Holomorphic Anosov Systems, Foliations and Fibering

Holomorphic Anosov systems, foliations and ﬁbering complex manifolds

B. Azevedo Sc´ardua Instituto de Matem´atica Universidade Federal do Rio de Janeiro. C.P. 68530 Rio de Janeiro-RJ. 21945-970 - Brazil

Abstract We study possible versions of Tichler’s ﬁbration theorem ([21]) for the complex case. Holo- morphic Anosov ﬂows are studied as well, under the hypothesis that they preserve a codimension one holomorphic foliation.

1 Introduction

The well-known Theorem of Tischler [21] states that a closed real differentiable manifold M fibers overs the circle S1 if, and only if, M supports a non-singular closed differentiable 1-form Ω. Such a 1-form defines a codimension one foliation F, without holonomy, which is invariant under the transverse flow ϕt of a vector field X, defined on M, and satisfying Ω · X = 1. Conversely, any foliation F of codimension one which is invariant under the transverse flow ϕ of a vector field X on M must be given by a closed 1-form Ω with Ω · X = 1 (see [8] pages 45-47 and [18] Proposition 2.3 pages 737-738). As a consequence, the foliation F is either a compact foliation or has all leaves dense in M (cf. [18] Proposition 2.7 page 741). In [18] the author applies these techniques in the study of differentiable Anosov flows. It is proved therein that any jointly integrable Anosov flow in M admits a smooth section and is topologically conjugate to the suspension of some Anosov diffeomorphism, which is a toral automorphism in the codimension one case (see Theorem 3.1 page 744 and Theorem 3.7 page 746). We recall that an u s T Anosov flow ϕt : M → M with corresponding splitting TM = E ⊕ E ⊕ E is jointly integrable if the bundle Eu ⊕ Es is integrable, generating therefore a codimension one foliation F which is invariant under the transverse flow ϕt ; giving this way the link with Tischler’s Theorem above. Finally, in [6] the author states the definition of holomorphic Anosov flow on a complex manifold in terms of actions of the multiplicative group C∗ (cf. [6] page 586). In this same work, holomorphic Anosov flows on compact complex 3-manifolds are classified. This is achieved using strongly the fact that the stable and unstable foliations, F s and F u, are holomorphic foliations with projective transverse structures. The holomorphy of F s is a consequence of the dimension 3 assumption. These are the main motivations for the present work. More precisely, we are interested in the

2000 Mathematics Subject Classification. Primary Primary 32L05; Secondary 37F75. Key words and phrases. Tischler fibration theorem, Anosov flow, holomorphic foliation.

1 following questions. (i) Let M be a compact complex manifold supporting a non-singular closed holomorphic 1-form Ω. Under which conditions M fibers over a complex 1-torus? Denote by F the codimension one foliation given by T F = Ker(Ω) ⊂ TM. What is the structure of the leaves of F? (ii) Consider now M a compact complex manifold admitting a codimension one holomorphic foliation F invariant under some holomorphic transverse flow ϕt. What is the transverse dynamics of F? Unlike the real case, (i) and (ii) above are not equivalent situations (cf. Example 3); a foliation F invariant under some transverse flow ϕ as in (ii) is given by a closed 1-form Ω as in (i). Given any leaf L0 of F (always as in (ii)), we have a holomorphic covering L0 × C → M, whose group A is isomorphic to the group of periods of Ω (Lemma 2). Thus, 2 ≤ rank A ≤ rank H1(M; Z) (Corollary 1) and therefore L0 is compact if, and only if, A is (naturally isomorphic to) a rank 2 lattice in C (Lemma 2); in this case F is compact (Corollary 2). This is certainly the case if rank H1(M; R) = 2 (Corollary 2 and 4). When F is compact the manifold M is actually a holomorphic fibre bundle over a one-dimensional complex torus C/Λ, where Λ ⊂ C is some lattice, and the fibers of the bundle are the leaves of F (Theorem 1). Definition 1 (Holomorphic fibre bundle). We shall say that a holomorphic manifold M is a holomorphic fibre bundle over a complex manifold N if there exists a holomorphic submersion f : M → N which defines a locally holomorphically trivial bundle structure. Let M be a connected complex manifold. Assume that M admits a proper holomorphic submersion f : M → R onto a complex manifold R. In this case, according to Ehresmann Theorem [8] M is a locally trivial C∞-fibre bundle over R, whose fibers are the connected components of the levels f −1(y), y ∈ R of f. In particular, these levels are C∞-diffeomorphic; usually they are not holomorphically equivalent. In general a compact manifold M equipped with a non-singular closed 1-form admits a C∞-fibration over the real torus S1 × S1 (Proposition 1) and this is the best result (cf. Remark 3 and Example 4). It turns out that the questions addressed above may admit more precise answers in the case M is a compact Kählersurface or a compact homogeneous manifold; whose basic framework is introduced in §2. We prove for instance that (cf. Theorem 2) if M n+1 is a compact Kählermanifold supporting a codimension one foliation F invariant under the transverse holomorphic flow of X and such that the one dimensional foliation FX defined by X has a trivial normal bundle; i.e., FX is defined by n holomorphic 1-forms ω1, ..., ωn on M; then M fibers over a complex torus of dimension one or M is a torus itself (and F is linear). For the case M is a compact homogeneous manifold, we observe that F is compact or transversely projective in M (Proposition 4). The techniques introduced in §§2 and 3 apply to the study of jointly-integrable holomorphic Anosov flows and codimension one holomorphic Anosov diffeomorphisms. This is done in §4. After recalling some basic examples and introducing some features from [6] we prove that if a compact Kählermanifold M supports a codimension one transitive holomorphic Anosov diffeomorphism f : M → M then M is (holomorphically conjugate to) a complex torus C/Λ, for some lattice Λ ⊂ Cn, and f is (holomorphically conjugate to) a linear automorphism of Anosov type (Theorem 3 §4). The study of jointly integrable holomorphic Anosov flows is carried out in §§4.2

2 and 4.3 where we obtain the expected classification result stating that the manifold is a holomorphic torus fibre bundle over some 1-torus C/Λ and the flow is topologically conjugate to the suspension of a holomorphic toral Anosov diffeomorphism (cf. Theorem 4 §4.3). The questions addressed in the previous sections (§§2,3 and 4) are also interesting in the non- compact case which is considered in §5. The first case to be considered is the case of foliations with singularities on compact manifolds. Let therefore F be a codimension one singular holomorphic foliation on a complex manifold M. We may assume that the singular set sing(F) of F has codimension at least 2 in M (recall that sing(F) ⊂ M is always an analytic subset). Using Hartogs’ Extension Theorem [10] we first observe that if F invariant under some holomorphic transverse flow on M\ sing(F) then indeed F is non-singular (sing(F) = ∅) and we are in the same situation of Lemma 1 of §2 (see Lemma 8 §5.1). Motivated by this and by some concrete examples as, for instance, linear foliations on complex projective spaces (cf. Example 6 in §5.1) we are led to consider the following situation: F is a codimension one foliation with singularities on M, there exist an analytic codimension one subset Γ ⊂ M and a holomorphic vector field Z in M such that Z is transverse to F in M\Γ and F|M\Γ is invariant by the (transverse) flow ϕt of Z in M\Γ. The non-trivial case arises when Γ is simultaneously F and Z invariant (Remark 11 §5.1). For this situation we prove a result analogous to Lemma 1, stating that F is given by a closed meromorphic

1-form Ω on M with (Ω)∞ = Γ and Ω · Z = 1 in M (Lemma 9). Let also M be compact and L0 ⊂ M\Γ any leaf of F. We prove that (analogously to Lemmas

2 and 3 of §2) the dynamics of F|M\Γ is classiﬁed by the structure of the group of periods of the 0 restriction Ω|M\Γ : there exists a holomorphic covering L0×C → M\Γ whose group A is (isomorphic to) the group of periods Per(Ω|M\Γ). In particular, if H1(M\Γ, R) = 0 then Per(ΩM\Γ) = 0 and F is a singular compact foliation given by a holomorphic map f : M → CP (1) (Proposition 5). ∞ The leaf space of F|M\Γ is C -diﬀeomorphic to a complex torus if and only if Per(Ω|M\Γ) is a lattice in C, in which case F|M\Γ has closed leaves on M\Γ. These leaves are also closed in M if rank Per(Ω|M\Γ) = 1. In any other case the leaves of F are not closed in M\Γ.

2 Invariant ﬂows and closed 1-forms

Throughout this section F will denote a (non-singular) codimension one holomorphic foliation on a complex connected manifold M of dimension n ≥ 2.

Definition 2. Let ϕ: C×M →¯M be a holomorphic flow on M. We say that ϕ is a flow transverse ¯ ∂ϕ ¯ to F if the vector field Z = ∂t ¯ (where t ∈ C is the complex time) is transverse to (the leaves t=0 of) F. We say that F is invariant under the flow ϕ if each flow map ϕt : M → M takes leaves of F onto leaves of F. Also, we shall say that F is invariant under the transverse flow of Z, if Z is a complete holomorphic vector field on M, whose corresponding flow ϕ is transverse to F, and F is invariant under ϕ.

3 Example 1. Let M be a complex torus, M = Cn/Λ where Λ ⊂ Cn is some lattice. Let Fe be the foliation on Cn by hyperplanes parallel to a given direction Ze ∈ Cn. Then Fe induces a foliation F on the quotient M = Cn/Λ which is called a linear foliation on the torus M. Such a foliation is invariant under the transverse flow given by a vector field Z whose lift to Cn is Ze. As it is easily checked, F is given by a (non-singular) closed holomorphic 1-form Ω on M, with constant coefficients.

The following (classic real) result states the existence of Ω as a general fact:

Lemma 1. Let F be invariant by the transverse holomorphic flow ϕR of Z on M. Then F is t2 given by a non-singular closed holomorphic 1 form Ω characterized by Ω(ϕt(x)) · Z(ϕt(x)) dt = R t1 t2 t2 − t1, ∀ x ∈ M, ∀ t1, t2 ∈ C; where the integral is along any simple path γ : [0, 1] → C with t1 γ(0) = t1 γ(1) = t2 . In particular, the holonomy of F is trivial. This lemma is proved, in the present holomorphic framework, like Proposition 2.3 page 742 in [18]. It is a well-known fact that any holomorphic form on a compact Kählermanifold is closed, nevertheless not all Kählermanifolds support non singular closed holomorphic 1-forms (eg. M = CP (n) ).

