ON THE NEIGHBORHOOD OF A TORUS LEAF AND DYNAMICSOF HOLOMORPHIC FOLIATIONS

TAKAYUKI KOIKE1 AND NOBORU OGAWA2

Abstract. Let X be a complex surface and Y be an elliptic curve embedded in X.Assume that there exists a non-singular holomorphic foliation F with Y as a compactleaf, defined on a neighborhood of Y in X. We investigate the relation between Ueda’sclassification of the complex analytic structure of a neighborhood of Y and complexdynamics of the holonomy of F along Y .

1. Introduction

Let X be a complex manifold and Y be a compact complex submanifold of X. Assumethat there exists a non-singular holomorphic foliation F which has Y as a compact leafand is defined on a neighborhood V of Y in X. Our interest is the relation betweencomplex-analytic properties of small neighborhoods of Y and complex dynamics of theholonomy of F along Y .

In this paper, we consider the case where X is a complex surface and Y is an ellipticcurve as a compact leaf of F . It follows from the fundamental results in foliation theory([B],[M2]) that the normal bundle NY/X of Y is topologically trivial. On the other hand,the complex analytic structure on a neighborhood of Y may behave complicatedly evenif the normal bundle is holomorphically trivial. The neighborhood of an embedded curvewith topologically trivial normal bundle is analytically classified into three types (α), (β),and (γ) according to Ueda’s classification ([U], see also §2.2 below. Original Ueda’sneighborhood theory assumes neither that Y is an elliptic curve nor the existence of afoliation). A pair (Y,X) is said to be of type (β) if there exists a non-singular holomorphicfoliation G defined on a neighborhood of Y which has Y as a leaf and has U(1)-linearholonomy along Y . In this case, Y admits a system of pseudoflat neighborhoods. A pair(Y,X) is said to be of type (α) if, roughly speaking, it is different from the case of type(β) in n-jet along Y for some positive integer n (see §2.2 for the precise definition). Uedashowed that Y admits a system of strongly pseudoconcave neighborhoods in this case [U,Theorem 1]. The remaining case is called type (γ). So far, little is known about thiscase. The first example of the pair (Y,X) of type (γ) is constructed by Ueda [U, §5.4].Note that this example (of Ueda’s) satisfies our assumption on the existence of F on Vas above. In [K3], the first author investigated the complex-analytic properties of smallneighborhoods of Y for Ueda’s example (Y,X), whose generalization is one of the biggestmotivations of the present paper. Our purpose is to determine the type of (Y,X) fromthe information of holonomy of F along Y .

2010 Mathematics Subject Classification. 37F75, 37F50.Key words and phrases. Ueda’s neighborhood theory, dynamics of holomorphic foliations, hedgehogs.

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To state our main result, we recall the notion of holonomy of F . Take a closed smoothcurve c : [0, 1]→ Y with p = c(0) = c(1) and a transversal τ for F through p, i.e. holomor-phically embedded open disk of dimC = 1 which intersects the leaves of F transversely.If τ is contained in a foliated chart (W ; (z, w)), τ is parametrized by the transversecoordinate w. We cover c by finitely many foliated charts (Wj; (zj, wj))j such thatWj ∩ Y = wj = 0, j = 1, . . . , N , and W1 contains τ . By moving from one chart tothe next chart, we have a local holomorphic diffeomorphism on the w-coordinate. If wego around along c, the composition of the finitely many transition maps gives rise to alocal holomorphic diffeomorphism on τ . Under fixing a parametrization of τ by w1, thecomposition determines an element hF(c, τ, Wj) of Diff(C, 0). Here, Diff(C, 0) standsfor the group of germs at 0 of local holomorphic diffeomorphisms on C fixing 0. Thegerm hF(c, τ, Wj) is called the holonomy of F along c. This is not changed under aleafwise homotopy, keeping p fixed, of c inside the charts. Moreover, it does not dependon the choice of foliated charts by the cocycle condition on wj’s. Thus, the germ dependsonly on its leafwise homotopy class γ := 〈c〉 keeping p fixed. The difference of the holo-nomy by the choice of transversals can be expressed as the conjugation of the elements inDiff(C, 0), so that the holonomy homomorphism hF : π1(Y, p) → Diff(C, 0) is defined, upto conjugation.

We say the holonomy of F along c is said to be linearizable (resp. non-linearizable) ifthe corresponding map hF(γ) is linearizable (resp. non-linearizable) at 0, where γ := 〈c〉 ∈π1(Y, p). Also, say the holonomy is said to be rationally (resp. irrationally) indifferentif the fixed point 0 of hF(γ) is rationally (resp. irrationally) indifferent. See §2 forthe definitions. Note that, these properties of hF(γ) are invariant under conjugation onDiff(C, 0), so that the definitions do not depend on the choice of transversals. The mainresult is the following.

Theorem 1.1. Let X be a complex surface and Y be an elliptic curve embedded in X.Assume that there exist a non-singular holomorphic foliation F with Y as a compact leaf,defined on a neighborhood of Y in X, and a closed curve c in Y along which the holonomyof F is irrationally indifferent and non-linearizable. Then, (Y,X) is of type (γ).

This result includes Ueda’s example, see §3.3.2. Ueda used the small cycle propertyto show that it is of type (γ). Compare with this, we will use a complete invariant setaround irrationally indifferent fixed point 0, the so-called hedgehog, which was introducedby Perez-Marco [P4].

By using Theorem 1.1, one can generalize [K3, Theorem 1.1]. As we will explain fullygeneralized variant of our result in §3.1, here we explain one of our conclusions under theconfiguration as in [K3] for simplicity in order to make our contribution clear. Considerthe following two conditions: (1) there exists a neighborhood V of Y and a holomorphicsubmersion π : V → Y such that π|Y is the identity map, and (2) there exists γ1 ∈ π1(Y, p)such that hF(γ1) is the identity map. Then, as a generalization of [K3, Theorem 1.1], wehave the following statement.

Theorem 1.2. Assume that (Y,X;F) satisfies the conditions (1) and (2) above. Denoteby g := hF(γ2) where π1(Y, p) ∼= Zγ1 ⊕ Zγ2. Then, the following hold:

(i) The pair (Y,X) is of type (α) if and only if g has a rationally indifferent fixedpoint 0 and is not of finite order, i.e. gn 6= id for any n > 0.

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(ii) The pair (Y,X) is of type (β) if and only if g is linearizable at 0.(iii) The pair (Y,X) is of type (γ) if and only if g has an irrationally indifferent fixed

point 0 and is non-linearizable at 0.

See §3 for the result in more generalized configurations and their proofs.

Theorem 1.1 will be proved in §4.2. We describe outline of the proof here. First, itfollows from a standard observation of the normal bundle of Y that the pair (Y,X) isnot of type (α). Second, we exclude the possibility of type (β). If the pair (Y,X) is oftype (β), there exists a pluriharmonic function Φ: V \ Y → R for a sufficiently smallneighborhood V of Y such that Φ(p) = O(− log dist(p, Y )) as p→ Y , where “dist” is thelocal Euclidean distance. Theorem 1.1 is deduced from Theorem 1.3.

Theorem 1.3. Let (Y,X;F) be as in Theorem 1.1, V a neighborhood of Y in Xand Φ: V → R ∪ ∞ a continuous function which is pluriharmonic on V \ Y , whereR ∪ ∞ is homeomorphic to the standard (0, 1] ⊂ R. Then, Φ is bounded from above ona neighborhood of Y .

