# A Measurable Stability Theorem for Holomorphic Foliations Transverse to Fibrations

Authors: Bruno Scardua
Journal: International Journal of Differential Equations
 (2012)

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## 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.


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## Body

## 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.Letη=(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 toη if: (1) for each p∈E, the leaf Lp of ℱ with p∈Lp 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.
Letℱ 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 Theorem1.1 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]: LetGbe a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifoldF. If each element ofGhas finite order, then the subgroups with a common fixed point are finite.We recall that a subsetX⊂M of a differentiable m-manifold has zero measure onM if M admits an open cover by coordinate charts φ:U⊂M→φ(U)⊂ℝm such that φ(U∩X) has zero measure with respect to the standard Lebesgue measure in ℝm. For sake of simplicity, if X⊂ℝn 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∈ℕ‍Xn of subsets Xn⊂M then X has zero measure in M if and only if Xn has zero measure in M for alln∈ℕ. In terms of our notation, we have therefore μ(X)>0 if and only if μ(Xn)>0 for somen∈ℕ.In this paper, we improve Theorem1.1 above by proving the following theorems.Theorem 1.2.
Letℱ 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.
LetG 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 subgroupG⊂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

Letℱ 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 pointp∈E, put b=π(p)∈B and denote by Fb=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,Lp) the holonomy group of the leaf Lp through p obtained by lifting to the leaves of ℱ, locally, closed paths in Lp based on p, transversely to ℱ (see [1] for the construction of holonomy). Let us denote by Diff(Fb,p) the group of germs at p of holomorphic diffeomorphisms of Fb fixing p∈Fb. The group Diff(Fb,p) is then identified with the group Diff(ℂk,0) of germs at the origin 0∈ℂk of complex diffeomorphisms, where k=dimF.This holonomy groupHol(ℱ,Lp) is formally defined as a conjugacy class of equivalence under diffeomorphism germs conjugation. Let us denote by Hol(Lp,Fb,p)⊂Diff(Fb,p), its representative given by the local representation of this holonomy calculated with respect to the local transverse section induced by Fb at the point p∈Fb. The group Hol(Lp,Fb,p) is therefore a subgroup of Diff(Fb,p) identified with a subgroup of Diff(ℂk,0).


### 2.2. Global Holonomy

As it is well known, the fundamental groupπ1(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 holonomyHol(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,Fb)⊂Diff(Fb) 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.
Letℱ 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 groupHol(L,Fb,p) of L is the subgroup of the global holonomy Hol(ℱ,Fb)⊂Diff(Fb) of those elements that have p as a fixed point.(2)
Given another intersection pointq∈L∩Fb, there is a global holonomy map h∈Hol(ℱ,Fb) such that h(p)=q.(3)
Suppose that the global holonomyHol(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)
Ifℱ has a compact leaf L0∈ℱ then each point p∈Fb∩L0 has periodic orbit in the global holonomy Hol(ℱ). In particular, there are ℓ∈ℕ and p∈F such that hℓ(p)=p for every h∈Hol(ℱ).


## 2.1. Holonomy

For a given pointp∈E, put b=π(p)∈B and denote by Fb=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,Lp) the holonomy group of the leaf Lp through p obtained by lifting to the leaves of ℱ, locally, closed paths in Lp based on p, transversely to ℱ (see [1] for the construction of holonomy). Let us denote by Diff(Fb,p) the group of germs at p of holomorphic diffeomorphisms of Fb fixing p∈Fb. The group Diff(Fb,p) is then identified with the group Diff(ℂk,0) of germs at the origin 0∈ℂk of complex diffeomorphisms, where k=dimF.This holonomy groupHol(ℱ,Lp) is formally defined as a conjugacy class of equivalence under diffeomorphism germs conjugation. Let us denote by Hol(Lp,Fb,p)⊂Diff(Fb,p), its representative given by the local representation of this holonomy calculated with respect to the local transverse section induced by Fb at the point p∈Fb. The group Hol(Lp,Fb,p) is therefore a subgroup of Diff(Fb,p) identified with a subgroup of Diff(ℂk,0).