0 1 Example 2. Let M0 be a compact Kählermanifold with dimC H (M0;Ω ) = 0, for instance M0 = CP (m), and R a compact Riemann surface of genus 1, that is, a complex 1-torus C/Λ (we recall that R is holomorphically equivalent to a non-singular cubic in CP (2), given by an affine equation of the form y2 = x(x − 1)(x − λ), λ ∈ C the lattice Λ = {1, λ} generates R as a complex torus C/Λ). For the product M = M0 × R we have: Claim 1. A codimension one (non-singular) holomorphic foliation F on M is invariant under some holomorphic transverse flow if, and only if, F is the fibration M → R, (x, y) 7→ y.

Proof. Let F be given by a closed holomorphic 1-form Ω in M and let w ∈ H0(R;Ω1) be any generator. Denote by π : M → R the second coordinate projection π(x, y) = y and w∗ = π∗(w) ∈ H0(M;Ω1). By the Kunneth formula and Hodge Theory ([12]) we have £ ¤ £ ¤ 0 1 ∼ 0 0 0 0 1 0 0 H (M;Ω ) = H (M0;Ω ) ⊗ H (,R, Ω) ⊕ H (M0;Ω ) ⊗ H (R;Ω ) .

0 1 ∼ 0 1 0 1 Therefore H (M;Ω ) = H (R;Ω ) and in particular we have dimC H (M;Ω ) = 1. Thus, necessarily we have Ω = λ · w∗ for some λ ∈ C∗ and Ker w∗ = Ker Ω, so that T F = Ker w∗ what implies that F is the ﬁbration with projection π : M → R,(x, y) 7→ y. ¥ In particular, for M = CP (m) × R we get π1(M) ' π1(R) = Z ⊕ Z as well as H1(M; Z) ' H1(R; Z) ' Z ⊕ Z. Finally, for the groups of periods we have Per(Ω) ' Per(w) = Λ ' Z ⊕ Z. Compare this example with the result of Corollary 1.

4 Example 3. Let Ω = x dy + y dx in C2. Then Ω is closed holomorphic and singular at the origin. We may write Ω = d(xy) and look for vector fields Z with Ω · Z = 1. Let Z = ∂ , that is µ ¶ ∂(xy) 1 ∂f 1 1 ∂f 1 ∂f 2 Z(f) = 2 ∂(xy) = 2 y ∂x + x ∂y , for all germ f of holomorphic function at the origin 0 ∈ C . µ ¶ 1 1 ∂ 1 ∂ Then Z = 2 y ∂x + x ∂y so that Z is meromorphic with poles along the zeroes of the local first integral ϕ = xy for Ω. Clearly there exists no holomorphic vector field Z in a neighborhood of the origin such that Ω · Z = 1; Hartogs’ Extension Theorem [10] assures then that there exists no such holomorphic Z defined in a punctured neighborhood of the origin. Thus if we take M = C2 −{(0, 0)} then we see that it is not possible to state a straightforward converse of Lemma 1. Nevertheless, for the case of suitable complex surfaces we have the following converse: Claim 2. Let Ω be a non-singular closed holomorphic 1-form on a complex surface M 2 with H0,1(M) = 0 for the Doulbealt cohomology and assume that the foliation F defined by Ω is given by ∂ a non-singular holomorphic vector field ZF on M. Then there exists a holomorphic vector field Z in M such that Ω · Z = 1. If Z is complete then F is invariant under the holomorphic transverse flow defined by Z.

Proof. Since Ω is closed and non-singular we can find an open cover M¯= ∪jUj such that each Uj 2 is the domain of a holomorphic chart (xj, yj): Uj → C and such that Ω¯ = dyj in Uj. We define Uj ∂ therefore Zj = in Uj obtaining this way a local holomorphic vector field with Ω · Zj = 1 in Uj. ∂yj In each intersection Ui ∩ Uj 6= ∅ we have Ω · (Zi − Zj) = 0 and therefore Zi − Zj = aij.ZF for some holomorphic function aij : Ui ∩ Uj → C. Clearly the collection {aij} gives an additive cocycle on M and by the trivial cohomology hypothesis we may obtain holomorphic functions aj : Uj → C such that aij = ai − aj in Ui ∩ Uj 6= ∅. Define therefore the vector field Z on M by setting Z = Zi − aiZF in each Ui. Clearly Z is holomorphic and satisfies Ω · Z = 1. If Z is complete, then, since we have Ω · Z = 1, the Lie derivative of dΩ with respect to Z satisfies LZ (Ω) = d(Ω · Z) + iZ (dΩ) = d(1) = 0 because dΩ = 0. Therefore Ω = 0 is invariant under the transverse flow of Z. ¥ Now we return to the general case and state

Lemma 2. Let F, ϕ, Z, Ω be as in Lemma 1 and assume M is compact. Given any leaf L0 of F there exist a holomorphic covering σ : L0 × C → M, σ(x, t) = ϕt(x), and an exact sequence of σ# groups 0 −→ π1(L0 × C) −→ π1(M) −→ A −→ 0, where A is a ﬁnitely generated free abelian group. Moreover, L0 is compact if, and only if, A is (naturally) isomorphic to a rank 2 lattice on C.

Proof©. We follow theR sketch ofª the proof given for Theorem 2.4 page 238 in [18]. Consider the group H = [γ] ∈ π1(M); γ Ω = 0 , then H is a normal free subgroup of π1(M). Put A := π1(M)/H then A is ﬁnitely generated and abelian because H ⊃ [π (M), π (M)] (the subgroup of commutators R R 1 1 f of π1(M)) for γ∗δ Ω = δ∗γ ∀ δ, γ ∈ π1(M). Let P : M → M be the holomorphic covering of M, e e f corresponding to H. Let also F, Ω and ϕet be the lifting of F, Ω and ϕt to M respectively. By

5 e ˜ ˜ f ˜ ˜ construction Ω = df for some holomorphic function f : M → C and we have f(ϕet(˜x)) = t + f(˜x) f e f e e ∀ t ∈ C, ∀ x˜ ∈ M. Given any leaf L0 of F on M let L0 ⊂ M be a leaf of F such that P (L0) = L0 . e f Define the map g : L0 × C → M by setting g(˜x, t) = ϕet(˜x). e f Claim 3. The map g is a holomorphic diffeomorphism of L0 × C onto M. Proof. Since the flow of Z is transverse to F it follows that the flow ϕe is transverse to Fe so that g is e a local diffeomorphism in L0 × C. Now we notice that if g(˜x1, t1) = g(˜x2, t2) then ϕet1 (˜x1) = ϕet2 (˜x2) ˜ ˜ ˜ ˜ and f(ϕe(t1, x˜1)) = f(ϕe(t2, x˜2)) so that ϕet1−t2 (˜x1) =x ˜2 and t1 + f(˜x1) = t2 + f(˜x2). The points e e e x˜1 andx ˜2 belong to the same leaf Lx˜0 = L0 of F so that f(˜x1) = f(˜x2), this implies t1 = t2 and e thereforex ˜1 =x ˜2 . Therefore g is also injective and it is a diffeomorphism of L0 × C onto its image e f e f g(L0 × C) ⊂ M. It remains to prove that g(L0 × C) = M. It is enough to prove that this image f e f e of g is closed. Take any pointx ˜1 ∈ M belonging to the closure of g(L0 × C) in M. Let B1 3 x˜1 e e e S e be any open ball in the leaf L1 3 x˜1 . Let U be the “cylinder” U = ϕet(B1), and take any t∈C e e e x˜ ∈ U ∩ g(L0 × C). We havex ˜ ∈ ϕes(L0) for some s ∈ C and also there exists r ∈ C such that e e e x˜ ∈ ϕer(B1). Thusx ˜1 ∈ ϕes−r(L0) and hencex ˜1 ∈ g(L0 × C). This proves the claim. ¥ e f Claim 4. P | : L0 ⊂ M → L0 ⊂ M is a bijection and therefore a diffeomorphism. Le0 e Proof. P is injective, for ifx ˜1, x˜2 ∈ L0 are such that P (˜x1) = P (˜x2) then we may take a 1 pathα ˜ : [0, 1] → L0 of class C withα ˜(0) =x ˜1 andα ˜(1) =x ˜2 . This gives a projected path α = P ◦ α˜ : [0, 1] → L ⊂ M which is closed, i.e., α ∈ π (L ). We have 0 = d (f˜(˜α(t))) = 0 1 0 R dt R e 0 0 1 0 Ω(˜α(t)) · α˜ (t) so that Ω(α(t)) · α (t) = 0, ∀ t ∈ [0, 1]. Hence 0 = 0 Ω(α(t)) · α (t) dt = α Ω, so f that [α] ∈ H ⊂ π1(M). This givesα ˜(0) =α ˜(1) in M, i.e.x ˜1 =x ˜2 . ¥ f Let now ηt : L0 ¡× C → L × C be¢ given by ηt(x, s) := (x, s + t). Let also G: L0 × C → M be defined by G(x, s) := g (P | )−1(x), s . Consider the following diagram Le0

M ←−−−P Mf ←−−−G L × C   0     ϕty ϕety y ηt P f G M ←−−− M ←−−− L0 × C

The left side is commutative by construction. Now we observe that, given (x, s) ∈ L0 × C,

¡¡ ¢−1 ¢ ¡¡ ¢−1 ¢ ¡ ¡¡ ¢−1 ¢¢ G(ηt(x, s)) = g P | (x), s + t = ϕes+t P | (x) = ϕet ϕes P | (x) Le0 Le0 Le0 and therefore G(ηt(x, s)) = ϕet(G(x, s)) = (ϕet ◦ G)(x, s). Consequently the whole diagram is commutative.

6 Deﬁne now σ := P ◦ G: L0 × C → M by requiring that the diagram below is commutative:

M ←−−−σ L × C  0    ϕty y ηt σ M ←−−− L0 × C

In other words ϕt ◦ σ(x, s) = σ(x, s + t). Therefore, the map σ : L0 × C → M is a covering map.

