Abstracts
We study the classification of germs of differential equations in the complex plane giving a complete set of analytic invariants determining the analytic type of the underlying foliation. In particular we answer in affirmative a conjecture of S. Voronin, and generalize some previous results about dicritical singularities in a straightforward manner. Such problem has its origins in a conjecture proposed by R. Thom in the mid-1970s.
Singular foliations; resolution of singularities; holonomy; nonabelian cohomology
Estudamos a classificação de equações diferenciais analíticas em plano complexo fornecendo uma lista completa de invariantes analíticos que determinam o tipo analítico da folheação subjacente. Em particular respondemos afirmativamente a uma conjectura de S. Voronin e generalizamos de forma imediata alguns resultados preliminares a respeito de singularidades dicríticas. Tal problema tem suas orígens numa conjectura proposta por R. Thom em meados da década de 1970.
Folheações singulares; resolução de singularidades; holonomia; cohomologia não-abeliana
MATHEMATICAL SCIENCES
Invariants of germs of analytic differential equations in the complex plane
Leonardo M. Câmara
Departamento de Matemática, CCE-UFES, Av. Fernando Ferrari s/nº, Campus de Goiabeiras, 29060-900 Vitória, ES, Brasil
ABSTRACT
We study the classification of germs of differential equations in the complex plane giving a complete set of analytic invariants determining the analytic type of the underlying foliation. In particular we answer in affirmative a conjecture of S. Voronin, and generalize some previous results about dicritical singularities in a straightforward manner. Such problem has its origins in a conjecture proposed by R. Thom in the mid-1970s.
Key words: Singular foliations, resolution of singularities, holonomy, nonabelian cohomology.
RESUMO
Estudamos a classificação de equações diferenciais analíticas em plano complexo fornecendo uma lista completa de invariantes analíticos que determinam o tipo analítico da folheação subjacente. Em particular respondemos afirmativamente a uma conjectura de S. Voronin e generalizamos de forma imediata alguns resultados preliminares a respeito de singularidades dicríticas. Tal problema tem suas orígens numa conjectura proposta por R. Thom em meados da década de 1970.
Palavras-chave: Folheações singulares, resolução de singularidades, holonomia, cohomologia não-abeliana.
1 INTRODUCTION
We study the classification of differential equations of the form Adx + bdy in two variables, by their underlying foliations. As it is well known, any analytic differential equation in (2, 0) has a nonvoid set of integral curves through the origin, called separatrix set, and denoted by Sep(
). In Câmara 2001 we use the blow-up method in order to identify a complete set of invariants describing the analytic type of a generic subset of the above differential equations: the generalized curves. By definition a generalized curve has only a finite number of separatrices and no saddle-nodes along its minimal resolution (Seidenberg 1968). In fact, we introduce a new kind of resolution of foliations (called rectifier resolutions) taking care not only of the topological behavior of the singularities along the ''solved'' foliation but paying attention to the disposition of the strict transform of Sep(
), say Sep(
). In fact, if Dj is an irreducible component of the exceptional divisor of
and Sep(
j) is the restriction of Sep(
) to a neighborhood of Dj, then Sep(
j) will be a subset of fibers of a fibration transversal to Dj. Thence we determine the analytic type of each Hopf component (see definition below), say
j, of the solved foliation by its set of singular points (up to homographic conjugacy class), the first jets of their singularities and the conjugacy class of the holonomy group in Diff(
, 0). Indeed, we use classical topological methods of path lifting due to Ehersmann and analytic extension technics for nondegenerate reduced singularities (cf. Mattei and Moussu 1980, Martinet and Ramis 1983). Finally, the remaining invariants appear as characteristic classes which arise naturally from this geometric construction.