Theorem 1.3 is shown by leading a contradiction to the maximum principle for therestriction of Φ on a dense leaf L of F in an invariant set containing Y . The existence ofsuch an invariant set is guaranteed by Theorem 1.4, see §4.1 for more details.

Theorem 1.4 ([P2], [P4], [P5]). Let f(z) = e2π√−1θz + O(z2) and g(z) = e2π

√−1tz +

O(z2), θ, t ∈ R, be local holomorphic diffeomorphisms which satisfy f g = g f . Assumethat t is an irrational number. Then, the following statements hold.

(i) For arbitrary small neighborhood U ⊂ C of 0, there exists a compact connectedsubset K of U such that 0 ∈ K, K 6= 0, C \ K is connected, and that K iscompletely invariant under f and g, i.e. f(K) = f−1(K) = g(K) = g−1(K) = K.

(ii) If g is linearizable at 0, then f is also linearizable by the linearization map of g.In this case, K can be chosen as the closure of a domain in U with 0 ∈ Int (K).

(iii) If g is non-linearizable at 0, then K contains 0 as the boundary point.(iv) There exists a point x0 of ∂K such that the orbit (gn(x0))n∈Z is dense in ∂K.

Theorem 1.4 is a combination of several facts in the theory of commuting holomorphicgerms fixing the origin 0. See the end of §2.1.

By imposing an additional condition on the holonomy of F , we have the statementabout plurisubharmonic functions. Compare with Theorem 1.3.

Theorem 1.5. Let (Y,X;F) be as in Theorem 1.1, V a neighborhood of Y in X andΦ: V → R ∪ ∞ a continuous function which is plurisubharmonic on V \ Y , whereR ∪ ∞ is homeomorphic to the standard (0, 1] ⊂ R. In addition, we assume that thereis a homotopically non-trivial closed curve c′ in Y along which the holonomy is of finiteorder. Then, Φ can be extended as a plurisubharmonic function on V which is boundedfrom above on a neighborhood of V .

Theorem 1.5 can be regarded as a weak analogue of Ueda’s theorem [U, Theorem 2]on the constraint of the increasing degree of plurisubharmonic functions. In [K2], the

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first author applied [U, Theorem 2] to show the non-semipositivity (i.e. non-existenceof a C∞ Hermitian metric with semi-positive curvature) of a line bundle L on X whichcorresponds to the divisor Y when (Y,X) is of type (α). As an application of Theorem1.5, we have the following:

Corollary 1.6. Assume that (Y,X;F) is as in Theorem 1.2. Let L be the line bundleon X which corresponds to the divisor Y . Then L is semi-positive (i.e. L admits a C∞

Hermitian metric with semi-positive curvature) if and only if the pair (Y,X) is of type(β).

We believe that Corollary 1.6 can be regarded as a supporting evidence of the followingconjecture for the line bundle [Y ] on X which corresponds to a divisor Y .

Conjecture 1.7 ([K4, Conjecture 1.1]). Let X be a complex surface and Y be acompact smooth curve holomorphically embedded in X such that the normal bundle istopologically trivial. The line bundle [Y ] admits a C∞ Hermitian metric with semi-positivecurvature if and only if the pair (Y,X) is of type (β).

Plan of the paper. The organization of the paper is as follows. In §2, we reviewsome fundamental results on linearization theorems, Siegel compacta, and Ueda theory.In §3, we divide the situation into ten cases and state a variant of some of our main resultsas Theorem 3.2. We here prove several cases in this theorem, which directly follow fromknown results. Moreover, we describe typical examples in some cases. The main part ofthis paper is §4. First, we show Theorem 1.3. Theorem 1.1 is deduced from this result.The proofs of Theorem 3.2 and Theorem 1.2 are also given here. In §5, we prove Theorem1.5. In §6, we prove Corollary 1.6. In §7, we discuss the dynamical behavior aroundnon-linearizable hedgehogs and give a slightly modified proof of the existence theorem ofcommon hedgehogs by commuting local holomorphic diffeomorphisms ([P5, Thm.III.14]).

Acknowledgment. The authors are grateful to Professor Eric Bedford for informingus about Perez-Marco’s theory on Siegel compacta. We also would like to thank ProfessorTetsuo Ueda for valuable comments and suggestions. Especially, some of the main the-orem could be improved by using Lemma 4.1, which we are taught by Professor TetsuoUeda. The first author is supported by Leading Initiative for Excellent Young Researchers(No. J171000201). The second author is partially supported by JSPS KAKENHI GrantNumber JP17K05283. They are also partly supported by Osaka City University Ad-vanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematicsand Theoretical Physics).

2. Preliminaries

2.1. Linearization theorems and Siegel compacta. In this section, we review somebasic properties of linearizability of local holomorphic diffeomorphisms around fixed points.Let f(z) = λz + O(z2) be a local holomorphic diffeomorphism fixing 0. We say that fis linearizable at 0 if there exist open neighborhoods U, V of 0, and a biholomorphismh : U → V such that (h f h−1)(z) = λz. It is classically known that the linearizabilityof f depends on the choice of λ. If |λ| 6= 0, 1, then f is linearizable (Koenigs’ linearization

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theorem). On the other hand, there are some obstructions for the case that |λ| = 1. Thefixed point 0 is said to be rationally indifferent (resp. irrationally indifferent) if |λ| = 1and λ is torsion (resp. non-torsion). In the irrationally indifferent case, f is linearizableif the argument (log λ)/2π

√−1 satisfies the Diophantine condition (Siegel’s linearization

theorem [Sie]). Further, this condition can be sharpened to the Brjuno condition. Themaximal linearization domain is biholomorphic to the unit disk, on which f is analyti-cally conjugate to a rotation. This is called the Siegel disk of f . Obviously, the domainis completely invariant under f , i.e. invariant by both of f and f−1.

On the other hand, in [P4], Perez-Marco showed the existence of completely invariantsets (not necessarily linearizable domains) around an indifferent fixed point. A Jordandomain U with C1-boundary is said to be admissible for f if f and f−1 are defined andare univalent on an open neighborhood of the closure U of U .

Theorem 2.1 ([P4, Theorem 1.1]). Let f(z) = λz + O(z2) be a local holomorphicdiffeomorphism with the indifferent fixed point 0. Let U be an admissible neighborhood of0. Then there exists a subset K in C which satisfies the following conditions :

(i) K is compact, connected, C\K is connected,(ii) 0 ∈ K ⊂ U ,

(iii) K ∩ ∂U 6= ∅, and(iv) K is completely invariant under f , i.e. f(K) = f−1(K) = K.

Moreover, assume that f is not of finite order. Then, f is linearizable if and only if0 ∈ IntK.

KU

z0

Figure 1. A hedgehog K of (U, f).

Perez-Marco called this completely invariant set a Siegel compactum, which can beregarded as a degeneration of Siegel disks in some sense. The Siegel compactum of thepair (U, f) is denoted by K(U,f).

Remark 2.2. At least, by Theorem 2.1 itself, the uniqueness of the Siegel compactumK(U,f) of the pair (U, f) is unclear. However, there is a canonical choice. Indeed, there is

a maximum of the set of all subset K as in Theorem 2.1, which is given by S, where

S = z ∈ U | fn(z) ∈ U for each integer n5

and S stands for the subset obtained by filling S with connected components of CP 1 \ Swhich do not contain ∞.