Clearly σ(x, t) = ϕt ◦ σ(x, 0) = ϕt(x), ∀ t ∈ C ∀ x ∈ L0 . Therefore σ satisﬁes the ﬁrst condition in the statement of LemmaR 2.R If for anyR [γ] ∈ π1(M) weR have [γ] = σ#([α]) in π1(M), for some ∗ ∗ [α] ∈ π1(L0 × C) then Ω = Ω = σ (Ω) = (ϕt) (Ω) = 0 for α ⊂ L0 × C and Ω = 0 γ σ◦α α⊂L0×C α f along L0, so that [γ] ∈ H. Conversely, if [γ] ∈ H then γ = P#(γe) for some γe ∈ π1(M) and therefore e we have γe = g#(αe) for some αe ∈ π1(L0 × C) so that γ = P#(g#(αe)) ⇒ γ = (P ◦ G)#(α) where α = P ◦ αe ∈ π1(L0 × C) is obtained in a natural way. AnotherR way of seeing the above equivalence is the following: if [γ] ∈ H then γ ∈ π (M) is such that Ω = 0. Therefore we may consider the 1 Rγ ∗ lifting α of γ by σ to L0 × C obtaining a path such that α(ϕt) (Ω) = 0 and therefore α is closed that is, [γ] = σ#([α]) for [α] ∈ π1(L0 × C). Therefore we have proved the following:

σ# Claim 5. The sequence 0 −→ π1(L0 × C) −→ π1(M) −→ A −→ 0 is exact. R Remark 1. We have a group homomorphism ξ : π1(M) → (C, +), [γ] 7→ γ Ω, whose kernel is H, and thus we have an injective homomorphism ξ : A = π1(M)/H → C. Hence, A is naturally identified to the subgroup Per(Ω) of (R2, +). We may now complete the proof of Lemma 2. First we assume that rank A = 0. In this f case H = π1(M) and P : M → M is the universal covering of M. Since A = {0} we have a σ diffeomorphism M ' L0 × C what is not possible because M is compact. Assume now that A has rank one, A ≈ Z. We may take a transformation T : L0 × C → L0 × C which corresponds to a generator of A = π1(M)/H (notice that the covering σ : L0 × C → M has group isomorphic to σ# π1(M)/H because of the exact sequence (0 → π1(L0 × C) −→ π1(M) → π1(M)/H → 0). The diagram T L0 × C −→ L0 × C σ &. σ M ¯ commutes. Let T0 := T ¯ : L0 × {0} → L0 × C. Therefore there exists t0 ∈ C − {0} such L0×{0} that T (L0 × {0}) = L0 × {t0}. If t0 = 0 then b(x, 0) = 0, ∀ x ∈ L0 and T (x, 0) = (a(x, 0), 0) and also form (*) a(x, 0) = x ∀ x ∈ L0 . Thus T (x, 0) = (x, 0), ∀ x ∈ L0 . This is not possible ∗ for T is a non trivial covering transformation. Define now a map f : M → C/t0Z ' C by setting f(x) := s (mod t0) where x ∈ ϕs(L0). Since the group of covering maps of σ is generated by T

7 ∗ we conclude that f : M → C = C/t0Z is well-defined. Since ϕt takes leaves of F onto leaves, f is constant along the leaves of F. Thus f is a holomorphic first integral for F. Since M is compact it follows that f is constant and Ω ≡ 0; this case does not occur therefore. Thus we must have rank A ≥ 2. If A has rank ≥ 3 then Per(Ω) is not a discrete subgroup of (R2, +) and therefore F has no closed leaf, in particular Lo is not compact. Thus if F has some compact leaf Lo then rank A = 2. Let therefore A have rank two, A ≈ Z ⊕ Z. As above we may prove that ∃ s0, t0 ∈ C such that T (L0 × {0}) = L0 × {t0} and S(L0 × {0}) = L0 × {s0} with t0 6= 0 6= s0 and s0, t0 linearly independent over Z. Suppose s0 and t0 are R-linearly independent. Then {s0, t0} ⊂ C defines a lattice. Define now f : M → R where R is the complex torus given by the lattice s0 Z⊕t0 Z ⊂ C; and f is given by f(x) := τ mod (s0 Z ⊕ t0 Z) where τ ∈ C is such that x ∈ ϕτ (L0). Then f : M → R is well-defined and holomorphic, constant along the leaves of F on M. Since M is compact, f is ∞ proper so that L0 (as well as any leaf of F) must be compact and f : M → R defines a C bundle structure for M with fiber (diffeomorphic to) L0 , over R. If s0 , t0 are R-linearly dependent then the group s0 Z + t0 Z ⊂ (C, +) is not discrete. Therefore L0 is not closed in M and indeed no leaf L of F is closed in M. Conversely, assume that L0 is a compact leaf of F. Since F has trivial holonomy the holomorphic version of Reeb’s Stability Theorem (see [4]) implies that all the leaves of F are compact and holomorphically equivalent. Now, the group A acts on L0 ×C taking leaves of σ∗(F) onto leaves of σ∗(F) is a natural way as above. Therefore, since F is a compact foliation, the ∗ leaves of σ (γ) are closed on L0 × C and therefore the action of A must be discrete so that indeed, A must correspond to a discrete subgroup of C and therefore rank A ≤ 2 as it is well-known. Thus we have proved that A is (isomorphic to) a rank 2 lattice iff L0 is compact iff all leaves of F are compact. ¥

Corollary 1. Let M be compact, F, ϕt, A as in Lemma 2. Then 2 ≤ rank A ≤ rank H1(M; Z). In particular, F is compact if rank H1(M; Z) = 2.

Proof. The inequalities follow immediately from the above proof for Lemma 2. If rank H1(M; Z) = 2 then A is a rank two lattice on C and, by Lemma 2, F is compact. ¥ Next step is the following:

Lemma 3. Let F, ϕt , Ω, A and M be as in Lemma 2. Let Per(L0) := {t ∈ C; ϕt(L0) = L0} and ηt : L0 × C → L0 × C be given by ηt(x, s) = (x, s + t). (i) If T : L0 × C → L0 × C is a covering transformation of the covering σ : L0 × C → M, (x, t) 7→

ϕt(x), then T (L0 × {t}) = ηt0 (L0 × {0}) for some t0 = t0(T ) ∈ C. (ii) The correspondence T 7→ t0(T ) deﬁnes an isomorphism A → Per(L0). (iii) Per(L0) is the group of periods of Ω. This lemma is proved like Proposition 2.6 page 740 in [18].

Corollary 2. Let F, Ω, ϕt , A, Per(Ω), M compact be as in Lemma 3 above. Then rank Per(Ω) ≥ 2 and the leaves of F are compact if, and only if, Per(Ω) ⊂ C is a rank two lattice on C.

8 Proof. It is enough to observe that we have a natural identification Per(Ω) = A. ¥ Besides the above results we have in general, for manifolds supporting non-singular closed 1- forms Lemma 4. Let M be a compact complex manifold supporting a non-singular closed holomorphic 1-form Ω whose group of periods Per(Ω) is a rank 2 lattice on C. Then M admits a holomorphic submersion over a complex torus C/ Per(Ω). The fibers (of this C∞ locally trivial fibre bundle) are the leaves of the foliation F determined by Ω. Proof. Fix any point x ∈ M and take generators ~α, β~ ∈ C for Per(Ω) as a rank two lattice on 0 R C. Let f : M → R = C/(~αZ + β~Z) be defined by f(x) = x Ω mod. (~αZ + β~Z). Notice that f is x0 well-defined and holomorphic by construction. Moreover, f(x) = f(y) for every x, y in a same leaf of F. We also have df = Ω so that f : M → R is a submersion whose connected components of the fibers are the leaves of F. ¥ The following is a corollary to the previous results. Corollary 3. Let M be a (connected) compact manifold. The following assertions are equivalent: (i) M admits a holomorphic submersion over a one dimensional complex torus.

(ii) M supports a compact foliation given by a non-singular closed holomorphic 1-form.

(iii) M supports a foliation given by a non-singular closed holomorphic 1-form and with one compact leaf. The conditions above are satisfied whenever M supports a (codimension one) foliation which has a compact leaf and is invariant under some (holomorphic) transverse flow. If one of the conditions above is satisfied then: (iv) M supports a compact foliation of codimension one with trivial holonomy. Proof. Given any holomorphic submersion f : M → R onto a one dimensional torus, we take any generator θ ∈ H0(R, Ω1) and its lift ω = f ∗(θ) to M. Since f is a submersion ω is a non-singular closed 1-form on M; clearly the foliation ω = 0 coincides with the fibration f : M → R and it is therefore compact, thus (i) implies (ii). Since (ii) trivially implies (iii) it remains to show that (iii) implies (i). Assume that M supports a foliation F given by a non-singular closed 1-form Ω and having some compact leaf Lo; then Lo has trivial holonomy and by the complex version of Reeb Stability Theorem for codimension one holomorphic foliations (cf. [4]) we conclude that F is a compact foliation. According to Corollary 2 and Lemma 4 it follows that Per(Ω) is a rank two lattice on C and therefore F defines a holomorphic submersion from M onto the complex one dimensional torus C/ Per(Ω). ¥ Remark 2. Notice that (iv) is not equivalent to (i), (ii), (iii). A product of two curves of general type gives a counter-example.

