A Measurable Stability Theorem for Holomorphic Foliations Transverse to Fibrations

Abstract

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.

1. Introduction

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 𝜂=(𝐸,𝜋,𝐵,𝐹) be a (locally trivial) fibration with total space 𝐸, fiber 𝐹, base 𝐵, and projection 𝜋∶𝐸→𝐵. A foliation ℱ on 𝐸 is transverse to 𝜂 if: (1) for each 𝑝∈𝐸, the leaf 𝐿_𝑝 of ℱ with 𝑝∈𝐿_𝑝 is transverse to the fiber 𝜋⁻¹(𝑞), 𝑞=𝜋(𝑝); (2) dim(ℱ)+dim(𝐹)=dim(𝐸); (3) for each leaf 𝐿 of ℱ, the restriction 𝜋|_𝐿∶𝐿→𝐵 is a covering map. A theorem of Ehresmann ([1] Chpter V) [2]) assures that if the fiber 𝐹 is compact, then conditions (1) and (2) together already imply (3). Such foliations are conjugate to suspensions and are characterized by their global holonomy ([1], Theorem 3, page 103 and [2], Theorem 6.1, page 59).

The codimension one case is studied in [3]. In [4], we study the case where the ambient manifold is a hyperbolic complex manifold. In [5], the authors prove a natural version of the stability theorem of Reeb for (transversely holomorphic) foliations transverse to fibrations. A foliation ℱ on 𝑀 is called a Seifert fibration if all leaves are compact with finite holonomy groups.

The following stability theorem is proved in [5].

Theorem 1.1. Let ℱ be a holomorphic foliation transverse to a fibration 𝜋∶𝐸−→^𝐹𝐵 with fiber 𝐹. If ℱ has a compact leaf with finite holonomy group then ℱ is a Seifert fibration.

It is also observed in [5] that the existence of a trivial holonomy compact leaf is assured if ℱ is of codimension 𝑘 has a compact leaf, and the base 𝐵 satisfies 𝐻¹(𝐵,ℝ)=0,𝐻¹(𝐵,GL(𝑘,ℂ))=0.

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 [5]: Let 𝐺 be a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold 𝐹. If each element of 𝐺 has finite order, then the subgroups with a common fixed point are finite.

We recall that a subset 𝑋⊂𝑀 of a differentiable 𝑚-manifold has zero measure on 𝑀 if 𝑀 admits an open cover by coordinate charts 𝜑∶𝑈⊂𝑀→𝜑(𝑈)⊂ℝ^𝑚 such that 𝜑(𝑈∩𝑋) has zero measure with respect to the standard Lebesgue measure in ℝ^𝑚. For sake of simplicity, if 𝑋⊂ℝ^𝑛 is not a zero measure subset, then we will say that 𝑋 has positive measure and write 𝜇(𝑋)>0. This may cause no confusion since, Indeed, we notice that if 𝑋⊂𝑀 writes as a countable union 𝑋=⋃_𝑛∈ℕ𝑋_𝑛 of subsets 𝑋_𝑛⊂𝑀 then 𝑋 has zero measure in 𝑀 if and only if 𝑋_𝑛 has zero measure in 𝑀 for all 𝑛∈ℕ. In terms of our notation, we have therefore 𝜇(𝑋)>0 if and only if 𝜇(𝑋_𝑛)>0 for some 𝑛∈ℕ.

In this paper, we improve Theorem 1.1 above by proving the following theorems.

Theorem 1.2. Let ℱ be a transversely holomorphic foliation transverse to a fibration 𝜋∶𝐸−→^𝐹𝐵 with fiber 𝐹 a connected complex manifold. Denote by Ω(ℱ)⊂𝐸 the union of all compact leaves of ℱ. Suppose that one have 𝜇(Ω(ℱ))>0. Then 𝐹 is a Seifert fibration with finite global holonomy.

Parallel to this result we have the following version for groups.

Theorem 1.3. Let 𝐺 be a finitely generated subgroup of holomorphic diffeomorphisms of a complex connected manifold 𝐹. Denote by Ω(𝐺) the subset of points 𝑥∈𝐹 such that the 𝐺-orbit of 𝑥 is periodic. Assume that 𝜇(Ω(𝐺))>0. Then 𝐺 is a finite group.

As an immediate corollary of the above result, we get that, for a finitely generated subgroup 𝐺⊂Diff(𝐹) of a complex connected manifold 𝐹, if the volume of the orbits gives an integrable function for some regular volume measure on 𝐹 then all orbits are periodic and the group is finite. This is related to results in [6].

2. Holonomy and Global Holonomy

Let ℱ be a codimension 𝑘 transversely holomorphic foliation transverse to a fibration 𝜋∶𝐸−→^𝐹𝐵 with fiber 𝐹, base 𝐵, and total space 𝐸. We always assume that 𝐵, 𝐹, and 𝐸 are connected manifolds. The manifold 𝐹 is a complex manifold.