## 2.2. Global Holonomy

As it is well known, the fundamental groupπ1(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 holonomyHol(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,Fb)⊂Diff(Fb) 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.
Letℱ 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 groupHol(L,Fb,p) of L is the subgroup of the global holonomy Hol(ℱ,Fb)⊂Diff(Fb) of those elements that have p as a fixed point.(2)
Given another intersection pointq∈L∩Fb, there is a global holonomy map h∈Hol(ℱ,Fb) such that h(p)=q.(3)
Suppose that the global holonomyHol(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)
Ifℱ has a compact leaf L0∈ℱ then each point p∈Fb∩L0 has periodic orbit in the global holonomy Hol(ℱ). In particular, there are ℓ∈ℕ and p∈F such that hℓ(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. LetG be a group with identity eG∈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 gm=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|≤ℓk2.(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 subgroupG⊂Diff(ℂk,0) is necessarily finite.(2)
A (not necessarily finitely generated) subgroupG⊂Diff(ℂk,0) of finite exponent is necessarily finite.(3)
LetG⊂Diff(ℂk,0) be a finitely generated subgroup. Assume that there is an invariant connected neighborhood W of the origin in ℂk such that each point x is periodic for each element g∈G. Then G is a finite group.(4)
LetG⊂Diff(ℂk,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 subgroupG⊂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.
LetG⊂Diff(F) be a (not necessarily finitely generated) subgroup of holomorphic diffeomorphisms of a connected complex manifold F.(1)
IfG 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 pointp∈F which is fixed by G and a fundamental system of neighborhoods {Uν}ν of p in F such that each Uν is invariant by G, the orbits of G in Uν are periodic (not necessarily with uniformly bounded orders). Then G is a finite group.(3)
Assume thatG has a periodic orbit {x1,…,xr}⊂F such that for each j∈{1,…,r}, there is a fundamental system of neighborhoods Uνj of xj with the property that Uν=⋃j=1r‍Uνj is invariant under the action of G, Uνj∩Uνℓ=∅ if j≠ℓ, and each orbit in Uν is periodic. Then G is periodic.Proof.
In order to prove (1), we consider the case whereG has a fixed point p∈F. We identify the group 𝒢p⊂Diff(F,p), of germs at p∈F of maps in G⊂Diff(F), with a subgroup of Diff(ℂn,0) where n=dimF. If G is finitely generated and periodic, the group 𝒢p is finitely generated and periodic. By Lemma 3.2 (1), the group 𝒢p 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 𝒢p is periodic of finite exponent. By Lemma 3.2(2), the group 𝒢p is finite and by the Identity principle the group G is also finite of same order. This proves (1).
As for (2), sinceUν is G-invariant, each element g∈G induces by restriction to Uν an element of a group Gν⊂Diff(Uν). It is observed in [5] (proof of Lemma 3.5) that the finiteness of the orbits in Uν implies that Gν 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).
LetG 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 pointx∈F with finite orbit, we can write 𝒪G(x)={x1,…,xk} with xi≠xj if i≠j. Given any diffeomorphism f∈G, we have 𝒪G(f(x))=𝒪G(x) so that there exists an unique element σ∈Sk of the symmetric group such that f(xj)=xσf(j), for all j=1,…,k. We can therefore define a map
(3.1)η:G→Sk,η(f)=σf
Now, if f,g∈G are such that η(f)=η(g), then f(xj)=g(xj), for all j and therefore h=fg-1∈G fixes the points x1,…,xk. In particular, h belongs to the stabilizer Gx. By Proposition 3.3(1) and (2) (according to G is finitely generated or not), the group Gx is finite. Thus, the map η:G→Sk is a finite map. Since Sk is a finite group, this implies that G is finite as well.


## 4. Measure and Finiteness

The following lemma paves the way to Theorems1.2 and 1.3.Lemma 4.1.
LetG 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 𝒪G(x) is periodic. If μ(Ω(G))>0 then G is a periodic group of finite exponent.Proof.
We haveΩ(G)={x∈F:#𝒪G(x)<∞}=⋃k=1∞‍{x∈F:#𝒪G(x)≤k}, therefore there is some k∈ℕ such that
(4.1)μ({x∈F:#OG(x)≤k})>0.
In particular, given any diffeomorphism f∈G we have
(4.2)μ({x∈F:#Of(x)≤k})>0.
In particular, there is kf≤k such that the set X={x∈F:fkf(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 fkf=Id in F. This shows that G is periodic of finite exponent.Proof of Theorem1.2.
Fix a base pointb∈B. By Proposition 2.1, the compact leaves correspond to periodic orbits of the global holonomy Hol(ℱ,Fb). Therefore, by the hypothesis the global holonomy G=Hol(ℱ,Fb) 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 Theorem1.3 from Theorem 1.2.Proof of Theorem1.3.
SinceG 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.



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