9 Remark 3. There are situations where F is not compact, even for M compact Kähler(or homogeneous) manifold; for instance take M = Cn/Λ a complex torus for some lattice Λ ⊂ Cn. We may choose Λ in such a way that M has algebraic dimension zero (cf. [22] page 215), then it admits no non constant meromorphic function, and therefore no holomorphic submersion to a one dimensional complex torus. On the other hand M is Kählerand supports (linear) foliations F which are invariant under (linear) transverse flows (cf. Example 1); this is not the only example (cf. Example 4). The following result may therefore be regarded as a version of Tischler’ Theorem for compact complex manifolds (notice that, in the statement below, the holomorphic fibration is locally holomorphically trivial, as in Definition 1). Theorem 1. Let M be a connected compact complex manifold equipped with a codimension one compact foliation F which is invariant under a holomorphic transverse flow. Then M is a holomorphic fibre bundle over a complex one dimensional torus. Proof. We already know that F is given by a non-singular closed holomorphic 1-form Ω on M. According to Corollary 2 and Proposition 4 the manifold M admits a C∞ fibration with holomorphic projection over a one dimension complex torus C/Λ; the fibers of this fibration are the leaves of F. Now, according to the Local-triviality theorem of Grauert-Fischer (cf. [2] page 29) a smooth family of compact complex manifolds is locally trivial if and only if all fibers are holomorphically equivalent. This is the case for the leaves of F; indeed the flow maps give holomorphic equivalence for the leaves, hence we conclude that F defines a locally holomorphically trivial fibration M → C/Λ. ¥ Proposition 1. Let M be a compact connected complex manifold admitting a non-singular codimension one holomorphic foliation F which is invariant under some holomorphic transverse flow ϕ. Then there exist, arbitrarily C∞-close to F, real codimension two compact foliations, of class C∞, given by 2-forms ξ = ξ1 ∧ ξ2, where ξ1, ξ2 are closed real 1-forms which are everywhere transverse. In particular, M admits a C∞-fibration over the real 2-torus f : M → S1 × S1. Proof. Let Ω be a non-singular closed 1-form defining F on M; using the complex structure J : TM → TM we can√ introduce the 1-forms ξ1 and ξ2 as the real part and the imaginary part of Ω, so that Ω = ξ1 + −1ξ2. A simple computation shows that ξ1, ξ2 are closed and ξ1 ∧ ξ2 6= 0 everywhere on M. Denote by Fk the real codimension one foliation defined by ξk = 0 on M; clearly F = F1 ∩F2 as a real codimension two foliation in the usual sense. Now, choose loops γj, j = 1, ..., r 1 corresponding to a basis of the free partR of H (M; Z) for which we have a basis {wj, j = 1, ..., r} 1 of the group H (M; R), and satisfying wi = δij where δij is the Kronecker delta. We may write γj r R P k ∞ k k ξk = λ wj + dfk for some C function fk : M → R and some λ ∈ R given by λ = ξk. Let j j j γj j=1 r k k r 0 P now (µ1, . . . , µr ) ∈ R be such that ξk := µj.wj + dfk is close enough to ξk, hence it is also non- j=1 k k singular (recall that ξk is non-singular and M is compact) and the subgroup hµ1, . . . , µr i of (R, +)

10 k k 0 is a rank 2 discrete lattice (it is enough to choose {µ1, . . . , µr } ⊆ Q of rank 2). Then ξk defines a 1 0 fibration of M over the circle S and we obtain (by transversality of ξ1 and ξ2, and therefore of ξ1 0 ∞ 1 1 ∞ and ξ2) a C fibration M → S × S , which is arbitrarily C -close to the foliation F, as a real codimension two foliation on M. ¥

Proposition 2. Let M be a compact complex manifold supporting a holomorphic flow ϕ transverse to the kernel of some (non-singular) closed holomorphic 1-form η on M. Then the flow ϕ admits a C∞-compact section; i.e., there exists a codimension two compact real foliation R, of class C∞, which is everywhere transverse to the flow ϕ.

We highlight that we are not assuming that the foliation F given by η = 0 is invariant under the flow ϕ . Proof. Let F be the holomorphic foliation defined by η = 0 on M. Using Proposition 1 we may approximate F (as a C∞ real codimension two foliation on M) by some compact real foliation R and we obtain, for any R close enough to F, that R is also transverse to the flow ϕ . Hence, any leaf of R gives a compact section of the flow ϕ. ¥

3 K¨ahlermanifolds and homogeneous manifolds

Let M be a complex compact manifold with Albanese torus Alb(M) = Cg/ Per(M) and Albanese map µ: M → Alb(M) (cf. [22],[12]). Because M is compact the group Aut(M), of holomorphic automorphisms of M, is a complex Lie group acting holomorphically on M ([12]). The manifold M is called homogeneous if this action is transitive. For the case of K¨ahlermanifolds we have the following Albanese Theorem:

Lemma 5 ([15]). Let M be a connected compact (not necessarily homogeneous) K¨ahlermanifold 1,0 1 with Albanese map µ: M → Alb(M). Then g := dimC Alb(M) = h (M) = 2 b1(M). Moreover, every holomorphic 1-form Ω on M is the lift Ω = µ∗(θ) by µ of some linear form θ on Alb(M). In particular, Ω is closed.

Remark 4. Given a compact K¨ahlermanifold M, despite the existence of the Albanese map, it is not clear that M admits a holomorphic submersion over some torus C/Λ; for instance we may have b1(M) = 0, what is the case of M = CP (m) (cf. [12]). Thus the result of Theorem 1 above is a real restriction on the (structure of the) manifold M.

1,0 1 Assume now that g = h (M) = 2 b1(M) = 1. Then the Albanese variety Alb(M) = C/Λ is a one-dimensional compact complex torus and the Albanese map may explicit as follows: let η ∈ H0(M;Ω1) be any non trivial holomorphic 1-form on M, then any η is closed and is the lifting ∗ µ (α) of some 1-form α on C/Λ which is linear with constant coeﬃcients. Given any basis {γ1, γ2} R p for the homology H1(M; Z) we have µ: M → C/Λ given by µ(p) = η (mod Λ) where Λ ⊂ C is p0

11 © R R ª the lattice η, η ⊂ C. We shall refer to the foliation given by this fibration as the Albanese γ1 γ2 fibration of M; this fibration does not depend on the choice of the representative η ∈ H0(M;Ω1). Therefore we have proved:

Corollary 4. Let M be a compact Kählermanifold with rank H1(M; R) = 2. Let F be a holomorphic foliation of codimension one on M, invariant under some holomorphic transverse flow ϕ on M. Then F is the Albanese fibration of M, in particular, F is compact.

Question 1. Given a compact Kählermanifold M supporting a codimension one foliation F invariant under some transverse holomorphic flow, decide, in terms of the pair (M, F), whether there exists a holomorphic fibre bundle M → R of M over a complex torus R.

This question is not so easy (cf. Remark 3); even in the case such a ﬁbration exists we may have foliations that cannot be approximated by compact ﬁbrations as in the example below:

Example 4. Let Rj, j = 1, ..., k be complex one dimensional tori and let M = R1 × ... × Rk. For 0 1 each Rj choose a generator θj ∈ H (Rj, Ω ) and a vector field Xj on Rj with θj · Xj = 1. Then k k P ∗ P −1 ∗ for ω = λjωj , and X = λj Xj , where ωj (respectively Xj) is the lift of θj (respectively Xj) j=1 j=1 ∗ to M by the natural projection M → Rj, and λj ∈ C generic, we have ω · X = k so that ω = 0 defines a foliation F on M, which is invariant under the transverse flow of X. For generic λj this foliation is not compact; nevertheless the algebraic dimension of M is k (indeed, M is an algebraic projective manifold). Moreover, M fibers over complex one dimension tori, but, always for generic λj, F cannot be approximated by compact holomorphic foliations. Despite the above examples and remarks we may, using the Albanese Theorem (Lemma 5), state some versions of Tischler’ Theorem for suitable Kählersurfaces; we consider the following situation: M 2 is a compact complex Kählersurface, supporting a non-singular holomorphic vector field X. The one dimensional foliation FX , defined by X on M, is supposed to have trivial normal bundle; i.e., there exists a holomorphic 1-form ωX such that ωX = 0 defines F, and therefore ωX · X ≡ 0. Let F be a foliation (of dimension one) on M which is invariant under the transverse flow of X and let ω be the holomorphic 1-form defining F and satisfying ω · X = 1. Denote by g = h0(M, Ω1) the dimension of the space of holomorphic 1-forms on M; clearly g ≥ 2. Suppose g ≥ 3; in this case there exists ω1 which is not linearly dependent with {ω, ωX }. Since ω1 · X is holomorphic on M, which is compact, it follows that, up to multiplying ω1 by a nonzero constant, we have ω1 · X ≡ 1 or ω1 · X ≡ 0. If ω1 · X ≡ 0 then, since M has dimension two, it follows that ωX ∧ ω1 ≡ 0. By a classical result of Castelnuovo-De Franchis (see [3]) there exists a holomorphic map π : M → R onto an algebraic curve R of genus ≥ 2, such that ωX and ω1 are pull-backs by π of holomorphic 1-forms on R. Since ω is non-singular it follows that π is in fact a submersion. Assume now that ω1 · X ≡ 1, then (ω −ω1)·X ≡ 0 and therefore (ω −7ω1)∧ωX ≡ 0. Since ωX is (clearly) not linearly dependent with ω − ω1 it follows again by Castelnuovo-De Franchis that there exists a map π : M → R as

12 above, which is a submersion. Finally, assume that g = 2; in this case we may choose ω1 such that 0 1 ω and ω1 are everywhere linearly independent, and it follows that H (M, Ω ) has {ω, ω1} as a basis and Alb(M) = C2/Λ is a 2-torus. Since M has dimension two, by hypothesis, and since a finite covering of a torus is a torus itself, it follows that M 2 = Alb(M) = C2/Λ, i.e. M is a 2-torus itself. As in Remark 3 and Example 4 the foliation F may be compact or not. Summarizing we reobtain the following well-known result for surfaces supporting non-singular vector fields: Proposition 3. Let M 2 be a compact Kählersurface supporting a non-singular holomorphic vector field X whose corresponding foliation FX has trivial normal bundle. Then M is itself a complex torus (and F is linear cf. Example 1), or M admits a holomorphic submersion over a one dimensional complex torus. Proof. We preserve the above notation; if g = 1 then Alb(M) is a complex one dimensional torus and the holomorphic projection is already given by the Albanese map. Assume g ≥ 2. According to the above discussion it remains to show that M supports a foliation F of dimension one which is invariant under the transverse flow of X; this is easy for given any non-singular holomorphic 1-form ω on M with ω linearly independent with ωX we may assume that ω · X ≡ 1, and therefore ω = 0 defines F (such a 1-form is given by the Albanese map, notice that ω is necessarily closed). ¥ Using the same proof we obtain for Kählermanifolds of dimension n + 1 ≥ 2: Theorem 2. Let M n+1 be a compact Kählermanifold supporting a codimension one foliation F invariant under the transverse holomorphic flow of X. Suppose the one dimensional foliation FX defined by X has a trivial normal bundle; i.e., FX is defined by n holomorphic 1-forms ω1, ..., ωn on M. Then M fibers over a complex torus of dimension one or M is a torus itself (and F is linear). For the case of homogeneous (not necessarily Kähler)manifolds we have the following consequence of [4] and [7] Proposition 4. Let F be a holomorphic foliation on a compact homogeneous manifold M, given by a non-singular closed holomorphic 1-form. Then F is compact or it is transversely projective in M.