In order to extend the previous construction for any foliation, we have the following hindrances to overcome:
First consider the case of a nondicritical foliation with a Hopf component
j defined in the Hopf bundle![](https://minio.scielo.br/documentstore/1678-2690/VK3BJ8zt9DS7MLDgKwgZQLC/0847e80e887e3911094b14a4cc3cfc52eee0596f.gif)
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Now, suppose that
j has just nondegenerate singularities and at least one saddle-node with central variety (see definition below) siting on Dj. In this case the conjugacy class of the projective holonomy does not determine the analytic type of the saddle-node in general (Martinet and Ramis 1982). But exploring the ''symmetries'' of saddle-nodes with central variety, we can define an equivalence relation (with analytic moduli space) such that the new class of the projective holonomy determines the analytic type of![](https://minio.scielo.br/documentstore/1678-2690/VK3BJ8zt9DS7MLDgKwgZQLC/c4be35727e60a56c29bb45bc239cd47d0ecb4d41.gif)
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On the other hand, we recall that until now, the dicritical singularities i.e., those with an infinite number of separatrices are not well understood (cf. Cerveau 1999). In fact, their extended resolutions (Camacho et al. 1984) are composed by nondicritical and dicritical Hopf components (see definition below) satisfying:
(1) The nondicritical components are solved, that is, they have just reduced singularities;
(2) The dicritical components are pair-wise nonintersecting, and have no singular or tangential points in the zero section of its Hopf bundle.
We apply the same reasonings in order to obtain an extended rectifier resolution. First remark that the zero section of a Hopf bundle corresponding to a dicritical component is transversal to the foliation. It follows that the only invariants provided by it are the corners and the Chern class of its Hopf bundle (we verify this by path lifting). In particular, it follows that any two analytically equivalent dicritical foliations, namely i, i = 1, 2, with dicritical components
,0 intercepting just the nondicritical Hopf components, say
,k, k = 1 ¼ ni, must have the same global conjugacy in PGL(2,
) which does not only carry corners of
,0 into corners of
,0 but simultaneous conjugate all the projective holonomies of
,k evaluated over the transversal section
,0 the zero section of the Hopf bundle determined by the dicritical component
, 0 (see figure 2). So, we cannot choose the projective holonomies of the nondicritical Hopf components freely, in the extended rectifier resolution of a germ of dicritical foliation. In other words, the cellular objects of the resolution of a dicritical singularity are no longer just the Hopf components, but the following ones:
(1) The union of a dicritical component together with the nondicritical components intercepting it, say
;
(2) The nondicritical Hopf components of the foliation nonintersecting any dicritical foliation.
In fact the analytic types of these objects are determined by the projective holonomies of the nondicritical Hopf components of the foliation in the extended rectifier resolution and by the tree of corners and singularities of each cell. It follows that this objects are suitable for the definition of a good covering for the first cohomology set of the isotropy group of a foliation model as in Câmara 2001. This work was presented in the International Conference Geometry and Foliations 2003 held at Ryukoku University, Kyoto, Japan.
2 BASIC DEFINITIONS AND NOTATIONS
A germ of singular foliation, say (: w = 0), in (
2, 0) of codimension 1 is, roughly, the set of integral curves of a given germ of 1-form w Î W1(
2, 0) which may be assumed to have just an isolated singularity at the origin. Let Diff(
k, 0) (respect.
(
k, 0)) be the group of germs of analytic (respect. formal) diffeomorphisms of (
k, 0) fixing the origin and DiffI(
k, 0) Ì Diff(
k, 0) (respect.
I(
k, 0) Ì
(
k, 0)) be the subset of such diffeomorphisms, tangent to the identity. We say that two germs of foliations in (
2, 0), namely (
j : wj = 0), j = 1, 2, are analytically conjugate if there is F Î Diff(
2, 0), such that F sends leaves of
1 into leaves of
2. We say that h1, h2Î Diff(
, 0) are analytically conjugate if there is f Î Diff(
, 0) such that Ad(f-1)(h2) := f1 ° h2 ° f = h1. We denote by Iso(
) the isotropy group of the germ of foliation (
: w = 0) in (
2, 0), defined by
Further, let us denote the Hopf bundle of Chern class k by (k) : (p(k) :
(k) ®
(1)), or just by its total space
(k). Recall, from the theory of Algebraic Curves, that whenever p : (
, D) ® (
2, 0) is a map resulting from the iteration of finite number of blow-ups, with exceptional curve D = p1(0), whose irreducible components are Dj, j = 1 ¼ n, with Chern classes kj, then a suitable neighborhood of D in
results from pasting together suitable neighborhoods of the zero sections of
(kj). Thus, for each Hopf bundle component p :
j ® Dj of a given resolution, we shall denote by
j, the germ of foliation at Dj Ì
j induced by the restriction of
, and call it the jth Hopf component of the resolution. Further, recall that
j is nondicritical if Dj is an invariant set of the foliation, and dicritical otherwise. In the former case, the holonomy of
j with respect to a section S transversal to D @
(1) is called the projective holonomy of
j, and denoted by HolS(
j, Dj), furthermore one says that it is solved if it has just reduced singularities (cf. Mattei and Moussu 1980), and rectified if Sep(
j) is contained in a fibration transversal to Dj in (
j, Dj). In both cases one calls the set of singularities of
j the first tangent cone of
j, and, in the later case, the set of tangencies between
j and Dj the second tangent cone of
.
