of holomorphic 1-form in the origin ofC2 having Wy generic as y-jet, v > 2 and let w' be a stable deformation of w. Suppose that w has a multiform integrating factor of the form f = n^i f j3' fj e ^ ^j ^ C*.
Then w' has a multiform integrating factor of the same type.
The proposition follows from what we have remarked above, from Proposition 5.1 and from the two following remarks:
(a) Let G and G" be subgroups of Bih(C,0) topologically conjugated. Then G is solvable if and only if G' is solvable.
(b) Let w == A dx + B dy where w is as in Proposition 5.1. Then w has an integrating factor of the form / = Hj f^\ fj G V2, \j G C* if and only if the projective holonomy T~iw of w is a solvable group.
We supply a proof for (b): Assume that u has such an integrating factor /. Then T] = ^ = ^. dj- is an adapted form to uj along the separatrices set |j .{fj =0}. Therefore it follows that the holonomy of the projective P1 arising in the desingularization of uj is
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