In the irrationally indifferent case, such an invariant set K is called a hedgehog if itcontains a relatively compact linearization domain in U or f is non-linearizable. Perez-Marco showed the existence of a hedgehog by perturbing the local diffeomorphism f andconsidering the Hausdorff limit of their Siegel disks. According to [P5, Theorem III.8],[P4, §V], and [P1, Theorem 5], sufficiently near the irrationally indifferent point 0, non-linearizable hedgehogs have no interior points and are not locally connected at any pointdifferent from 0. By considering the associated analytic circle diffeomorphisms, he studiedsome properties of Siegel compacta and found several applications (see [P4]). In particular,with regard to the dynamics on Siegel compacta, the following fact is remarkable.

Theorem 2.3 ([P4, Theorem IV.2.3.]). Let f be as in Theorem 2.1. For µK-a.e. pointz in K, the orbit of z is dense in ∂K. In particular, if f is non-linearizable, the orbit isdense in K. Here, µK is the harmonic measure at ∞ of K in CP 1.

We mention some results about commuting holomorphic germs fixing the origin 0.

Proposition 2.4 (e.g. [P2, §I.4]). Let f(z) = λz + O(z2), g(z) = µz + O(z2) be localholomorphic diffeomorphisms fixing 0 which satisfy f g = g f . Assume that µ ∈ C∗ isnon-torsion, i.e. µm 6= 1 for any m > 0. Then the following statements hold.

(1) If g is linearizable at 0, then f is also linearizable by the linearization map of g.

(2) If g is non-linearizable at 0, then λ = e2π√−1θ for some θ ∈ R. Moreover,

(a) f is linearizable at 0 if θ ∈ Q,(b) f is non-linearizable at 0 if θ ∈ R \Q.

The first part of the statement follows by comparing the coefficients of power series off and g. The second part is obtained by using Koenigs’ linearization theorem. The restpart also follows by the standard arguments. For more details, see e.g. [P2, §I.4].

In [P2] and [P5], Perez-Marco studied hedgehogs for commuting local holomorphicdiffeomorphisms. We state a slightly weaker version of his result, which is sufficient forour purpose. We give a modified proof in §7.

Theorem 2.5 (a part of [P5, Thm.III.14]). Let f(z) = λz + O(z2) and g(z) = µz +O(z2) be local holomorphic diffeomorphisms with the irrationally indifferent fixed point 0which satisfy f g = g f . Assume that g is non-linearizable at 0. Then, for any openneighborhood W of 0, there exists a compact subset K in W which is a common hedgehogof f and g. More precisely, there exist admissible neighborhoods U and V in W for f andg respectively such that K(U,f) = K(V,g) holds.

Proof of Theorem 1.4. For any neighborhood V of 0, we take a sufficiently small openneighborhood U which is admissible for f and g. If necessary, we will replace a smallerneighborhood. First, we assume that g is linearizable at 0. Proposition 2.4 (1) showsthat f is also linearizable by the linearization map of g. Thus, a Siegel disk K ⊂ U of gis also invariant under f . The subset K satisfies the statement (i) and (ii). Since g|K isconjugate to the irrational rotation, the statement (iv) follows in this case.

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Second, we assume that g is non-linearizable at 0. By Theorem 2.1, there exists ahedgehog K ⊂ U of (U, g). If θ is a rational number, then by Proposition 2.4 (2) (a), f isconjugate to the rational rotation on a domain W 3 0, in particular, f |W is of finite order,i.e. (f |W )N = id for some N > 0. As replacing a smaller neighborhood U if necessary, wemay assume that U is contained in the invariant domain W of f . Since f and g commute,f(K) is invariant under g, and it is a hedgehog of (f(U), g). By applying Lemma 7.4 to Kand f(K), we have K ⊂ f(K) or f(K) ⊂ K, and may assume that f(K) ⊂ K. Iteratingby f , we have

K ⊃ f(K) ⊃ f 2(K) ⊃ · · · ⊃ fN(K) = K

for some N , so that f(K) = f−1(K) = K. Thus, the subset K satisfies the statement (i)and (iii) by Theorem 2.1. If θ is an irrational number, Theorem 2.5 show the existence ofa common hedgehog K ′ of (U ′; f, g), U ′ ⊂ U , which satisfies the statement (i) and (iii).The remaining case of the statement (iv) follows directly from Theorem 2.3.

2.2. Review of Ueda’s neighborhood theory. Let X be a complex surface and Ya compact curve with the topologically trivial normal bundle NY/X . Fix a finite opencovering Uj of Y . Since Y is compact and Kahler, NY/X is U(1)-flat, i.e., the transitionfunctions on Ujk can be represented by U(1)-valued constant functions tjk (U(1) =t ∈ C | |t| = 1). Here Ujk = Uj ∩ Uk. Take an open neighborhood Vj of Uj in X andset V :=

⋃j Vj. As shrinking Vj, we can choose the defining function wj of Uj in Vj such

that (wj/wk)|Ujk≡ tjk.

For a system of such defining functions, the expansion of tjkwk|Vjk in the variable wj iswritten as

tjkwk = wj + f(n+1)jk (zj) · wn+1

j +O(wn+2j )

for n ≥ 1. Such a system is said to be of type n. Then it follows that (Ujk, f (n+1)jk )

satisfies the cocycle conditions (see [U, §2]). Denote the cohomology class by

un(Y,X) := [(Ujk, f (n+1)jk )] ∈ H1(Y,N−nY/X),

which is called the n-th Ueda class of (Y,X). The n-th Ueda class is an obstruction toexistence of a system of type (n+1). Indeed, it is not difficult to see that a type n systemcan be refined to be of type (n+ 1) if (and only if) un(Y,X) = 0. Therefore, the followingtwo cases occur:

• There exists a positive integer n such that the following holds:For any m ≤ n, there is a defining system of type m such that um(Y,X) = 0 form < n and un(Y,X) 6= 0.• For any positive integer n, there exists a defining system of type n such thatun(Y,X) = 0.

In the former case, the pair (Y,X) is said to be of finite type or of type (α) (more precisely,of type n). The latter case, we say, the pair (Y,X) is infinite type. For example, if Y admitsa holomorphic tubular neighborhood in X, then (Y,X) is infinite type. Here a holomorphictubular neighborhood means a neighborhood of Y in X which is biholomorphic to thatof the zero section of the normal bundle NY/X . More generally, consider the case wherethere exists a system wj as

tjkwk = wj.

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Namely, the U(1)-flat structure on the normal bundle NY/X can be extended to [Y ] aroundY , where [Y ] is the line bundle which corresponds to the divisor Y , i.e., there exists aneighborhood V of Y in X such that [Y ]|V is U(1)-flat. In such a case, we say that (Y,X)is of type (β). Note that, in this case, Y admits a pseudoflat neighborhoods system inX, that is, a neighborhoods system with Levi-flat boundary. The remaining case is calledtype (γ).

Remark 2.6. It does not change whether the type is finite or infinite after taking afinite covering space of a tubular neighborhood of Y , though the smallest number n ofnon-vanishing Ueda classes varies.

Ueda showed the following result:

Theorem 2.7. ([U, Theorem 3]) Suppose that the pair (Y,X) is of infinite type. If thenormal bundle NY/X ∈ Pic0(Y ) is torsion or satisfies the Diophantine condition, then Yis of type (β), that is, it admits a pseudoflat neighborhood system in X.

As previous results, Arnol’d first studied neighborhoods of elliptic curves embedded ina surface with topologically trivial normal bundle (see [A]). By regarding it as a kind oflinearization problem, he applied the technique of Siegel’s linearization theorem to thisproblem. Ueda’s theorem is a partial generalization of Arnol’d theorem.