Let
i, i = 1, 2, be two germs of singular nondicritical foliations at![](https://minio.scielo.br/documentstore/1678-2690/VK3BJ8zt9DS7MLDgKwgZQLC/5fcc2480475eaa28d0d1ab7aceadffeabb1381b7.gif)
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We say that a resolution of a nondicritical foliation is simple if its exceptional divisor has only one projective line with three or more singularities (see figure 1), which is called principal projective line of the exceptional divisor; its holonomy is called the projective holonomy of the foliation. In particular the Hopf component of a simple nondicritical foliation about the principal project line is called the principal Hopf component.
Now recall that a saddle-node is a singularity with a degenerated nonvanishing linear part which presents two invariant curves passing through the origin: the strong variety which is analytic and a formal one, which is called central variety when it converges. Their holonomies are called strong holonomy in the former case, and weak holonomy in the later. Let be a nondicritical solved foliation defined around the zero section of a Hopf bundle
: (p :
® D), with saddle-nodes along the divisor. Then one says that a saddle-node defined in (
, D) is transversal if its strong variety is transversal to D. Otherwise one say that the saddle-node is parallel.
Recall from Dulac 1904 that every saddle-node ( : w = 0) can be analytically reduced to the form (
p,A : wp,A = 0) where
with A Î 2 and A(0, y) = y. That is, there is Fp,AÎ Diff(
2, 0) such that
which will be called an analytic Dulac reduction. On the other hand each singularity of the form (p,A : wp,A = 0) has a formal normal form (
p,l : wp,l = 0) where
In other words, there is (
p,A) Î
I(
2, 0) given by
(
p,A)(x, y) = (x, g(x, y)), with g(x, y) = y + åj³ 1 aj(y)xj, where all aj converge with the same radius of convergence, and such that
which will be called a formal Dulac reduction. Furthermore, for each Uj = {(x, y) Î 2 : arg(x)
p Î] p/p, p/p[, 0 < |y| < r, r ~ 0), with j Î
2p, there is an unique holomorphic transformation of Uj, namely Nj(
p,A), such that Nj(
p,A) ®
(
p,A), i.e.
(
p,A) is the asymptotic expansion of each Nj(
) (in fact p-summable, cf. Hukuhara et al. 1961, Martinet and Ramis 1982) satisfying
Now let
j :=![](https://minio.scielo.br/documentstore/1678-2690/VK3BJ8zt9DS7MLDgKwgZQLC/26c22b2740c68f7713a15851aed31d2251bf8c9d.gif)
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is a well defined biholomorphism, where Sp,l is the orbit space under analytic equivalence of the set of saddle-nodes with formal normal form (p,l : wp,l = 0). In particular we have that (
: w = 0) has central variety if and only if Q(
) = (I, N2,1 (
), ¼ , I, N2p,2p1(
)). Note that to determine the analytic type of a saddle-node, it suffices to determine Nj(
) for all j Î
2p (Martinet and Ramis 1982). So we say that (
j, Nj(
)) is a sectorial chart and that
(
j, Nj(
)) is a sectorial atlas for the saddle-node (
: w = 0). Now let
1,
2Î Sp,l be two saddle-nodes with central variety and formal normal form
p,l. Let Sa = (x = a),
,
, and f be a conjugation between
and
. Then one says that f is p-periodic in sectorial charts if
Henceforth, we say that a saddle-node with central variety has a q-symmetric Stokes phenomenon or has q-order of symmetry if q is the greater integer for which the map Rq(I) = exp() · z commutes with Nj+1,j for every j º 1 mod 2. Let
and
' be two analytically equivalent, up to order one (for reduced nondegenerate singularities) and formal type (for saddle-nodes), solved foliations defined in the Hopf bundle
: (p :
® D), with
and
being transversal saddle-nodes, such that
= j(tj), where j Î Diff(D) » PGL(2,
) is their tree isomorphism, and with
and
'