Proof: Suppose F has some compact leaf L0 ⊂ M. Since the holonomy of F is trivial, it follows from the complex version of Reeb’s Stability Theorem ([4]) that F is compact. Hence we may assume that F has no compact leaf; according to [7] Theorem 1.4, F is transversely projective in the complement of the set of compact leaves of F and, therefore, in the whole manifold M. ¥

4 Holomorphic Anosov ﬂows

4.1 Definition and unstable foliations Let M be a real smooth Riemannian manifold and ϕ: M × R → M a non-singular differentiable flow on M. We say that ϕ is an Anosov flow if there is a ϕt-invariant splitting of the tangent bundle

13 uu ss T T u TM = E ⊕ E ⊕ E , where E is the line bundle tangent to the non-singular ﬂow ϕt and E and Es satisfy the following conditions:

uu t (i) There exist constants A > 0, µ > 1 such that ∀ t ∈ R, v ∈ E implies ||(ϕt)∗(v)|| ≥ Aµ ||v||.

ss (ii) There exist constants B > 0, λ < 1 such that ∀ t ∈ R and v ∈ E we have ||(ϕt)∗v|| ≤ Bλt||v||.

Such a splitting is called hyperbolic and its hyperbolicity is independent of the Riemannian metric on M for the case M is compact ([1], [20]). This deﬁnition has been settled for the complex holomorphic case by E. Ghys [6] as follows:

Definition 3. Let M be a compact complex manifold. A holomorphic Anosov flow on M is given by a holomorphic action ψ : C∗ × M → M with the following property: there exists a splitting uu ss T uu ss TRM = E ⊕ E ⊕ E where TRM is the real tangent bundle of M, E , E are (real) sub- T bundles of TRM and E is the (real) two dimensional bundle tangent to the orbits of ψ. Moreover, −α ss for some given hermitian metric || · || on M we have ||dψT · v|| ≤ c · |T | ||v||, ∀ v ∈ E , and α uu ∗ ||dψT · w|| ≤ c · |T | ||w||, ∀ w ∈ E , all this ∀ T ∈ C and for some fixed constants c > 0, α > 0. In [6] we find the classification of holomorphic Anosov flows on compact 3-manifolds. As far as we know there is no such classification result for dimension n ≥ 4.

Example 5 (Suspension of an Anosov holomorphic diffeomorphism). Let M be a compact complex n-dimension manifold and f : M → M a holomorphic Anosov diffeomorphism. This means the following: we shall equip the holomorphic fibre bundle TCM with an auxiliary hermitian metric, s u || · ||. There exist, then, a splitting TRM = E ⊕ E of TRM, and constants c > 0, µ > 1 such that ||Df k(v)|| ≤ c µ−k ||v|| and ||Df −k(w)|| ≤ c µ−k ||w||, ∀ v ∈ Es, ∀ w ∈ Eu, ∀ k ∈ Z, k ≥ 0. The sub-bundles Es, Eu are called stable and unstable respectively and both are continuous and integrable. Denote by F s, F u the foliations tangent to Es and Eu respectively. These are ∞ s continuous foliations whose leaves are C immersed√ real submanifolds of M. Since the bundles E , Eu are clearly stable under multiplication by −1 ∈ C it follows that the leaves of F s, F u are immersed holomorphic submanifolds of M. Now we shall construct the suspension of f. Let w ∈ C\S1 and put F : M ×C∗ → M ×C∗ as the holomorphic diffeomorphism F (x, t) = (f(x), tw). Then F defines a group action of Z on M × C∗, this action given by Z × (M × C∗) → M × C∗, (n, (x, t)) 7→ F n(x, t) = (f n(x), twn). Clearly this action is free, properly discontinuous and commutes with the action ψ : C∗ × (M × C∗) → M × C∗, (s, (x, t)) 7→ (x, ts). Therefore, the quotient V = [M × C∗]/Z is a (n + 1)-dimensional compact complex manifold, equipped with a holomorphic Anosov flow ψ : C∗ × V → V given by n n ∗ ψ(s, [(x, t)]) = [ψs(x, t)] = [(x, ts)] = [(f (x), stw )], ∀ n ∈ Z . Each flow map ψs : V → V is given by ψs([x, t]) = [(x, ts)], in particular ψw : V → V is given by ψw([x, t]) = [(x, wt)]. The manifold V is a bundle over C∗/Z, where the quotient of C∗ is given by the Z-action (n, t) 7→ twn and

14 having fiber M. The projection map is π : V → C∗/Z, π : [(x, t)] 7→ [[t]] 3 C∗/Z. Thus we have π([x, t]) = π([x0, t0]) ⇔ ∃ n ∈ Z such that x0 = f n(x), t0 = t wn. The first return map is therefore given by ψw|M×{1} : M → M which is just (up to conjugacy) the diffeomorphism f : M → M. The flow ψ on V is Anosov (holomorphic) as it is easily verified, and we shall refer to ψ as the suspension of the diffeomorphism f.

Remark 5. Given a holomorphic Anosov flow as defined above say, ψ : C∗ × M → M on the uu ss T hermitian compact manifold (M; h i) with decomposition TCM = E ⊕ E ⊕ E , we may define a C-action on M by setting ϕ: C × M → M, ϕ(t, x) = ψ(et, x). The Anosov flow condition for ψ is ( ||dψ · v|| ≤ C|T |−α ||v|| ∀ v ∈ Ess T ∀ T ∈ C∗ α uu ||dψT · w|| ≤ C|T | ||w|| ∀ w ∈ E with C > 0, α > 0 constants; therefore we obtain ( ||dϕ · v|| ≤ C e−α Re(t) ||v|, ∀ v ∈ Ess t ∀ t ∈ C. α Re(t) uu ||dϕt · w|| ≤ C e ||w||, ∀ w ∈ E

In particular, from this last inequality, we obtain

−α Re(t) ||dϕ−t · w| ≤ C e ||w||, ∀ t ∈ C;

1 −α Re(t) uu but replacing w by dϕt · w we obtain therefore ||w|| ≤ C e ∀ w ∈ E , ∀ t ∈ C (notice that uu E is dϕt invariant so that we actually may replace w by dϕt · w as above). Therefore ϕ defines a complete holomorphic vector field Z on M whose one-parameter subgroups {ϕt ∈ Aut(M), t ∈ C} α Re(t) uu −α Re(t) ss satisfies ||dϕt · w|| ≥ A e ||w||, ∀ w ∈ E and ||dϕt · v|| ≤ B e ||v||, ∀ v ∈ E , ∀ t ∈ C for some constants α, A, B > 0. In particular the restriction ξ := ϕ|R×M : R × M → M defines a real flow such that

α t −α t ||dξt · w| ≥ A(e ) ||w|| and ||dξt · v|| ≤ B(e ) ||v||

uu ss ∀ w ∈ E , ∀ v ∈ E , ∀ t ∈ R; so that {ξt; t ∈ R} defines a real differentiable Anosov flow on

M. On the other hand, the restriction η := ϕ|iR×M : iR × M → M defined by η(t, x) = ϕ(it, x); defines a real differentiable flow on M whose flow maps satisfy ||dηt · w|| ≥ A||w| and ||dηt · v|| ≤ uu ss B||v||, ∀ w ∈ E , ∀ v ∈ E , ∀ t ∈ R; and it is not Anosov.√ The corresponding real (analytic) vector fields X and Y to ξ and η respectively satisfy Z = X + −1 Y and [X,Y ] = 0. All these facts will be used, often implicitly, in what follows.

∞ ∞ uu ss uu T Given a real C Anosov ﬂow ϕt on a C real manifold M the bundles E , E , E ⊕ E =: Es and Ess ⊕ ET =: Es are, in general, only continuous and uniquely integrable ([14]). The

15 corresponding continuous foliations are denoted by F uu, F ss, F u and F s respectively. These are continuous foliations whose leaves are C∞ immersed submanifolds of M.

Let now M be a complex manifold. Denote by TCM the complex tangent bundle of M and ∗ equip TCM with an additional hermitian metric || · ||. Let ψ : C × M → M be a holomorphic Anosov flow on M with corresponding holomorphic vector field Z whose flow is ϕ: C × M → M, ϕ(s, x) = ψ(es, x). We have by definition a decomposition

ss uu TRM = E ⊕ E ⊕ C · Z.

The foliations F s and F u generated by the bundles Es := Ess ⊕ C · Z and Eu := Euu ⊕ C · Z (respectively) are called central stable and central unstable foliations, respectively. According to [6] u s (Lemma 2.1 page 589) the sub-bundles E , E are complex sub-bundles of TCM so that each leaf of F s or F u is an immersed holomorphic submanifold of M. Also ([6] Proposition 2.2 page 589) if Euu has complex dimension one (i.e., if Es has complex codimension one) then F s is also transversely holomorphic. In this case, according to Hartogs’ Theorem for foliations, F s is a codimension one holomorphic foliation on M. Remark 6. If ψ is a holomorphic Anosov flow on (compact) M such that F s has codimension one then: (i) There exists a unique way to equip each leaf of the (strong) unstable foliation F uu with a uu complex affine structure so that each flow map ψT acts in an affine way on the leaves of F ([6], Proposition 3.1, page 590).

(ii) The holonomy pseudo-group of F s acts projectively with respect to the above complex aﬃne structures on the leaves of F uu ([6], Proposition 4.1, page 593).