When (Y,X) is of type (α), there are some results about the existence of strictlyplurisubharmonic functions on a neighborhood of Y and the constraint of its increasingdegree.

Theorem 2.8. ([U, Theorem 1, 2]) Suppose that the pair (Y,X) is of type n. Then thefollowing hold:

(i) For any real number a > n, there exist a neighborhood V of Y and a strictlyplurisubharmonic function Φ defined on V \ Y such that Φ(p) = O(dist(p, Y )−a)as p→ Y .

(ii) Let V be a neighborhood of Y . For any positive real number a < n and anyplurisubharmonic function Ψ defined on V \ Y such that Ψ(p) = o(dist(p, Y )−a)as p→ Y , there is a neighborhood W of Y in V such that Ψ|W\Y is constant.

Remark 2.9. In contrast, by definition, the curve Y of type (β) admits a holomorphicfoliation defined on an open neighborhood of Y and the holonomy along the compact leafY is U(1)-linear. Thus, there is a pluriharmonic function defined on V \Y which divergeslogarithmically toward Y .

Remark 2.10. The Ueda type cannot be specified without the unitarity conditionfor the linear part of the holonomy, even if the holonomy is linearizable. In [CLPT,remark 2.2], they constructed examples in the case III or VIII in §3 without the unitaritycondition, although they are of type (α).

3. Classification of the holonomies and examples

3.1. Classification of the holonomies. Let X be a complex surface and Y be an ellipticcurve embedded in X. Assume that there exist a non-singular holomorphic foliation F

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with Y as a compact leaf, defined on a neighborhood of Y in X. Take a point p ∈ Y andgenerators γ1 and γ2 of π1(Y, p) and we fix them. Consider the holonomies f := hF(γ1)and g := hF(γ2) of F along γ1 and γ2 respectively, with respect to a transversal τ at p.

According to the observation in [CLPT, Remark 2.2], it is natural to focus on the casewhere both λ = f ′(0) and µ = g′(0) are elements of U(1) (see Remark 2.10). In this case,the situation can be classified into ten cases I, II, . . . , X below in accordance with the(non-) torsionness of λ and µ and the (non-) linearizability of f and g. As the determiningproblem of Ueda’s classification types (α), (β), and (γ) is stable under the change of themodels by taking finite etale coverings (see Remark 2.6), we may assume that λ (resp. µ)is equal to 1 if λ (resp. µ) is a torsion element of U(1) in what follows.

Case I: λ = µ = 1, and f and g are linearizable.Case II: λ = µ = 1, f is linearizable, and g is non-linearizableCase III: λ = 1, µ is non-torsion, and both f and g are linearizableCase IV: λ = 1, µ is non-torsion, f is linearizable, and g is non-linearizableCase V: λ = µ = 1, and both f and g are non-linearizableCase VI: λ = 1, µ is non-torsion, f is non-linearizable, and g is linearizableCase VII: λ = 1, µ is non-torsion, and both f and g are non-linearizableCase VIII: both λ and µ are non-torsion, and both f and g are linearizableCase XI: both λ and µ are non-torsion, f is linearizable, and g is non-linearizableCase X: both λ and µ are non-torsion and both f and g are non-linearizable

µ ∈ U(1): torsion µ ∈ U(1): non-torsiong: linearizable g: non-linearizable g: linearizable g: non-linearizable

λ ∈ U(1) f : linearizable I II III IV: torsion f : non-linearizable V VI VIIλ ∈ U(1) f : linearizable VIII IX

: non-torsion f : non-linearizable X

Remark 3.1. For example in Case I, III and VIII, just the existence of the linearizationmaps ϕ, ψ ∈ Diff(C, 0) such that ϕ−1 f ϕ(w) = λ · w and ψ−1 g ψ(w) = µ · w isassumed and nothing on the relationship between ϕ and ψ is assumed literally. Howeverin reality, it turns out that f and g can be linearized simultaneously in these cases (i.e.we have that ϕ = ψ, see §2).

It suffices to consider only these ten cases from the symmetry. Each case is invariantunder conjugations, in particular, it does not depend on the choice of transversals. Onthe other hand, it does depend on the choice of generators of π1(Y, p).

Theorem 3.2. For the quadruple (Y,X;F , γ1, γ2), the following statements hold.

(i) In Case I, the pair (Y,X) is of type (β).(ii) In Case II, the pair (Y,X) is of type (α).

(iii) In Case III, the pair (Y,X) is of type (β).(iv) In Case IV, the pair (Y,X) is of type (γ).(v) In Case V, the pair (Y,X) is of type (α) or (β). Both pairs of types (α) and (β)

exist.(vi) No pairs (Y,X) is in Case VI.

(vii) No pairs (Y,X) is in Case VII.

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(viii) In Case VIII, the pair (Y,X) is of type (β).(ix) No pairs (Y,X) is in Case IX.(x) In Case X, the pair (Y,X) is of type (γ).

3.2. Proof of Theorem 3.2 except (iv) and (x). Here we prove Theorem 3.2 exceptthe assertions (iv) and (x). The proof of (iv) and (x) are one of the main parts in thepresent paper. We describe the proof of them in §4. By using the generators γ1, γ2 ofπ1(Y, p), here we identify the elliptic curve Y with the quotient C/〈1, τ〉 for a modulusτ ∈ H := z ∈ C | Im z > 0. In §3.2 and §3.3, we fix the generators of π1(Y, p)corresponding to the curves c(t) := z + t and c′(t) := z + τt, and keep the notations γ1and γ2 for them.

First, we show the assertion (i). Assume that (Y,X;F , γ1, γ2) is in Case I. In thiscase, both f and g are the identity. By considering the foliation chart corresponding tothis, we obtain a system wj of local (or even global) defining functions of Y . Thus thepair (Y,X) is of type (β).

Next, we show the assertion (ii). Assume that (Y,X;F , γ1, γ2) is in Case II. In thiscase, f is the identity. Note that NY/X is holomorphically trivial in this case. Let

g(w) = w +∞∑ν=2

bν · wν

be the expansion of g. Denote by n the minimum element of the set ν ∈ Z | ν ≥2, bν 6= 0. Then, the foliation chart of F gives a system wj of local defining functionsof type n − 1, which means that the pair (Y,X) is of type greater than or equal ton − 1. By definition, the (n − 1)-th Ueda class corresponds to (the conjugacy class of)the representation

ρ : π1(Y, p) ∼= Z⊕ Zτ → Cdefined by ρ(1) = 0 and ρ(τ) = bn under the natural identification H1(Y,N−n+1

Y/X ) =

H1(Y,OY ) = H0,1(Y,C) and the injection H0,1(Y,C) → H1(Y,C). Thus, we have thatun−1(Y,X) 6= 0, which means that the pair (Y,X) is of type n − 1. Therefore, the pair(Y,X) is of type (α).

The assertion (iii) is shown by the same argument as the proof of (i) above. The proofof (v) is given in §3.3. The assertions (vi), (vii), (viii), and (ix) follow from Proposition2.4.