having the same order of symmetry, say qj. Then one says that a holomorphic conjugation between HolS(
, D) and HolS'(
', D) commutes with the stokes phenomenon of
and
' if for each pair of equivalent points (tj,
), f is pj-periodic in sectorial charts, where pj is a divisor of gcd(
). Let
be a singular foliation in (
2, 0) with dicritical component, and
its extended resolution, then one says that each nondicritical Hopf component that does not intercept a dicritical Hopf component is a nondicritical cell. Further, if
has a dicritical component, then the union of each dicritical Hopf component with the nondicritical ones intercepting it, is called a dicritical cell, and denoted by [
j =
j,0
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Let = 1, 2, be two germs of dicritical cells, where each
is defined at the zero section of the Hopf bundle
k : (pk :
k ® Dk). Then one says that
HolD0(
, Dk) and
HolD0(
, Dk) have a Möbius conjugation if there is f Î Diff(D0) @ PGL(2,
) such that Ad(f)(HolD0(
, Dk)) = HolD0(
, Dk). Furthermore, let jkÎ Diff(Dk) @ PGL(2,
) be an isomorphism between Sing(
) and Sing(
) for k = 1 ¼n and between the corners of
and
. Then we say that a Möbius conjugation is subordinated to j :=
if it is subordinated to each jk as a local conjugation, for k = 1 ¼n, and is such that f|corners of
= j|\textcorners of
. Further we set
: =
0
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Further, let be a simple nondicritical branch of a solved and rectified foliation. Then we say that a dicritical singular foliation, say
, is a companion fibration for
, if
satisfies the following properties:
(1)
and
have the same chain of projective lines (cf. Câmara 2001);
(2)
is simple and its (only) dicritical component is a fibration transversal (cf. Câmara 2001) to the principal Hopf component of
;
We say that a singular foliation ( : w = 0) has an (extended) rectifier resolution, say (
:
= 0) if each of its Hopf components, namely
j, satisfies one of the following conditions:
(1)
j is nondicritical, solved, rectified and does not have parallel saddle-nodes;(2)
j has just two singularities, and in antipodal points of Dj »(1): a parallel saddle-node and a (corner) resonant saddle with index -1 with respect to Dj;
(3)
j is dicritical and has void first and second tangent cones.
Notice that, similarly to the case of Seidenberg's resolution, we can talk about a minimal rectifier resolution, although this is not unique in general. Two rectifier resolutions, namely
1 and![](https://minio.scielo.br/documentstore/1678-2690/VK3BJ8zt9DS7MLDgKwgZQLC/c4be35727e60a56c29bb45bc239cd47d0ecb4d41.gif)
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3 STATEMENT OF THE MAIN RESULTS
The above definitions of rectifier resolution are supported by the following.
PROPOSITION 1. Every germ of analytic foliation in (
2, 0) has an (extended) rectifier resolution. Let ( : w = 0) be a germ of foliation in (
2, 0), then pick a foliation
º cell-wise isomorphic to
, which shall be called a fixed model for
, and consider the elements FjÎ Diff(
j ,
), which shall be called cell charts for the jth cell of the fixed model. Then it is straightforward that for each cell
j and each fixed model cell
, there exists only one cell chart, up to left composition by an element of Iso(
). So, consider the sheaf of nonabelian groups Lº:= Iso(
º), then we say that
: = È
j is a good covering for Lº if Uj are neighborhoods of Dj Ì
j or of BDj Ì
j, respectively for nondicritical or dicritical cells. Therefore, consider the first cohomology set H1(
, Lº) associated to the good covering
, and set H1(D, Lº) as the direct limit of H1(
, Lº) for the good coverings for of Lº, associated to D = ÈDj, the exceptional divisor of a minimal rectifier resolution of
.
