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International Journal of Differential Equations
Volume 2012 (2012), Article ID 585298, 6 pages
A Measurable Stability Theorem for Holomorphic Foliations Transverse to Fibrations
Instituto de Matematica, Universidade Federal do Rio de Janeiro, CP 68530, 21945-970, Rio de Janeiro, RJ, Brazil
Received 22 May 2012; Accepted 22 July 2012
Academic Editor: Kanishka Perera
Copyright © 2012 Bruno Scardua. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We prove that a transversely holomorphic foliation, which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not a zero measure subset. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold is finite provided that the set of periodic orbits is not a zero measure subset.
Foliations transverse to fibrations are among the very first and simplest constructible examples of foliations, accompanied by a well-known transverse structure. These foliations are suspensions of groups of diffeomorphisms and their behavior is closely related to the action of the group in the fiber. For these reasons, many results holding for foliations in a more general context are first established for suspensions, that is, foliations transverse to a fibration. In this paper, we pursue this idea, but not restricted to it. We investigate versions of the classical stability theorems of Reeb [1, 2], regarding the behavior of the foliation in a neighborhood of a compact leaf, replacing the finiteness of the holonomy group of the leaf by the existence of a sufficient number of compact leaves. This is done for transversely holomorphic (or transversely analytic) foliations.
Let be a (locally trivial) fibration with total space , fiber , base , and projection . A foliation on is transverse to if: (1) for each , the leaf of with is transverse to the fiber , ; (2) ; (3) for each leaf of , the restriction is a covering map. A theorem of Ehresmann ( Chpter V) ) assures that if the fiber is compact, then conditions (1) and (2) together already imply (3). Such foliations are conjugate to suspensions and are characterized by their global holonomy (, Theorem 3, page 103 and , Theorem 6.1, page 59).
The codimension one case is studied in . In , we study the case where the ambient manifold is a hyperbolic complex manifold. In , the authors prove a natural version of the stability theorem of Reeb for (transversely holomorphic) foliations transverse to fibrations. A foliation on is called a Seifert fibration if all leaves are compact with finite holonomy groups.
The following stability theorem is proved in .
Theorem 1.1. Let be a holomorphic foliation transverse to a fibration with fiber . If has a compact leaf with finite holonomy group then is a Seifert fibration.
It is also observed in  that the existence of a trivial holonomy compact leaf is assured if is of codimension has a compact leaf, and the base satisfies .
Since a foliation transverse to a fibration is conjugate to a suspension of a group of diffeomorphisms of the fiber, we can rely on the global holonomy of the foliation. As a general fact that holds also for smooth foliations, if the global holonomy group is finite then the foliation is a Seifert fibration. The proof of Theorem 1.1 relies on the local stability theorem of Reeb [1, 2] and the following remark derived from classical theorems of Burnside and Schur on finite exponent groups and periodic linear groups : Let be a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold . If each element of has finite order, then the subgroups with a common fixed point are finite.
We recall that a subset of a differentiable -manifold has zero measure on if admits an open cover by coordinate charts such that has zero measure with respect to the standard Lebesgue measure in . For sake of simplicity, if is not a zero measure subset, then we will say that has positive measure and write . This may cause no confusion since, Indeed, we notice that if writes as a countable union of subsets then has zero measure in if and only if has zero measure in for all . In terms of our notation, we have therefore if and only if for some .
In this paper, we improve Theorem 1.1 above by proving the following theorems.
Theorem 1.2. Let be a transversely holomorphic foliation transverse to a fibration with fiber a connected complex manifold. Denote by the union of all compact leaves of . Suppose that one have . Then is a Seifert fibration with finite global holonomy.
Parallel to this result we have the following version for groups.
Theorem 1.3. Let be a finitely generated subgroup of holomorphic diffeomorphisms of a complex connected manifold . Denote by the subset of points such that the -orbit of is periodic. Assume that . Then is a finite group.
As an immediate corollary of the above result, we get that, for a finitely generated subgroup of a complex connected manifold , if the volume of the orbits gives an integrable function for some regular volume measure on then all orbits are periodic and the group is finite. This is related to results in .
2. Holonomy and Global Holonomy
Let be a codimension transversely holomorphic foliation transverse to a fibration with fiber , base , and total space . We always assume that , , and are connected manifolds. The manifold is a complex manifold.
For a given point , put and denote by the fiber of over , which is a complex biholomorphic to . Given a point , we denote by the holonomy group of the leaf through obtained by lifting to the leaves of , locally, closed paths in based on , transversely to (see  for the construction of holonomy). Let us denote by the group of germs at of holomorphic diffeomorphisms of fixing . The group is then identified with the group of germs at the origin of complex diffeomorphisms, where .
This holonomy group is formally defined as a conjugacy class of equivalence under diffeomorphism germs conjugation. Let us denote by , its representative given by the local representation of this holonomy calculated with respect to the local transverse section induced by at the point . The group is therefore a subgroup of identified with a subgroup of .
2.2. Global Holonomy
As it is well known, the fundamental group acts on the group of holomorphic diffeomorphisms of the manifold , by what we call the global holonomy representation. This consists of a group homomorphism , obtained by lifting closed paths in to the leaves of via the covering maps , where is a leaf of . The image of this representation is the global holonomy of , and its construction shows that is conjugated to the suspension of its global holonomy (, Theorem 3, page 103). Given a base point , we will denote by the representation of the global holonomy of based at .