3.3. Examples. Here we give some examples.

3.3.1. Examples of the case V. We describe examples of the case V to show the assertion(v) of Theorem 3.2. As NY/X is torsion in this case, we can apply Theorem 2.7 to concludethat the type of the pair is whether (α) or (β). By considering the example of [CLPT,Remark 2.2] with general choice of the representation c•, we obtain an example of thepair of type (α) in Case V. In what follows, we construct a pair of type (β) in Case V.Define an affine bundle V → Y over Y = C/〈1, τ〉 by

V = C2/

⟨(11

),

(ττ

)⟩,

10

or equivalently, V is the quotient C2/ ∼ of C2 with coordinates (x, ξ) by the relationgenerated by (x, ξ) ∼ (x+ 1, ξ + 1) ∼ (x+ τ, ξ + τ) (see also §3.3.2). The projection to Yis the one induced by the first projection (x, ξ) 7→ x. Let X be the ruled surface over Ywhich is a compactification of V by adding the infinity section. We will identify the infinitysection with Y by the natural manner and also denote it by Y , i.e. X = V ∪ Y . Denoteby F the foliation on X whose leaves are locally defined by the equation ξ = (constant).By regarding w = 1/ξ as a local defining function of Y , the holonomies f = hF(γ1) andg = hF(γ2) can be expressed as

f(w) =1

1w− 1

=w

1− wand

g(w) =1

1w− τ

=w

1− τw.

It follows from the direct calculation that this example (Y,X;F , γ1, γ2) is in Case V.

On the other hand, by considering another coordinate (x, ξ) of C2 defined by x = x and

ξ = ξ− x, we can easily see that X is biholomorphic to Y ×CP 1 and that Y correspondsto the subvariety Y × ∞ of Y × CP 1. Thus, we conclude that the pair is of type (β).

3.3.2. Suspension construction. We consider the suspension construction over Y = Cx/〈1, τ〉.Let f, g be elements in Diff(C, 0) which satisfies f g = g f and U an admissible domainfor both of f and g. Then, we take the quotient space of Cx × Cξ by the equivalencerelation generated by (x, ξ) ∼ (x + 1, f−1(ξ)) ∼ (x + τ, g−1(ξ)) for any x ∈ Cx andξ ∈ U ⊂ Cξ. Denote by π the projection to Y induced by the first projection (x, ξ) 7→ x.Then, we can choose a smooth tubular neighborhood X0 of Y with respect to π. Thisequips the holomorphic foliation F0 induced by ξ = constant which has Y as a compactleaf. The space X0 is not necessarily invariant by the leaves. The holonomies of F0 alongγ1 (resp. γ2) is given by f (resp. g) with respect to the transversal π−1(0). This modelautomatically satisfies the condition (1) which is appeared in Theorem 1.2.

As C∞ smooth foliations, the isomorphism classes of foliation germs along the compactleaf Y is determined by the conjugacy classes of holonomy homomorphisms (see e.g. [CC,Theorem 2.3.9]). Or equivalently, a foliation germ along the compact leaf Y is isomorphicto that of (X0, Y ;F0) whose holonomy is conjugate to the former one. However, such astatement does not hold in holomorphic settings. In [K3], the first author investigatedexamples of (X, Y ;F) which satisfy the conditions (1) and (2) appeared in Theorem 1.2.Under these conditions, the isomorphism classes of holomorphic foliation germs along Ycan be determined by information of the holonomies (see [K3, §2.2] for example).

At the end of this section, we describe some examples obtained by using the suspensionconstruction. We restrict ourselves to the case that f is the identity map (the condition(2)). First, if g is linearizable, then (X0, Y ;F0, γ1, γ2) belongs to Case I or III. Next,we provide two examples of the case where g is non-linearizable. If g is defined as

g(w) =w

1− w,

(X0, Y ;F0, γ1, γ2) gives a typical example of Case II, which is known as Serre’s example.See e.g. [K3, Example 4.1] for the detail. In [U, §5.4], Ueda constructed the first example

11

of type (γ). This can be explained as the case that g is a (monic) polynomial of degreed ≥ 2 such that µ is a non-torsion element of U(1) which satisfies “the strong Cremercondition” lim inf

n→∞An · |1 − µn|1/(dn−1) = 0. Ueda showed that, for each neighborhood Ω

of the origin, there exists a periodic cycle

w, g(w), g2(w), . . . , g`−1(w), g`(w) = wwhich is included in Ω for such a polynomial g (the existence of small cycles). As a fixedpoint with Siegel disk never have such a property, it is clear that such g is non-linearizable.Therefore, it follows that Ueda’s example belongs to Case IV.

4. Proof of main theorems

4.1. Proof of Theorem 1.3. Let c be a closed curve in Y as in Theorem 1.1. Take atransversal τ of F with a fixed parametrization and set f := hF(γ1) ∈ Diff(C, 0) whereγ1 = 〈c〉 ∈ π1(Y, p). By the assumption, the holonomy f is irrationally indifferent andnon-linearizable. As we mentioned in §1, the properties does not depend on the choice ofτ . Denote by g := hF(γ2) where π1(Y, p) ∼= Zγ1⊕Zγ2. Since f and g commute, it followsfrom Proposition 2.4 (2) (applying by switching the roles of f and g) that the holonomyg is either

(i) linearizable and rationally indifferent, or(ii) non-linearizable and irrationally indifferent.

Here we remark that, in the case (i), by changing the coordinate on τ and taking afinite covering space of X, one may read the following proof by assuming that g is theidentity.

In both of the cases, there exists a complete invariant set K (hedgehog) in τ under theaction of the subgroup Γ of Diff(τ, p) ∼= Diff(C, 0) generated by f and g. The existence isguaranteed by Theorem 1.4. As replacing a smaller transversal τ , we can get an arbitrarysmall invariant set K in τ . We may assume that the saturated set

F(K) :=⋃x∈K Lx

of K is included in a relatively compact subset W of V , where Lx is the leaf of F throughx. The set F(K) includes Y . By Theorem 1.4 (iii) and (iv), there is a leaf Lx0 such that

Lx0 = F(∂K) 3 Y,where F(∂K) :=

⋃x∈∂K Lx.

For sufficiently small K, the conformal type of leaves in F(K) is determined by thefollowing lemma, which was taught by Professor Tetsuo Ueda.

Lemma 4.1. For sufficiently small K, any leaf L of F(K) is parabolic, i.e. the universal

covering space L is biholomorphic to C.

Proof of Lemma 4.1. By replacing W with a smaller neighborhood of Y , we can choose asmooth retraction φ : W → Y whose restriction φ|L : L→ Y on each leaf L is an orienta-tion preserving local diffeomorphism. If necessary, we here retake a smaller transversal τ

12

to achieve that F(K) is included in W . Then, the dilatation

DF ,φ(p) :=φz(p) + φz(p)

φz(p)− φz(p)∈ [1,∞]

of φ along leaves is defined, where z is a complex coordinate on the leaf direction. (Moreprecisely, we fix a holomorphic universal covering map C→ Y and consider the derivatives

φz and φz of a lift φ : W → C of φ.) Note that the derivatives satisfy φz(p) → 1 andφz(p)→ 0 as p→ Y . Since Y is compact, the dilatation is bounded on F(K). Thus, therestriction φ|L : L→ Y on a leaf L ∈ F(K) is a quasiconformal.

The universal covering space L is biholomorphic to either D or C. The dilatation of a lift

φ|L : L→ Y = C is also bounded, so that it is a quasiconformal diffeomorphism. However,there are no quasiconformal homeomorphisms from D to C (see e.g. [N]). Therefore, it

follows that L is biholomorphic to C.