Thus we define the map
where . Note that Q does not depend on the fixed models up to cell-wise conjugacy class. Then, as a consequence of path-lifting construction, one has that:
THEOREM 1. Let ,
' be two germs of singular foliations in (
2, 0), and
,
' one of their minimal rectifier resolutions respectively. Then
,
' are analytically equivalent if and only if they satisfy:
(1)
,
' Î
for some 1-form w Î W1(
2, 0) (holomorphic one-forms). In particular we denote by j = (jj) : Gs ®
the component-wise isomorphism between the trees of singularities of their minimal rectifier resolution.
(2) Two equivalent (up to formal type) transversal saddle-nodes
= jj (ti,j) have the same order of symmetry.
(3) For each nondicritical cell
j and their projective holonomies have an analytic conjugation, namely fj, subordinated to jj, and commuting with the Stokes phenomenon of the transversal saddle-nodes of j and .(4) For each dicritical cell
j = j,0 j,k and = ,0 ,k, we have that HolDj,0(j,k, Dj,k ) and HolDj,0(
,k, Dj,k) have a Möbius conjugation, namely fj, subordinated to j, and commuting with the Stokes phenomenon of the transversal saddle-nodes of j,k and ,k.
(5) Q(
.) = Q(
'), for one (and every) fixed model of
Notice that (2) above answers in affirmative way the following conjecture of S. M. Voronin (cf. Elizarov et al. 1993): Beyond the resolution tree of the minimal resolution and the projective holonomies there are some further numerical invariants for singular foliations in (2, 0). In fact, these new invariants are the order of symmetry of each transversal saddle-node.
Recall that in Câmara 2001, for f(x, y) = (yp ljxq), with 1 < p < q Î N*, g.c.d.(p, q) = 1, and ljÎ
*, we classify the set
(the subset of generalized curves of
) by its projective holonomy, generalizing a result of Cerveau and Moussu 1988 concerning the subset
. Here we give a slight generalization of this result, including in particular a result of Meziani 1996 concerning
.
THEOREM 2. Let f(x, y) = (yp ljxq), where 1 < p < q Î N*, g.c.d.(p, q) = 1, and ljÎ
*. Then, for every
Î
we have that
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(1) If j is the isomorphism between the trees of
and ' with= j(tij), then tij and have the same order of symmetry, whenever they are transversal saddle-nodes;
(2) Their projective holonomies have a conjugation subordinated to j and commuting with the Stokes phenomenon of the transversal saddle-nodes of
0 and .
We would like to stress the importance of the fixed separatrix type in the above theorem, remarking that in general
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THEOREM 3. Let
be a germ of dicritical foliation which has a simple minimal rectifier resolution, say , whose nondicritical branches have a companion fibration. Then Q(![](https://minio.scielo.br/documentstore/1678-2690/VK3BJ8zt9DS7MLDgKwgZQLC/36c7e684fb3cc7a6d55be062c5c4651502730ac5.gif)
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(1) If j is the isomorphism between the trees of
and ' with= j(tij), then
|(Di,) and '|(Di,) have the same order of symmetry, whenever tij and are transversal saddle-nodes;
(2) For any dicritical cells
j=j,0 j,k and
=
j,0 ,k , we have that HolDj,0 (
j,k , Dj,k ) and HolDj,0(,k, Dj,k) have a Möbius conjugation, namely f, subordinated to j, and commuting with the Stokes phenomenon of the transversal saddle-nodes of j,k and ,k.
A particular instance of a foliation in the hypothesis of the above theorem happens when the minimal (Seidenberg) resolution of a dicritical singularity is simple and has just tangencies and the kind of singularities we study in theorem 2, in the dicritical component, generalizing some results of Klughertz 1988. We would like to remark that the stated theorems are proved in Câmara 2003. Furthermore, in a forthcoming work we shall describe more precisely H1(D, Lº) showing that, roughly speaking, it can be split in the first cohomology group of the coherent sheaf of vector fields tangent to the model º and the first cohomology group of a (in general finitely generated) sheaf of abelian groups generated by the centralizers of the projective holonomy groups of
. In particular we will present some examples for which the cohomology is nontrivial.
Manuscript received on August 13, 2004; accepted for publication on September 9, 2004; presented by MÁRCIO SOARES
E-mail: camara@cce.ufes.br
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Publication Dates
-
Publication in this collection
02 Feb 2005 -
Date of issue
Mar 2005
History
-
Accepted
09 Sept 2004 -
Received
13 Aug 2004