2.1. Holonomy

For a given point 𝑝∈𝐸, put 𝑏=𝜋(𝑝)∈𝐵 and denote by 𝐹_𝑏=𝜋¹(𝑏)⊂𝐸 the fiber of 𝜋 over 𝑏, which is a complex biholomorphic to 𝐹. Given a point 𝑝∈𝐸, we denote by Hol(ℱ,𝐿_𝑝) the holonomy group of the leaf 𝐿_𝑝 through 𝑝 obtained by lifting to the leaves of ℱ, locally, closed paths in 𝐿_𝑝 based on 𝑝, transversely to ℱ (see [1] for the construction of holonomy). Let us denote by Diff(𝐹_𝑏,𝑝) the group of germs at 𝑝 of holomorphic diffeomorphisms of 𝐹_𝑏 fixing 𝑝∈𝐹_𝑏. The group Diff(𝐹_𝑏,𝑝) is then identified with the group Diff(ℂ^𝑘,0) of germs at the origin 0∈ℂ^𝑘 of complex diffeomorphisms, where 𝑘=dim𝐹.

This holonomy group Hol(ℱ,𝐿_𝑝) is formally defined as a conjugacy class of equivalence under diffeomorphism germs conjugation. Let us denote by Hol(𝐿_𝑝,𝐹_𝑏,𝑝)⊂Diff(𝐹_𝑏,𝑝), its representative given by the local representation of this holonomy calculated with respect to the local transverse section induced by 𝐹_𝑏 at the point 𝑝∈𝐹_𝑏. The group Hol(𝐿_𝑝,𝐹_𝑏,𝑝) is therefore a subgroup of Diff(𝐹_𝑏,𝑝) identified with a subgroup of Diff(ℂ^𝑘,0).

2.2. Global Holonomy

As it is well known, the fundamental group 𝜋₁(𝐵) acts on the group of holomorphic diffeomorphisms of the manifold 𝐹Diff(𝐹), by what we call the global holonomy representation. This consists of a group homomorphism 𝜑∶𝜋₁(𝐵,𝑏)→Diff(𝐹), obtained by lifting closed paths in 𝐵 to the leaves of ℱ via the covering maps 𝜋|_𝐿∶𝐿→𝐵, where 𝐿 is a leaf of ℱ. The image of this representation is the global holonomy Hol(ℱ) of ℱ, and its construction shows that ℱ is conjugated to the suspension of its global holonomy ([1], Theorem 3, page 103). Given a base point 𝑏∈𝐵, we will denote by Hol(ℱ,𝐹_𝑏)⊂Diff(𝐹_𝑏) the representation of the global holonomy of ℱ based at 𝑏.

From the classical theory [1], chapter V and [5] we have the following.

Proposition 2.1. Let ℱ be a foliation on 𝐸 transverse to the fibration 𝜋∶𝐸→𝐵 with fiber 𝐹. Fix a point 𝑝∈𝐸, 𝑏=𝜋(𝑝) and denote by 𝐿 the leaf that contains 𝑝.(1)The holonomy group Hol(𝐿,𝐹_𝑏,𝑝) of 𝐿 is the subgroup of the global holonomy Hol(ℱ,𝐹_𝑏)⊂Diff(𝐹_𝑏) of those elements that have 𝑝 as a fixed point. (2)Given another intersection point 𝑞∈𝐿∩𝐹_𝑏, there is a global holonomy map ℎ∈Hol(ℱ,𝐹_𝑏) such that ℎ(𝑝)=𝑞.(3)Suppose that the global holonomy Hol(ℱ) is finite. If ℱ has a compact leaf then it is a Seifert fibration, that is, all leaves are compact with finite holonomy group.(4)If ℱ has a compact leaf 𝐿₀∈ℱ then each point 𝑝∈𝐹_𝑏∩𝐿₀ has periodic orbit in the global holonomy Hol(ℱ). In particular, there are ℓ∈ℕ and 𝑝∈𝐹 such that ℎ^ℓ(𝑝)=𝑝 for every ℎ∈Hol(ℱ).