Let Φ: V → R ∪ ∞ be a continuous function which is pluriharmonic on V \ Y andi : Lx0 → F(K) ⊂ W (b V ) the inclusion, which is a holomorphic immersion. Then, the

pullback ϕ := (π i)∗Φ of Φ is a harmonic function on Lx0∼= C, where π is the universal

covering map π : Lx0 → Lx0 . Since ϕ is bounded from below, Liouville’s theorem showsthat ϕ is constant. Thus, Φ is constant on Lx0 = F(∂K) 3 Y . By the continuity of Φaround Y , Φ is bounded from above on a neighborhood of Y .

4.2. Proof of Theorem 1.1. Let Y be an elliptic curve embedded in a complex surfaceX and F a holomorphic foliation as in Theorem 1.1, defined on a neighborhood V of Y .By the Serre duality, for m ∈ Z,

H1(Y,N−mY/V ) ∼=

C (if N−mY/V = 1Y )

0 (if N−mY/V 6= 1Y ),

where we denote by 1Y the holomorphically trivial line bundle. It follows from the holo-nomy assumption and Proposition 2.4 (2) that NY/V is non-torsion, i.e. N−mY/V 6= 1Y for

any m > 0 (Consider a similar argument in the proof of the assertion (ii) of Theorem 3.2.Here, we use Proposition 2.4 (2) for assuring that the representation π1(Y, p)→ C whichcorresponds to the normal bundle is a unitary representation). Thus, Y is either of type(β) or of type (γ).

By assuming that (Y,X) is of type (β), we lead to contradiction. For an arbitrary smallneighborhood of Y , say V again, there exists a pluriharmonic function Φ: V \ Y → Rsuch that Φ(p) = O(− log dist(p, Y )) as p→ Y . In particular, this is bounded from belowon a compact neighborhood V ′ ⊂ V of Y . Then, Theorem 1.3 shows that Φ is boundedfrom above on V ′ ⊃ Y . This contradicts to the growth condition of Φ.

4.3. Proof of Theorem 3.2. The assertions (iv) and (x) follows from Theorem 1.1. See§3.2 for the proof of the others.

13

4.4. Proof of Theorem 1.2. As any g can be classified into only one of the assertions(i), (ii), or (iii), “only if” part follows from “if” part. Therefore here we only show “if”-part of each of the assertions.

The case where the modulus of µ := g′(0) is not equal to 1, the assertion is shown in[K3] (Indeed, it is shown that the pair is of type (β) in this case by using the assumption(1)). Therefore we may assume that |µ| = 1, and so the situation is as in §3, in whatfollows.

First, assume that µ is a torsion element of U(1). If g is of finite order, i.e. gn = id forsome n > 0, the situation is reduced to Case I in §3 by taking a finite covering space ofa tubular neighborhood of Y . In this case, it follows from Theorem 3.2 (i) that the pair(Y,X) is of type (β). If g is not of finite order, then it is non-linearizable at 0. Namely, itis in Case II. Thus, it follows from Theorem 3.2 (ii) (and the similar argument as above)that the pair is of type (α).

Next, assume that µ is a non-torsion element of U(1). If g is linearizable at 0, thesituation is in Case III. In this case, the pair is of type (β) by Theorem 3.2 (iii). If g isnon-linearizable at 0, the situation is in Case IV. It follows from Theorem 3.2 (iv) thatthe pair (Y,X) is of type (γ).

5. Proof of Theorem 1.5.

Let (Y,X;F) be as in Theorem 1.5. Consider a continuous function Φ: V → R∪∞,defined on a neighborhood V of Y , which is plurisubharmonic on V \ Y . Set

Ω := p ∈ Y | Φ(p) <∞.

First, we show that Ω 6= ∅ by assuming a contradiction. Second, we check that Ω = Yand Φ is extended as a plurisubharmonic function.

We take a transversal τ of F at p0 ∈ Y . Let c be a closed curve as in Theorem 1.1 andc′ as in the assumption of Theorem 1.5. After taking a finite covering space if necessary,we may assume that c and c′ generate the fundamental group π1(Y, p0). Note that theconclusion is not changed by this operation. Set f := hF(〈c〉) and g := hF(〈c′〉), where fis irrationally indifferent and non-linearizable and g is of finite order. As in the proof ofTheorem 1.3, there exists a hedgehog K of an admissible neighborhood U in τ which iscompletely invariant under the action of Γ = 〈f, g〉. Also as in the proof of Theorem 1.3,take a relatively compact neighborhood W of Y in V such that the saturated set F(K)is included in W . By the assumption, the set

VM := p ∈ V | Φ(p) > M,

is a neighborhood of Y in V , where M is a constant. By enlarging M , we may assumethat VM b W and UM := VM ∩ τ b U . Since K ∩∂U 6= ∅ (see Theorem 2.1), K \UM 6= ∅.

There exists a point x0 of ∂K such that the forward orbit

O+f (x0) := fn(x0) | n ∈ N

14

of f is dense in ∂K. Indeed, it follows from Theorem 1.4 (iii), (iv) (applying by switchingthe roles of f and g) and the recurrence property of orbits in K (see [P3, Corollaire 1]).By the holonomy condition, the leaf Lx0 through x0 is diffeomorphic to an open annulus.

Choose a point y0 ∈ K ∩ ∂U (note y0 /∈ UM) and a sufficiently small neighborhood D

of y0 in τ such that D is included in Int(W \ VM). Here D is the set defined as follows:When g is the identity,

D := (z, w) | z ∈ c′, w ∈ D,where we used coordinates (z, w) in a Stein neighborhood of Y \q0 in X where the pointq0 ∈ Y does not lie on the closed smooth curve c′, z is a local coordinate on Y \ q0, andeach leaf of F is defined by w = constant there (Or one may replace Y \ q0 with any

open neighborhood of c′ in Y ). When g is in general, D is defined as the same manner.The existence of such a neighborhood is guaranteed by Siu’s theorem [Siu]. Since 0 and

y0 is contained in O+f (x0), we can take positive integers n1 < n2 < n3 such that

fn1(x0), fn3(x0) ∈ D and fn2(x0) ∈ UM .

We here take the lift of the closed curve c to Lx0 through fn1(x0), denoted by c. Threepoints fn1(x0), f

n2(x0), fn3(x0) lie on the path c in this order with respect to the natural

orientation. Also, take the lift of the closed curve c′ to Lx0 through fnj(x0), j = 1, 3,which is expressed as

cj′ = (z, w) | z ∈ c′, w = fnj(x0).

It is included in D. Denote by A the subset of Lx0 bounded by c1′ and c3

′. It follows fromthe above construction that fn2(x0) lies on IntA.

Let i : Lx0 → W be the inclusion, which is a holomorphic immersion. Then, the functioni∗Φ is subharmonic and which satisfies i∗Φ|∂A < M by the construction. However, sincefn2(x0) ∈ VM , we obtain supA i

∗Φ|A > M . This contradicts to the maximum principle.

Next, we show that Ω = Y . By [D, Theorem 5.24], Φ is plurisubharmonic on a neigh-borhood of p in V for each p ∈ Ω. Since Y is connected and Ω is a non-vacuous opensubset of Y , it suffices to show that Ω is closed. Take a point q ∈ Ω and a coordinate(z, w) around q in V so that (z, w) = (0, 0) at q and Y = w = 0 on the locus. Considera sequence qν = (zν , 0)ν ⊂ Ω which tends to q as ν →∞. Then, for a sufficiently smallε > 0,

Φ(q) = Φ(0, 0) = limν→∞

Φ(zν , 0) ≤ limν→∞

1

2π

∫ 2π

0

Φ(zν , εe√−1θ) dθ.