(iii) If F uu is also a holomorphic foliation then F s is transversely projective ([6], Prop.2.2, page 589). For the case of homogeneous Kählermanifolds we immediately obtain Theorem 3. Let M be a compact homogeneous Kählermanifold supporting a transitive holomorphic Anosov diffeomorphism f : M → M such that F s has (complex) codimension one. Then M is a complex torus and f is a linear automorphism of Anosov type. Proof. The foliation F s is a continuous foliation with holomorphic leaves. Since f has codimension one it follows that indeed F s is a holomorphic foliation; thus M supports a (non-singular) codimension one holomorphic foliation. According to [7] M is a product M = Q × T of a flag manifold Q and a torus T . On the other hand, by Theorem B page 585 of [6] M is homeomorphic to a torus and f is topologically conjugate to a linear toral automorphism. This implies that actually M = T and f : T → T is a linear toral automorphism. ¥

16 4.2 Jointly integrable holomorphic Anosov flows Given a holomorphic Anosov flow ψ on a compact complex manifold M we consider the correspond- uu ss T ing holomorphic flow ϕ: C × M → M, defined by ϕt = ψet and denote by E , E , E ⊂ TCM uu ss T the sub-bundles of the decomposition TCM = E ⊕ E ⊕ E as in Definition 3 §4.1.

uu ss Deﬁnition 4. We shall say that the ﬂow ψ is jointly integrable if the sub-bundle E ⊕ E ⊂ TCM is integrable. In this case we have a codimension one (non-singular) holomorphic foliation F(ψ)t with the following property:

Lemma 6. Let ψ be jointly integrable, the foliation F = F(ψ)t is invariant under the transverse flow ϕt = ψet on M. Proof. Indeed, given any point x ∈ M and any flow map ϕ : M → M we have (ϕ ) (Euu) ⊂ Euu t t ∗ x ϕt(x) and (ϕ ) (Ess) ⊂ Ess as a consequence of the flow property ϕ ◦ ϕ = ϕ . Thus for the leaf L t ∗ x ϕt(x) s t s+t x of F through x we have (ϕ ) (T L ) = (ϕ ) (Euu ⊕ Ess) = Euu ⊕ Ess = T L . In t ∗ x x t ∗ x x ϕt(x) ϕt(x) ϕt(x) ϕt(x) ∗ other words (ϕt)∗(T F) = T F so that ϕt (F) and F coincide. This means that ϕt takes F-leaves onto F-leaves. ¥

The first example of a jointly integrable holomorphic Anosov flow is the suspension of an Anosov holomorphic diffeomorphism f : N → N as in Example 5. Indeed, the suspension manifold is M = [N × C∗]/Z where the Z-action on N × C∗ is given by n, (x, t) 7→ (f n(x), t0wn) for some fixed w ∈ C\S1 and ∀ n ∈ Z,(x, t) ∈ N ×C∗. The second coordinate projection N ×C∗ → C∗,(x, t) 7→ t induces therefore a holomorphic submersion π : M → C∗/(w) of M onto the compact complex torus C∗/(w), this submersion defines a compact foliation F on M, which is the foliation corresponding ss uu to the bundle E ⊕ E ⊂ TCM. Therefore the flow induced by f on M is jointly integrable with compact corresponding foliation.

Question 2. Is any jointly integrable holomorphic Anosov ﬂow topologically or holomorphically conjugate to a suspension?

Collecting the preceding results and those of §2 we obtain:

Remark 7. Let ψ be a jointly integrable holomorphic Anosov ﬂow on a compact manifold M.

(i) The resulting foliation F is given by a closed holomorphic (non-singular) 1-form Ω on M; it is a compact foliation or it has all leaves non-closed, the foliation is compact provided that H1(M; Z) has rank two.

(ii) If F is compact then ψ admits a holomorphic section (given by a leaf L0 of F and a ﬂow map

ϕt0 with ϕt0 (L0) = L0) and M is a holomorphic ﬁbre bundle over a complex torus C/Λ.

17 (iii) Assume that ψ is transitive of codimension one and admits some holomorphic section N; let f : N → N be the induced Anosov holomorphic diﬀeomorphism on N. Then N is homeomorphic to a complex torus Cn−1/Λ and f is topologically conjugate to a toral automorphism. In particular if, moreover, F is compact then the leaves are homeomorphic to a ﬁxed torus and (M, ψ) is topologically conjugate to a suspension of a hyperbolic toral automorphism.

4.3 Anosov flows and homology Let N be a compact complex manifold and f : N → N a holomorphic diffeomorphism. We recall the construction of the suspension manifold of f in our setting. Let Zf,λ be the action of Z in M × C∗ given by (n, (x, t)) 7→ (f n(x), λnt) where n ∈ Z,(x, t) ∈ N × C∗ and 0 6= λ ∈ C\S1 is µ a fixed complex number, say λ = e for some µ ∈ C \ R. Then Zr,λ is a properly discontinuous ∗ holomorphic action and the quotient manifold M = Mf,λ := [N × C ]/Zf,λ is a compact complex manifold with the following properties: (i) M is a holomorphic fibre bundle over the complex torus C∗/(λ) := C/{1, µ} with fiber diffeomorphic to N. (ii) The group of the bundle M −→ C∗/(λ) is isomorphic to Z, naturally conjugate to the N suspension of the diffeomorphism f : N → N. (iii) M is equipped with a holomorphic C∗-action which defines a time one mapping naturally conjugate to f : N → N, this action has orbits transverse to the fibration and its maps act on M taking fibers onto fibers.

Now we must add one additional property to those above.

Lemma 7. Let M be the suspension manifold of the holomorphic diﬀeomorphism f : N → N. We have rank H1(M; R) = 2 if, and only if, the homomorphism f ∗ : H1(N; R) → H1(N; R) does not have 1 as an eigenvalue.

Proof. Let ψ : C∗ × M → M be the C∗-action corresponding to the suspension of the holomorphic diffeomorphism f : N ←-. We may define a holomorphic flow on M by setting

Φ: C × M → M, Φ(T, x) = ψ(eT , x), ∀ T ∈ C, ∀ x ∈ M.

This complex flow “splits” in two commuting real flows ϕt and ϕit on M, given by ϕt(x) := Φ(t, x) and ϕit(x) := Φ(it, x), ∀ t ∈ R, ∀ x ∈ M. The flow ϕit is associated to the circle group action 0 1 ψ : S × M → M. Since Φ is a suspension this action is free and the quotient M1 of M by this action is a dim N + 1 real manifold equipped with the real flow ξt induced by ϕt (recall that ϕt and ϕit commute in M). Now, ξt is the real suspension of the real diffeomorphism f : N → N, that is, M1 is obtained as the quotient of N × R by the Z-action generated by the diffeomorphism (x, t) 7→ (f(x), t − 1). In particular, we have the orbit fibration M −→ M1 whose corresponding S1

18 exact homology sequence implies that

rank H1(M; Z) = rank H1(M1; Z) + 1.

Now, since M1 is the (real) suspension manifold of f : N → N it follows from [18] (Lemma 3.3 1 ∗ 1 1 page 744) that rank H (M1; R) = 1 if, and only if, f : H (N; R) → H (N; R) does not have 1 as 1 an eigenvalue. The lemma follows from these remarks and from the fact that rank H (M1; R) = rank H1(M1; Z). ¥ We obtain now a result already obtained by J. Plante for the case of real flows [18]: Corollary 5. Let ψ : C∗ × M → M be a jointly integrable holomorphic Anosov flow on a compact manifold M. Assume that the flow admits some (holomorphic) section N ⊂ M. Denote by f : N → N the Anosov diffeomorphism induced on N. If f ∗ : H1(N; R) → H1(N; R) does not have 1 as an eigenvalue then the foliation given by Eu ⊕ Es is compact. Proof. Indeed, the above hypothesis on f ∗ implies, via Lemma 7, that H1(M; R) has rank 2 and by Corollary 1, the foliation of Eu ⊕ Es is compact. ¥ Remark 8. Let M be a compact manifold equipped with a jointly integrable holomorphic Anosov flow ψ. According to Proposition 2 the flow ψ admits a C∞ (real codimension two compact) section ∞ NR ⊂ M, denote by f : NR → NR an induced Anosov diffeomorphism (real of class C ). One may try to go further in this study by proving that the flow ψ on M is (from the C∞ viewpoint) a suspension of f on NR; and therefore, by an application of Corollary 5 above, conclude that the only case where Eu ⊕ Es may generate a non-compact foliation would be the case where it admits some section NR ⊂ M with Anosov map having 1 as an eigenvalue. Corollary 6. Let ψ : C∗ × M → M be as holomorphic Anosov flow on a compact manifold M such that ψ is jointly integrable and admits some (holomorphic) section. Denote by π : Mf → M the universal covering of Mf. Assume that: f (i) H∗(M; Z) is finitely generated

(ii) π1(M) has a polycyclic subgroup of finite index. Then Eu ⊕ Es generates a compact foliation. Proof. By hypothesis the flow admits some section N ⊂ M; therefore, the universal covering Mf of M is diffeomorphic to the universal covering of N and we have an exact sequence

∗ 0 → π1(N) → π1(M) → π1(C /Z) → 0

π1(M) f ∼ e what implies that ' Z ⊕ Z. Since H∗(M; Z) = H∗(N × C; Z) it follows from the Kunneth π1(N) formula ([19]) and (i) in the statement that: f (i)’ H∗(M) is ﬁnitely generated.

19 Now, (ii) and the isomorphism π1(M) ' Z ⊕ Z imply that π1(N)

(ii)’ π1(N) has a polycyclic subgroup of ﬁnite index. It follows therefore from M. Hirsch [13] (see also [18] Theorem 3.2 page 744) that the (induced) Anosov diﬀeomorphism f : N → N is such that f ∗ : H1(N; R) → H1(N; R) does not have a root of unity as an eigenvalue. Applying Corollary 5 we conclude the proof. ¥

Remark 9. Let us spend some words about the results above: (1) A group G (abstract group) is said to be polycyclic if there exists a sequence of subgroups G = G0 ⊃ G1 ⊃ · · · ⊃ Gk = {e} such that Gi+1 is normal in Gi and the quotient Gi/Gi+1 is cyclic (see [17]). A group G is virtually polycyclic if it has a polycyclic subgroup of finite index. It is proved in [18] that a virtually polycyclic finitely generated group G either has exponential growth or it has polynomial growth and a nilpotent subgroup of finite index ([18], Theorem 1.9 page 522). Such groups arise naturally in the study of real codimension one foliations on compact manifolds (see [17] and [18]). Hirsch’s Theorem [13] mentioned above may be stated as follows: Let f be an Anosov (differentiable) diffeomorphism of a (real) compact manifold N such that π1(N) is virtually polycyclic and the universal covering Ne of N has finitely generated homology then the corresponding suspension manifold V has rank H1(V ; Z) = 1. (2) Let f be a holomorphic Anosov diffeomorphism of a compact manifold N and assume that N is homeomorphic to a torus Cn/Λ and f is topologically conjugate to a linear automorphism n n e ∼ n A: C /Λ → C /Λ of this torus. Then, we have isomorphisms H∗(N) = H∗(C ) = {0} and ∼ n ∗ 1 π1(N) = π1(C /Λ). Therefore, by [18] Theorem 3.2 page 744, f : H (N; R) ←- does not have a ∗ root of 1 as an eigenvalue. This implies that A : H1(Cn/Λ; R) ←- does not have a root of 1 as an eigenvalue. We also notice that the linear mapping A: Cn → Cn, which induces A on Cn/Λ, must have no root of 1 as an eigenvalue.