3. Periodic Groups and Groups of Finite Exponent

First we recall some facts from the theory of Linear groups. Let 𝐺 be a group with identity 𝑒_𝐺∈𝐺. The group is periodic if each element of 𝐺 has finite order. A periodic group 𝐺 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 𝑚∈ℕ with 𝑔^𝑚=1 for all 𝑔∈𝐺 (cf. [5]). Because of this, a group which is periodic of bounded exponent is also called a group of finite exponent. Given 𝑅 a ring with identity, we say that a group 𝐺 is 𝑅-linear if it is isomorphic to a subgroup of the matrix group GL(𝑛,𝑅) (of 𝑛×𝑛 invertible matrices with coefficients belonging to 𝑅) for some 𝑛∈ℕ. 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, [7] A (not necessarily finitely generated) complex linear group 𝐺⊂GL(𝑘,ℂ) of finite exponent ℓ has finite order; actually we have |𝐺|≤ℓ^𝑘2.(2)Schur, [8] Every finitely generated periodic subgroup of GL(𝑛,ℂ) is finite.

Using these results, we obtain in [5].

Lemma 3.2 (see Lemmas 2.3, 3.2, and 3.3 [5]). About periodic groups of germs of complex diffeomorphisms one has the following. (1)A finitely generated periodic subgroup 𝐺⊂Diff(ℂ^𝑘,0) is necessarily finite.(2)A (not necessarily finitely generated) subgroup 𝐺⊂Diff(ℂ^𝑘,0) of finite exponent is necessarily finite.(3)Let 𝐺⊂Diff(ℂ^𝑘,0) be a finitely generated subgroup. Assume that there is an invariant connected neighborhood 𝑊 of the origin in ℂ^𝑘 such that each point 𝑥 is periodic for each element 𝑔∈𝐺. Then 𝐺 is a finite group.(4)Let 𝐺⊂Diff(ℂ^𝑘,0) be a (not necessarily finitely generated) subgroup such that for each point 𝑥 close enough to the origin, the pseudoorbit of 𝑥 is finite of (uniformly bounded) order ≤ℓ for some ℓ∈ℕ, then 𝐺 is finite.

Given a subgroup 𝐺⊂Diff(𝐹) and a point 𝑝∈𝐹 the stabilizer of 𝑝 in 𝐺 is the subgroup 𝐺(𝑝)⊂𝐺 of the elements 𝑓∈𝐺 such that 𝑓(𝑝)=𝑝. From the above one has the following.

Proposition 3.3. Let 𝐺⊂Diff(𝐹) be a (not necessarily finitely generated) subgroup of holomorphic diffeomorphisms of a connected complex manifold 𝐹.(1)If 𝐺 is periodic and finitely generated or 𝐺 is periodic of finite exponent, then each stabilizer subgroup of 𝐺 is finite.(2)Assume that there is a point 𝑝∈𝐹 which is fixed by 𝐺 and a fundamental system of neighborhoods {𝑈_𝜈}_𝜈 of 𝑝 in 𝐹 such that each 𝑈_𝜈 is invariant by 𝐺, the orbits of 𝐺 in 𝑈_𝜈 are periodic (not necessarily with uniformly bounded orders). Then 𝐺 is a finite group.(3)Assume that 𝐺 has a periodic orbit {𝑥₁,…,𝑥_𝑟}⊂𝐹 such that for each 𝑗∈{1,…,𝑟}, there is a fundamental system of neighborhoods 𝑈^𝑗_𝜈 of 𝑥_𝑗 with the property that 𝑈_𝜈=⋃^𝑟_𝑗=1𝑈^𝑗_𝜈 is invariant under the action of 𝐺, 𝑈^𝑗_𝜈∩𝑈^ℓ_𝜈=∅ if 𝑗≠ℓ, and each orbit in 𝑈_𝜈 is periodic. Then 𝐺 is periodic.

Proof. In order to prove (1), we consider the case where 𝐺 has a fixed point 𝑝∈𝐹. We identify the group 𝒢_𝑝⊂Diff(𝐹,𝑝), of germs at 𝑝∈𝐹 of maps in 𝐺⊂Diff(𝐹), with a subgroup of Diff(ℂ^𝑛,0) where 𝑛=dim𝐹. If 𝐺 is finitely generated and periodic, the group 𝒢_𝑝 is finitely generated and periodic. By Lemma 3.2 (1), the group 𝒢_𝑝 is finite and by the Identity principle the group 𝐺 is also finite of same order. If 𝐺 is periodic of finite exponent then the group 𝒢_𝑝 is periodic of finite exponent. By Lemma 3.2(2), the group 𝒢_𝑝 is finite and by the Identity principle the group 𝐺 is also finite of same order. This proves (1).
As for (2), since 𝑈_𝜈 is 𝐺-invariant, each element 𝑔∈𝐺 induces by restriction to 𝑈_𝜈 an element of a group 𝐺_𝜈⊂Diff(𝑈_𝜈). It is observed in [5] (proof of Lemma 3.5) that the finiteness of the orbits in 𝑈_𝜈 implies that 𝐺_𝜈 is periodic. By the Identity principle, the group 𝐺 is also periodic of the same order. Since 𝐺=𝐺(𝑝), (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 𝐺 be a subgroup of holomorphic diffeomorphisms of a connected complex manifold 𝐹. Assume that(1)𝐺 is periodic of finite exponent or 𝐺 is finitely generated and periodic,(2)𝐺 has a finite orbit in 𝐹.Then 𝐺 is finite.