For a fixed ε > 0, the sequence max0≤θ<2π Φ(zν , εe√−1θ)ν is bounded from above, so that

we have q ∈ Ω, i.e. Ω is closed. Therefore, Ω = Y holds, and the proof is complete.

6. Proof of Corollary 1.6

By [K2], L is not semi-positive when the pair (Y,X) is of type (α). Assume thatthe pair (Y,X) is of type (β). Then there exists a neighborhood V of Y such that Ladmits a unitary flat metric hV on a neighborhood V of Y (i.e. hV is a C∞ Hermitianmetric on L|V whose Chern curvature is 0, see §2.2). On the other hand, L admits asingular Hermitian metric hsing such that hsing|X\Y is a C∞ Hermitian metric on L|X\Y ,hsing →∞ holds when a point approaches to Y , and that the Chern curvature of hsing|X\Y

15

is 0. Indeed, the singular Hermitian metric defined by |fY |2hsing ≡ 1 satisfies this property,

where fY ∈ H0(X,L) is a section with div(fY ) = Y . A C∞ Hermitian metric h on L withsemi-positive curvature can be constructed by using the regularized minimum constructionfor these two metrics hV and hsing, which is the same construction as we used for proving[K1, Corollary 3.4]. This proves the semi-positivity of L when the pair (Y,X) is of type(β).

Therefore all we have to do is to show that L is not semi-positive assuming that thetriple (Y,X,F) is in Case IV, which is done by the same manner as in the proof of themain theorem in [K2] by using Theorem 1.5 instead of [U, Theorem 2].

In the cases we described in §3, Case X is the most interesting case from the viewpointof Conjecture 1.7:

Question 6.1. Does L admit a C∞ Hermitian metric with semi-positive curvaturewhen the pair (Y,X) is in Case X?

7. hedgehogs for commuting holomorphic diffeomorphisms

As we mentioned in §2.1, Theorem 2.5 (a part of [P5, Thm.III.14]) is useful for ourpurpose. In this section, we give a slightly modified proof, based on [P5]. To do this,we need several propositions. Proposition 7.1 below is obtained in [P3] and [P5], whichis a key proposition. See the references and also [Y] for the proof. Theorem 7.3 is alsoshown, which is one of the main applications of Proposition 7.1. We give a sketch of theproof. Furthermore, we prepare Lemma 7.2 and Lemma 7.4. Finally, Theorem 2.5 willbe proved by using Lemma 7.4 and Theorem 7.3. Note that this can be proved withoutTheorem III.4 in [P5]. In this section, we consider CP 1 with the Fubini-Study metric gFSas the ambient space.

Proposition 7.1 ([P3, Proposition 1], [P5, Proposition II.3]). Let g(z) = µz+O(z2) bea local holomorphic diffeomorphism with the irrationally indifferent fixed point 0. Assumethat g is non-linearizable at 0. Let U be an admissible domain of g and K a hedgehog of(U, g). Then, for each n ∈ N, there exists a quintuple (Ωn, Bn, ηn, Rn, An) associated withK, where Ωn is an open neighborhood of K in CP 1, Bn is a closed annulus in Ωn \ Kseparating ∂Ωn from K, ηn is a Jordan closed curve in the interior of Bn separatingtwo boundary components of Bn, Rn is a closed quadrilateral in Bn, and An is a closedannulus in Ωn \ (K ∪ Rn) separating Rn from K whose modulus tends to ∞ as n → ∞.The quintuple satisfies the following conditions:

(1) For each qj (j = 0, . . . , n), the iterations g±qj are defined on Ωn, where qn is givenby the continued fractional approximation (pn/qn)n∈N of the irrational number α =(log µ)/2π

√−1.

(2) For any point z in ηn, there exists an iteration gmn(z) which is contained in Rn.(3) For any point z in the component of CP 1 \ Bn, if there is an integer k such that

gk(z) is contained in the other component of CP 1\Bn, then there exists an iterationgkn(z) which is contained in Rn.

This is a rewrite of [P3, Proposition 1] as the statement for the hedgehog K througha uniformization map ψ : D → CP 1 \ K. In fact, he used this version in [P3, §3 and

16

Rn

z0

ηn

Bn

KΩn

ηnBn

Rn

Rn+1

Rn+2

z0

z

gk(z)

gkn(z)

gkn+1(z)

Figure 2. The separating annulus Bn, the meridian curve ηn of Bn, andthe quadrilateral Rn associated with a hedgehog K of g : The trappedsubsequence (gkn(z))n∈N in the statement (3) is depicted in the right figure.The quadrilateral Rn converges to a point z0 in K ∩ ∂U . See Lemma 7.2.

§4]. Let us denote the corresponding quintuple in D by (Ωn, Bn, ηn, Rn, An)n∈N. Note thesymbols which we use. Compare with the original statement of [P3, Proposition 1], the

two boundary components of Bn correspond to curves γ(n)0 and γ

(n)1 , ηn corresponds to

γ(n). Also, Rn and An correspond to Rn and An respectively.

The statements (2) and (3) imply that the dynamics around the hedgehog behaves as“quasi-rotation”. The closed curve ηn is called a quasi-invariant curve, that is, after someiterations, the curve returns to near the initial position in the sense of Hausdorff distancewith respect to the Poincare metric on CP 1 \K. For more details, see [P5], [Y], and [P6].

Lemma 7.2. Let (U, g) and (Ωn, Bn, ηn, Rn, An)n∈N associated with K = K(U,g) be as inProposition 7.1. Then the annulus Bn converges to K in the sense of Hausdorff conver-gence with respect to the Fubini-Study metric. Moreover, for any point z0 in K ∩ ∂U , wecan take the quadrilateral Rn such that it converges to the point z0.

Proof of Lemma 7.2. It is clear from the construction of Bn in the proof of [P5, Proposi-

tion II.3] or [Y] that the annulus Bn converges to the boundary ∂D and the correspondingannulus Bn converges to the hedgehog K. Therefore, we show only the latter statement.Choose any point z0 in K∩∂U . Since ∂U is of class C1, there is a path γ : [0, 1)→ CP 1\Uwhich lands at z0, that is, the limit limt→1 γ(t) exists and is z0. For a uniformization mapψ : D → CP 1 \ K, it follows that the path γ maps under ψ−1 to a path γ in D whichlands at some point on ∂D (see e.g. [M1, Corollary 17.10.]). The landing point is denoted

by p. For sufficiently large n, the closed annulus Bn intersects with the path γ. We can

choose a point qn = γ(tn) in Bn∩ γ for each n such that (tn)n∈N is an increasing sequence.

According to the construction of Rn in [P5] or [Y], we can construct a quadrilateral Rn

which contains the point qn. Note that qn converges to p as n → ∞. Let us return to

CP 1 \K under ψ. The quadrilaterals Rn and the path γ map to Rn and γ. See Figure 3.

17

Then, we apply the modulus inequality (cf. [LV, §6.4])

Mod(An) ≤ 2π2

`2

to the separating annulus An surrounding Rn, where ` is the infimum of the length ofclosed curves separating two boundary components of An with respect to the Fubini-Study metric gFS on CP 1. Since the modulus Mod(An) tends to ∞ from Proposition7.1, it follows from the standard argument that the boundary beside Rn degenerates to asingle point. Hence, so does the quadrilateral Rn. By the choice of Rn’s, it converges tothe point z0.