Theorem 4. Let ψ : C∗ × M → M be a codimension one, transitive holomorphic Anosov ﬂow on a compact manifold M. Assume that ψ is jointly integrable, with foliation F, and admits a (holomorphic) section. Then:

(i) F is compact, M is a holomorphic fibre bundle over a 1-torus C/Λ having as fibers the leaves of F. In particular the leaf space of F is C∞-diffeomorphic to the torus S1 × S1.

∗ (ii) Given any leaf L0 of F and any t0 ∈ C such that ϕt0 (L0) = L0 (where ϕt = ψet is the corresponding ﬂow ϕt : C × M → M to ψ on M) then: L0 is homeomorphic to a torus n−1 n−1 C /Λ0 , for a ﬁxed lattice Λ0 ⊂ C , and f := ϕ | : L0 → L0 is topologically conjugate t0 L0 n−1 to a linear automorphism of C /Λ0 . In particular,

(iii) The ﬂow is topologically conjugate to a suspension of a hyperbolic toral automorphism.

20 Proof. Let N ⊂ M be a (compact) holomorphic section of the flow and denote by f : N → N an Anosov diffeomorphism induced on N. By the hypothesis on the flow it follows that f is transitive of codimension one. Therefore, according [6] Theorem B pages 585-586, we conclude that N is homeomorphic to a torus and f is topologically conjugate to a toral automorphism. In particular it follows (from Remark 9 (2) and Corollary 6) that F is compact, what proves (i). We may therefore take L0 = N above and f = ϕt0 for any t0 with the property that ϕt0 (L0) = L0 (see Lemma 3 where ∼ it is proved that Per(L0) = Per(Ω)). This proves (ii) and also (iii). ¥ It is a result of S. Newhouse that codimension one real differentiable Anosov diffeomorphisms, of real compact manifolds, are transitive [16]. Question 3. Is a codimension one jointly-integrable holomorphic Anosov flow (respectively, a Anosov diffeomorphism) always transitive? It is our belief that, probably as a consequence of the above proof of Corollary 7, any transitive jointly integrable holomorphic Anosov flow, on a compact manifold M, has compact corresponding foliation.

5 Foliations with singularities

5.1 Existence of closed meromorphic 1-forms We shall now investigate how to “extend” some of our preceding results to the case of foliations with singularities. All foliations are holomorphic of codimension one with singular set of codimension at least 2. A very basic remark is the following lemma that shows there is no new interest in the case of singular foliations invariant under global holomorphic transverse ﬂows:

Lemma 8. Let F be given on M (not necessarily compact) and assume that F|M 0 is invariant 0 under some holomorphic transverse ﬂow ϕt on M = M − sing(F). Then F is non singular, i.e., sing(F) = ∅ and we are in the situation of Lemma 1 of §2.

Proof. According to Lemma 1 we mayR construct a non-singular closed holomorphic 1-form Ω on 0 0 t2 0 M such that T F = Ker(Ω) in M and Ω(ϕt(x)) · Z(ϕt(x)) dt = t2 − t1 , ∀ x ∈ M , ∀ t1, t2 ∈ C, t1 0 where Z is the vector field corresponding to ϕt in M . Hartogs’ Extension Theorem assures that Ω and Z extend holomorphically to M. Clearly we have Ω·Z = 1 for these extensions in M. Therefore sing(Ω) = ∅ and also sing(Z) = ∅ in M. ¥ ¯ ¯ dy dx Example 6. Let Ω be the closed meromorphic 1-form given on CP (2) by Ω C2 = y − λ x in affine coordinates (x, y) ∈ C2, where λ ∈ C\{0}. Take Z = ay ∂ + bx ∂ , then Z defines a ∂y ∂x µ ¶ 1 y holomorphic vector field on CP (2) for all a, b ∈ C. For instance, for the chart (u, v) = x , x we

21 ∂ ∂ have Z(u, v) = −bu ∂u + (a − b)v ∂v . We have Ω · Z = a − λb so that we may choose a, b ∈ C such that Ω · Z = 1. Since it is linear, Z is complete. The projective foliation F defined by Ω (on CP (2)) is transverse to Z everywhere except along the two coordinate axis and the line at infinity CP (2)\C2. Thus F is invariant under the transverse holomorphic flow of Z in CP (2) minus these algebraic invariant curves, which are in fact the irreducible components of the polar divisor (Ω)∞ of Ω in CP (2). A converse of this example is given by the lemma below:

Lemma 9. Let F be a singular holomorphic foliation of codimension one on M which is invariant under the transverse flow in M\Γ of some holomorphic vector field Z in M and for some analytic codimension one subset Γ ⊂ M. Assume that F and Z are tangent along Γ. Then there exists a closed meromorphic 1-form Ω on M which defines F in M\Γ and such that (Ω)∞ = Γ and Ω·Z = 1.

Proof. Let Γ1 ⊂ Γ be any irreducible component of Γ. According to the Extension theorem of Levi it is enough to show that the closed holomorphic 1-form Ω given by in M\Γ, extends meromorphically to some neighborhood of some point p ∈ Γ1 . Let therefore p ∈ Γ1 be a regular point for F (and therefore for Γ) and also for Z. We assume for simplicity of notation that dim M = 2. Since F is regular at p and p ∈ Γ1 is F-invariant we may choose local coordinates (x, y) ∈ U for M, centered at p, such that Γ1 ∩U = {y = 0} and F|U : dy = 0. Thus Ω|U\{y=0} writes as Ω = g(x, y)dy for some holomorphic function g(x, y) in U\{y = 0}. Since Z is holomorphic, regular at p and Γ1 ∂ m ∂ is Z-invariant we may write Z|U = A(x, y) ∂x + y B(x, y) ∂y for some holomorphic A(x, y), B(x, y) in U and some m ∈ N − {0}. Therefore 1 = Ω · Z = ymB(x, y)g(x, y) in U\{y = 0}. This implies 1 that g extends meromorphically to U as g = ymB(x,y) · This proves the lemma. ¥ Remark 10. It follows from the proof above that the order of the pole of Ω along the component Γ1 of Γ is equal to the order of tangency of Z with this component. Also we must have g = g(y) B(y) because Ω is closed so that Ω = ym dy for some unity B(y) in U and by a suitable change of dy coordinates (in the variable y) we may write Ω = ym · This suggests some converse of the above result.

As in the real case (see [8] page 244) we shall say that a non-singular codimension one holomorphic foliation F on M is almost without holonomy if each non-compact leaf of F has trivial holonomy. This deﬁnition can be extended to the case sing(F) 6= ∅ as follows:

Deﬁnition 5. Let F be a codimension one foliation on M with singular set sing(F). A leaf L of F will be called compact singular leaf if L ⊂ L ∪ sing(F) and L is compact. We shall say that F is almost without holonomy if each leaf which is not a compact singular leaf of F, has trivial holonomy; F is a singular compact foliation if each leaf L of F is a compact singular leaf.

Remark 11. If the irreducible component Γ1 of Γ in the proof of Lemma 9 is not invariant by F or by Z then Ω extends holomorphically to Γ1 . Indeed, if Γ1 is not Z-invariant then m = 0 in the 1 proof above and therefore g = g(x,y) is holomorphic. Assume now that Γ1 is not F-invariant. Then

22 we may choose local coordinates (x, y) ∈ U centered at a regular generic point p1 ∈ Γ1 such that ∂ ∂ Γ1 ∩ U = {y = 0},, F|U : dx = 0. We write therefore Z(x, y) = A ∂x + B ∂y and Ω(x, y) = g(x)dx for some holomorphic g in U\{y = 0}. Since g depends only on x it follows that Ω is in fact holomorphic in U and therefore it extends holomorphically to Γ1 . This shows that the hypothesis that F and Z are tangent along Γ is natural.

According to the theorem of Remmert-Stein [10] a leaf L of F satisﬁes L ⊂ L ∪ sing(F) if and only if L is analytic of pure codimension one in M. Therefore, if M is compact, the compact singular leaves of F are naturally associated to invariant codimension one analytic subvarieties in M. An immediate consequence of Lemma 9 is the following:

Corollary 7. Let F be as in Lemma 9 above. Then F is almost without holonomy, indeed, any leaf of F in M\Γ has trivial holonomy.

The following omnibus proposition replaces Lemma 3 of §2 in this singular case:

Proposition 5. Let M be a compact complex manifold equipped with a codimension one singular holomorphic foliation F. Assume that there exist a holomorphic vector ﬁeld Z in M and an analytic codimension one subset Γ ⊂ M such that F is invariant under the transverse ﬂow ϕt of Z in M\Γ and Γ is both F-invariant and Z-invariant. Let Ω be the closed meromorphic 1-form on M such 0 0 that Ω · Z = 1, (Ω)∞ = Γ and F is given in M\Γ by Ω = 0. Let M = M\Γ. Let L0 ⊂ M be any leaf of F. Then:

0 (i) There exists a holomorphic covering σ : L0 ×C → M , σ(x, t) = ϕr(x), and an exact sequence σ# 0 0 0 of groups 0 −→ π1(L0 × C) −→ π1(M ) −→ A −→ 0, where A is a ﬁnitely generated free abelian group, naturally isomorphic to a subgroup Per(Ω) of (C, +).

(ii) If rank A0 = 0 then F admits a holomorphic first integral f : M → CP (1) without base points. 0 If rank A = 1 then F admits a meromorphic first integral f : M\Γ(2) → CP (1) where Γ(2) ⊂ Γ is the divisor of non-simple poles of Ω. This function f has essential singularities on each 0 irreducible component of Γ(2) . If rank A = 2 and Per(Ω) is not a lattice on C then L0 is not 0 closed in M . In case Per(Ω) is a lattice Λ on C the restriction F|M 0 admits a meromorphic first integral f : M 0 → C/Λ over a complex torus.