Proposition 2.1. Let be a foliation on transverse to the fibration with fiber . Fix a point , and denote by the leaf that contains .(1)The holonomy group of is the subgroup of the global holonomy of those elements that have as a fixed point. (2)Given another intersection point , there is a global holonomy map such that .(3)Suppose that the global holonomy is finite. If has a compact leaf then it is a Seifert fibration, that is, all leaves are compact with finite holonomy group.(4)If has a compact leaf then each point has periodic orbit in the global holonomy . In particular, there are and such that for every .
3. Periodic Groups and Groups of Finite Exponent
First we recall some facts from the theory of Linear groups. Let be a group with identity . The group is periodic if each element of has finite order. A periodic group is periodic of bounded exponent if there is an uniform upper bound for the orders of its elements. This is equivalent to the existence of with for all (cf. ). Because of this, a group which is periodic of bounded exponent is also called a group of finite exponent. Given a ring with identity, we say that a group is -linear if it is isomorphic to a subgroup of the matrix group (of invertible matrices with coefficients belonging to ) for some . We will consider complex linear groups. The following classical results are due to Burnside and Schur.
Theorem 3.1. With respect to complex linear groups one has the following.(1)Burnside,  A (not necessarily finitely generated) complex linear group of finite exponent has finite order; actually we have .(2)Schur,  Every finitely generated periodic subgroup of is finite.
Using these results, we obtain in .
Lemma 3.2 (see Lemmas 2.3, 3.2, and 3.3 ). About periodic groups of germs of complex diffeomorphisms one has the following. (1)A finitely generated periodic subgroup is necessarily finite.(2)A (not necessarily finitely generated) subgroup of finite exponent is necessarily finite.(3)Let be a finitely generated subgroup. Assume that there is an invariant connected neighborhood of the origin in such that each point is periodic for each element . Then is a finite group.(4)Let be a (not necessarily finitely generated) subgroup such that for each point close enough to the origin, the pseudoorbit of is finite of (uniformly bounded) order ≤ for some , then is finite.
Given a subgroup and a point the stabilizer of in is the subgroup of the elements such that . From the above one has the following.
Proposition 3.3. Let be a (not necessarily finitely generated) subgroup of holomorphic diffeomorphisms of a connected complex manifold .(1)If is periodic and finitely generated or is periodic of finite exponent, then each stabilizer subgroup of is finite.(2)Assume that there is a point which is fixed by and a fundamental system of neighborhoods of in such that each is invariant by , the orbits of in are periodic (not necessarily with uniformly bounded orders). Then is a finite group.(3)Assume that has a periodic orbit such that for each , there is a fundamental system of neighborhoods of with the property that is invariant under the action of , if , and each orbit in is periodic. Then is periodic.
Proof. In order to prove (1), we consider the case where has a fixed point . We identify the group , of germs at of maps in , with a subgroup of where . If is finitely generated and periodic, the group is finitely generated and periodic. By Lemma 3.2 (1), the group is finite and by the Identity principle the group is also finite of same order. If is periodic of finite exponent then the group is periodic of finite exponent. By Lemma 3.2(2), the group is finite and by the Identity principle the group is also finite of same order. This proves (1).
As for (2), since is -invariant, each element induces by restriction to an element of a group . It is observed in  (proof of Lemma 3.5) that the finiteness of the orbits in implies that is periodic. By the Identity principle, the group is also periodic of the same order. Since , (2) follows from (1). (3) is proved like the first part of (2).
The following simple remark gives the finiteness of finite exponent groups of holomorphic diffeomorphisms having a periodic orbit.
Proposition 3.4 (Finiteness lemma). Let be a subgroup of holomorphic diffeomorphisms of a connected complex manifold . Assume that(1) is periodic of finite exponent or is finitely generated and periodic,(2) has a finite orbit in .Then is finite.
Proof . Fix a point with finite orbit, we can write with if . Given any diffeomorphism , we have so that there exists an unique element of the symmetric group such that , for all . We can therefore define a map Now, if are such that , then , for all and therefore fixes the points . In particular, belongs to the stabilizer . By Proposition 3.3(1) and (2) (according to is finitely generated or not), the group is finite. Thus, the map is a finite map. Since is a finite group, this implies that is finite as well.
4. Measure and Finiteness
Lemma 4.1. Let be a subgroup of complex diffeomorphisms of a connected complex manifold . Denote by the set of points such that the orbit is periodic. If then is a periodic group of finite exponent.
Proof. We have , therefore there is some such that In particular, given any diffeomorphism we have In particular, there is such that the set has positive measure. Since is an analytic subset, this implies that (a proper analytic subset of a connected complex manifold has zero measure). Therefore, we have in . This shows that is periodic of finite exponent.
Proof of Theorem 1.2. Fix a base point . By Proposition 2.1, the compact leaves correspond to periodic orbits of the global holonomy . Therefore, by the hypothesis the global holonomy satisfies the hypothesis of Lemma 4.1. By this lemma, the global holonomy is periodic of finite exponent. Since this group has some periodic orbit, by the Finiteness lemma (Proposition 3.4) the global holonomy group is finite. By Proposition 2.1(3), the foliation is a Seifert fibration.
Proof of Theorem 1.3. Since is finitely generated, there are a compact connected manifold and a representation such that the image . The manifold is not necessarily a complex manifold, but this makes no difference in our argumentation based only on the fact that the foliation is transversely holomorphic. Denote by the suspension foliation of the fibre bundle with fiber which has global holonomy conjugate to . The periodic orbits of in correspond in a natural way to the leaves of which have finite order with respect to the fibration , that is, the leaves which intersect the fibers of only at a finite number of points. Thus, because the basis is compact, each such leaf (corresponding to a finite orbit of ) is compact. By the hypothesis, we have . By Theorem 1.2 the global holonomy is finite. Thus, the group is finite.
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