Proof . Fix a point 𝑥∈𝐹 with finite orbit, we can write 𝒪_𝐺(𝑥)={𝑥₁,…,𝑥_𝑘} with 𝑥_𝑖≠𝑥_𝑗 if 𝑖≠𝑗. Given any diffeomorphism 𝑓∈𝐺, we have 𝒪_𝐺(𝑓(𝑥))=𝒪_𝐺(𝑥) so that there exists an unique element 𝜎∈𝑆_𝑘 of the symmetric group such that 𝑓(𝑥_𝑗)=𝑥_{𝜎𝑓(𝑗)}, for all 𝑗=1,…,𝑘. We can therefore define a map 𝜂∶𝐺→𝑆_𝑘,𝜂𝑓=𝜎_𝑓 (3.1) Now, if 𝑓,𝑔∈𝐺 are such that 𝜂(𝑓)=𝜂(𝑔), then 𝑓(𝑥_𝑗)=𝑔(𝑥_𝑗), for all 𝑗 and therefore ℎ=𝑓𝑔⁻¹∈𝐺 fixes the points 𝑥₁,…,𝑥_𝑘. In particular, ℎ belongs to the stabilizer 𝐺_𝑥. By Proposition 3.3(1) and (2) (according to 𝐺 is finitely generated or not), the group 𝐺_𝑥 is finite. Thus, the map 𝜂∶𝐺→𝑆_𝑘 is a finite map. Since 𝑆_𝑘 is a finite group, this implies that 𝐺 is finite as well.

4. Measure and Finiteness

The following lemma paves the way to Theorems 1.2 and 1.3.

Lemma 4.1. Let 𝐺 be a subgroup of complex diffeomorphisms of a connected complex manifold 𝐹. Denote by Ω(𝐺) the set of points 𝑥∈𝐹 such that the orbit 𝒪_𝐺(𝑥) is periodic. If 𝜇(Ω(𝐺))>0 then 𝐺 is a periodic group of finite exponent.

Proof. We have Ω(𝐺)={𝑥∈𝐹∶#𝒪_𝐺(𝑥)<∞}=⋃^∞_𝑘=1{𝑥∈𝐹∶#𝒪_𝐺(𝑥)≤𝑘}, therefore there is some 𝑘∈ℕ such that 𝜇𝑥∈𝐹∶#𝒪_𝐺(𝑥)≤𝑘>0. (4.1) In particular, given any diffeomorphism 𝑓∈𝐺 we have 𝜇𝑥∈𝐹∶#𝒪_𝑓(𝑥)≤𝑘>0. (4.2) In particular, there is 𝑘_𝑓≤𝑘 such that the set 𝑋={𝑥∈𝐹∶𝑓^𝑘𝑓(𝑥)=𝑥} has positive measure. Since 𝑋⊂𝐹 is an analytic subset, this implies that 𝑋=𝐹 (a proper analytic subset of a connected complex manifold has zero measure). Therefore, we have 𝑓^𝑘𝑓=Id in 𝐹. This shows that 𝐺 is periodic of finite exponent.

Proof of Theorem 1.2. Fix a base point 𝑏∈𝐵. By Proposition 2.1, the compact leaves correspond to periodic orbits of the global holonomy Hol(ℱ,𝐹_𝑏). Therefore, by the hypothesis the global holonomy 𝐺=Hol(ℱ,𝐹_𝑏) 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.

The construction of the suspension of a group action gives Theorem 1.3 from Theorem 1.2.

Proof of Theorem 1.3. Since 𝐺 is finitely generated, there are a compact connected manifold 𝐵 and a representation 𝜑∶𝜋₁(𝐵)→Diff(𝐹) such that the image 𝜑(𝜋₁(𝐵))=𝐺. The manifold 𝐵 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 ℱ the suspension foliation of the fibre bundle 𝜋∶𝐸→𝐵 with fiber 𝐹 which has global holonomy conjugate to 𝐺. The periodic orbits of 𝐺 in 𝐹 correspond in a natural way to the leaves of ℱ which have finite order with respect to the fibration 𝜋∶𝐸→𝐵, that is, the leaves which intersect the fibers of 𝜋∶𝐸→𝐵 only at a finite number of points. Thus, because the basis is compact, each such leaf (corresponding to a finite orbit of 𝐺) is compact. By the hypothesis, we have 𝜇(Ω(ℱ))>0. By Theorem 1.2 the global holonomy Hol(ℱ) is finite. Thus, the group 𝐺 is finite.