γ

Rn

Bn

Bn+1

Bn+2

Rn+1 Rn+2

∂D

K

U

γ

Rn

z0

An

Figure 3. There is a (sequence of) closed quadrilateral Rn intersectingwith the path γ, which is surrounded by a closed annulus An whose modulustends to∞ as n→∞. The construction in D (under a uniformization mapψ : D→ CP 1 \K) is depicted in the left figure.

Theorem 7.3 ([P3, Theorem 1], [P5, Theorem III.12.]). Let g(z) = µz + O(z2) be alocal holomorphic diffeomorphism with the irrationally indifferent fixed point 0. Assumethat g is non-linearizable. Then, the sequence (gn(z))n∈N does not converge to 0 as n→∞for any point z distinct from 0.

Here, we give a sketch of the proof of Theorem 7.3 following [P5]. Let U be an admissibledomain for g and K a hedgehog of (U, g). First, we take a point z ∈ K \ 0. It is knownthat any orbits in K are recurrent ([P3, Corollaire 1]), so that the sequence (gn(z))n∈Ndoes not converge to 0. Second, take a point z 6∈ K where g is defined and a pointz0 ∈ K ∩ ∂U 6= ∅. Let Bn and Rn be as in Proposition 7.1 and Lemma 7.2. Afterenlarging n, we may assume that the point z belongs to the component of CP 1 \ Bn

which does not contain K. If the sequence (gk(z))k∈N converges to 0, then it follows fromProposition 7.1 (3) that there exists a subsequence (gkn(z))n∈N such that gkn(z) ∈ Rn foreach n. See Figure 2. Therefore, by Lemma 7.2, the sequence (gk(z))k∈N accumulates atz0. This contradicts the assumption, so that the statement follows.

To show Theorem 2.5, we prepare the following lemma.

18

Lemma 7.4. Let g be as Proposition 7.1. Let K and K ′ be two hedgehogs of (U, g) and(U ′, g), where U and U ′ are admissible domains of g. Then, K ⊂ K ′ or K ′ ⊂ K hold.

Proof of Lemma 7.4. We prove this by contradiction. Assume that K 6⊂ K ′ and K ′ 6⊂ Khold. Set Dr = z ∈ C | |z| < r, D = D1, and AR = z ∈ C | R < |z| < 1. First, wetake a uniformization map

ϕ : D→ CP 1 \K ′.Consider an open neighborhood of K ′ as

Vε = ϕ(A1−ε) ∪K ′

for ε > 0. Since K 6⊂ K ′ and K is compact, there is δ > 0 such that K ⊂ Vδ andK ∩ ∂Vδ 6= ∅. Choose a point z0 ∈ K ∩ ∂Vδ. Note that z0 6∈ Vδ/2. We take theconnected component of U ∩ Vδ whose closure contains K. After a suitable smoothing ofthe boundary, the domain is an admissible domain of g, denoted by V . Note that K ⊂ Vand z0 ∈ K ∩ ∂V 6= ∅ still hold. Therefore, K can be regarded as a hedgehog of (V, g).Now apply Proposition 7.1 and Lemma 7.2 to the pair (K = K(V,g), z0). For each n ∈ N,take ηn and Rn as in the proposition.

Let us show that there exists an integer N such that Rn ∩ Vδ/2 = ∅ and ηn ∩ K ′ 6= ∅hold for any n ≥ N . The former statement follows directly from Lemma 7.2. On theother hand, the latter statement is shown by leading a contradiction. Assume that thereis a subsequence (nk)k∈N so that ηnk

∩K ′ = ∅. Similarly to the previous paragraph, takea uniformization map

ψ : D→ CP 1 \K,an open neighborhood of K as

V ′ε = ψ(A1−ε) ∪Kfor ε > 0, and choose δ′ > 0 such that K ′ ⊂ V ′δ′ and K ′ ∩ ∂V ′δ′ 6= ∅ hold. Here we used theassumption K ′ 6⊂ K and the compactness of K ′. Also, we choose a point z′0 ∈ K ′∩V ′δ′ . ByLemma 7.2, ηnk

is contained in V ′δ′/2 for large k. Jordan closed curve theorem shows that

ηnkdecomposes CP 1 into two domains W0 and W∞ such that ∂W0 = ∂W∞ = ηnk

, where0 ∈ W0 and ∞ ∈ W∞ hold. Note that z′0 belongs to W∞. By the assumption above, K ′

is contained in W0 ∪W∞. However, 0 ∈ K ′ ∩W0 6= ∅ and z′0 ∈ K ′ ∩W∞ 6= ∅ hold. Thiscontradicts the connectivity of K ′, thus the latter statement follows.

For sufficiently large n, take a point z1 ∈ ηn ∩ K ′ 6= ∅. It follows from Proposition7.1 (2) that there exists an iteration gmn(z1) contained in Rn. On the other hand, sinceRn ∩ Vδ/2 = ∅, the point gmn(z1) lies outside of Vδ/2. This contradicts the invariance ofK ′ under g.

Proof of Theorem 2.5. For any open neighborhood W of 0, there is a small R > 0 suchthat DR = |z| < R ⊂ W and f, g, g f, and f g are defined on DR, further, f andg are univalent on an open neighborhood of DR. For sufficiently small ε > 0 and anyr ∈ (0, ε), f(Dr) ⊂ DR and f−1(Dr) ⊂ DR. By Theorem 2.1, for each r ∈ (0, ε), there is ahedgehog Kr of (Dr, g). Since f and g commute, f(Kr) is also a hedgehog of (f(Dr), g).As applying Lemma 7.4 to these hedgehogs, we have Kr ⊂ f(Kr) or f(Kr) ⊂ Kr. Wemay assume that f(Kr) ⊂ Kr holds by exchanging f and f−1.

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The rest part is the same as the proof of [P5, Thm.III.14]. Iterating by f , which iswell-defined, we have the nested sequence

Kr ⊃ f(Kr) ⊃ f 2(Kr) ⊃ · · · .The set L =

⋂n=0

fn(Kr) satisfies the following properties:

(1) L is compact, connected, and C\L is connected,(2) 0 ∈ L,(3) L 6= 0, and(4) L is invariant under f, f−1, g, and g−1.

It is not difficult to show (1),(2), and (4). Let us show (3). If L = 0, then, for anyz ∈ Kr \ 0, the sequence (fn(z))n∈N converges to 0 as n → ∞. However, f is alsonon-linearizable from Proposition 2.4, so that this contradicts Theorem 7.3. Therefore Lis a common hedgehog of f and g.

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[P4] R. Perez-Marco, Fixed points and circle maps, Acta Mathematica, 179 (1997), No.2, 243–294.[P5] R. Perez-Marco, Hedgehog dynamics, manuscript.[P6] R. Perez-Marco, On quasi-invariant curves, arXiv:1802.06726.[Sie] C. L. Siegel, Iterations of analytic functions, Ann. of Math., 43 (1942), 607–612.[Siu] Y. T. Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math., 38 (1976), 89–100.[U] T. Ueda, On the neighborhood of a compact complex curve with topologically trivial normal bundle,

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1 Department of Mathematics, Graduate School of Science, Osaka City University,3-3-138 Sugimoto, Sumiyoshi-ku,, Osaka 558-8585, Japan

E-mail address: tkoike@sci.osaka-cu.ac.jp

2 Department of Mathematics, Tokai University, 4-1-1 Kitakaname, Hiratsuka-shi,,Kanagawa 259-1292, Japan

E-mail address: nogawa@tsc.u-tokai.ac.jp

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