0 0 (iii) If rank A ≥ 3 then L0 is not closed in M . Proof. We set Ω0 := Ω| , and F 0 = F| so that Ω0 is closed holomorphic, and define H0 := © R ªM 0 M 0 0 0 0 [γ] ∈ π1(M ); γ Ω = 0 . Notice that we have a natural inclusion π1(M ) ⊂ π1(M) and this last 0 is finitely generated group, for M is compact. Also π1(M ) is finitely generated and we define 0 0 0 0 A := π1(M )/H . Notice that H is normal, free and abelian (as in the proof of Lemma 2). Let 0 f0 0 0 0 therefore P : M → M denote the covering of M, corresponding to the group H ⊂ π1(M ). 0 0 0 0 0 f0 0 0 0 0 Consider the set given by M , F ,Ω , ϕt in M , P : M → M , H ⊂ π1(M ) and A . From

23 now on we proceed as in the proof of Lemma 2. The argumentation is essentially the same and we 0 0 e e0 e mention only the basic steps. Given any leaf L0 ⊂ M of F we consider the lifting F, Ω , L0 , e f0 0 f0 0 ϕet and Z to M by P : M → M . Then we have:

(i) Ω0 = df˜0 for some holomorphic submersion f˜0 : Mf0 → C;

˜0 0 ˜0 0 0 f0 (ii) f (ϕet(˜x ) = t + f (˜x ), ∀ x˜ ∈ M , ∀ t ∈ C;

0 e f0 0 0 e (iii) g : L0 × C → M defined by g(˜x , t) = ϕet(˜x ) defines a holomorphic diffeomorphism of L0 × C with Mf0;

e0 e f0 0 (iv) P | : L0 ⊂ M → L0 ⊂ M is a holomorphic diﬀeomorphism; Le0 (v) If we deﬁne η : L × C → L × C by η (x0, s) = (x0, s + t), and G: L × C → Mf0 by ¡¡ t 0¢ ¢ 0 t 0 G(x0, s) = g P 0| −1(˜x), s then the following diagram is commutative Le0

0 M 0 ←−−−P Mf0 ←−−−G L × C   0     yϕt yϕet yηt

0 0 P f0 G M ←−−− M ←−−− L0 × C 0 0 0 0 (vi) Deﬁning σ : L0 × C → M as σ := P ◦ G the following diagram commutes

M 0 ←−−−σ L × C  0    ϕty yηt

0 M ←−−− L0 × C

0 0 0 0 0 (vii) Finally, σ : L0 × C → M is a covering map given also by σ(x , t) = ϕt(x ), ∀ x ∈ L0 , ∀ t ∈ C. (viii) We have an exact sequence of groups

σ# 0 0 0 → π1(L0 × C) −→ π1(M ) → A → 0 '' π1(M) A

Thus, there is a naturally deﬁned exact sequence of groups 0 −→ π1(L0 × C) −→ π1(M) −→ 0 A −→ 0, where the map π1(L0 × C) → π1(M) corresponds to σ# : π1(L0 × C) → π1(M ). Assume that A0 has rank zero. Thus A0 = {0} and we have Ω0 = df 0 for some holomorphic submersion f 0 : M 0 → C.

24 0 Claim 6. The map f extends to a meromorphic map f : M → CP (1) with polar divisor (f)∞ = Γ in M.

Proof. Indeed, choose any regular point p ∈ Γ1 for any given irreducible component Γ1 of Γ. There exist local coordinates (x1, . . . , xn−1, y) in a neighborhood U of p in M such that dy Γ ∩ U = Γ ∩ U = {y = 0}, F| : dy = 0, Ω(x , . . . , x , y) = 1 U 1 n−1 ym

0 Pn−1 ∂f ∂f for some m ∈ N. On the other hand Ω | = df = dxj + dy, so that f = f(y) U\{y=0} j=1 ∂xj ∂y 0 1 P k in U\{y = 0} and f (y) = ym in U\{y = 0}. Thus, the Laurent series f(y) = k∈Z fky satisﬁes P k−1 1 1−m 1 1−m k∈Z kfk f = ym what gives kfk = 0 for k − 1 6= −m and f = f1−my = 1−m y . Therefore, m 6= 1 and f extends meromorphically to U. Hartogs’ Theorem implies that f extends meromorphically to Γ1 and consequently, since Γ1 ⊂ Γ is arbitrary, f is meromorphic in M. We have also proved that (f)∞ = Γ in M. ¥ 0 ∗ Assume now that rank A = 1. In this case A is isomorphic to the group t0Z for some t0 ∈ C and, 0 0 ∗ 0 df 0 as in the proof of Lemma 2, we obtain a holomorphic map f : M → C ' C/t0Z such that Ω = f 0 0 in M . Let Γ1 ⊂ Γ be an irreducible component. We have two distinct cases to consider:

1st. case: Ω has simple poles along Γ1 . In this case we may choose local coordinates (x1, . . . , xn−1, y) ∈ dy ∗ U as above such that Ω(x1, . . . , xn−1, y) = λ y for some λ ∈ C , λ1 ∩ U = {y = 0}. We also have · ¸ df 1 Xn−1 ¡ ∂f ¢ ¡ ∂f ¢ Ω0| = = dx + dy U\{y=0} f f ∂x j ∂x j=1 j j

df(y) f 0(y) λ 0 so that f = f(y) and f(y) = f(y) dy = y dy in U\{y = 0}. Therefore yf (y) − λ f(y) = 0 and if P k P k P k we use Laurent series f(y) = fky then we obtain kfkf − λfky = 0 ⇒ (k − λ)fk = 0 k∈Z k∈Z k∈Z k0 ∀ k ∈ Z, and therefore λ = k0 ∈ N and f(y) = fk0 y so that it extends meromorphically to U and consequently to Γ1 .

2nd. case: Ω has poles of order m > 1 along Γ1 . Usingµ the above¶ notation we write Ω(x1, . . . , xn−1, y) = dy f 0(y) 1 1 1 ∗ ym and obtain f(y) = ym · This gives f = c. exp 1−m · ym−1 , for some constant c ∈ C in

U\{y = 0}. Clearly, such a function does not extend meromorphically to Γ1 .

We have proved that f extends meromorphically to M\Γ(2) where Γ(2) ⊂ Γ is the union of irreducible components Γ1 of Γ such that Ω has simple poles along Γ1 . Moreover f has essential singularities along Γ(2) .

25 Assume now that rank A0 = 2. Suppose first that A0 ⊂ C is discrete. Therefore A0 is (isomorphic to) some lattice Λ ⊂ C and we obtain a well-defined holomorphic submersion f 0 : M 0 → C/Λ such that f 0 is a holomorphic first integral for F 0 on M 0. In particular, F 0 has closed leaves in M 0. As we will see (Example 7 below) it is not necessarily true that the leaves of F are closed in M\ sing(F); 0 0 0 also f may do not extend meromorphically to M. If A ⊂ C is not discrete then clearly L0 ⊂ M is not closed in M 0, indeed, no leaf of F 0 is closed in M 0. Finally, we consider theR case rank A ≥ 3. Consider the natural embedding A,→ (C, +) given by 0 0 0 0 0 ξ : π1(M ) → (C, +), γ 7→ γ Ω and ξ : π1(M /H ) ,→ ξ(π1(M )) ⊂ (C, +) recalling that H = Ker(ξ) 0 0 0 0 0 and A := π1(M )/H . Also recall that A acts on F in the following way: any element T of the 0 0 group of covering transformations of σ : L0 ×C → M satisfies T (L0 ×{0}) = L0 ×{t0(T )}, for some 0 0 0 unique t0(T ) ∈ C and T 7→ t0(T ) defines an action of A on the leaf space of F in M . Therefore 0 0 F has some closed leaf L0 if and only if A embeds as a discrete subgroup of (C, +) and therefore it must have rank ≤ 2. This actually shows that rank A ≥ 3 ⇒ F 0 has all leaves non-closed in M 0 ⇒ F has a finite number of “compact” singular leaves (those contained in Γ) and all the other leaves are not closed in M\ sing(F). This ends the proof of Proposition 5. ¥

Example 7. Let Ω be the closed rational 1-form in the projective plane CP (2) given by Ω|C2 = dx − λ dy , λ ∈ C∗ as in Example 7. We deﬁne F and Γ as before, M = CP (2), M 0 = CP (2)\Γ = x y © 2 0 0 0 0 0 0 C \{xy R= 0}. Weª have π1(M ) ' Z ⊕ Z, thus rank A ≤ 2 where A = π1(M )/H and H = [γ] ∈ 0 0 0 π1(M ); γ Ω = 0 , Ω := Ω|M 0 .

2 TakeR simple positivelyR oriented loops α, β in C \{xy = 0} around the two coordinate axis. We know 0 0 0 Z⊕Z 0 0 hat α Ω = 2πi and β Ω = 2πi.λ. Therefore A ' 2πZ+2πiλZ · Thus rank A > 0 and rank A = 2 if, and only if, λ∈ / Q. If λ ∈ Q then rank A0 = 1 and it is well-known that in this case F exhibits a rational ﬁrst integral f = xnym (for some n, m ∈ Z) on CP (2) notice that Ω has simple poles in CP (2) and compare with Proposition 5, (ii)). If λ∈ / R then rank A = 2 and Λ = {1, λ} ⊂ C is a lattice. On the other hand, it is well-known that the leaf space of F|C2\{xy=0} is a complex torus. Now, for λ ∈ R\Q the subgroup generated by {1, λ} ⊂ C is not discrete, and it is known that λ the closure of any leaf L of F|C2\{xy=0} is a real 3-subvariety given by some equation |y| |x| = c, 2 0 c ∈ R+ so that these leaves one not closed in C \{xy = 0} = M .

Question 4. A foliation given by a non-singular closed holomorphic 1-form on a compact manifold is compact if and only if it exhibits some compact leaf (indeed, such a compact leaf has trivial holonomy so that we may apply the Stability Theorem proved in [4]). How can we generalize this fact to singular foliations?

0 0 0 Lemma 10. Let F, L0 , Ω, Z, M, Γ, M = M\Γ, F = F|M 0 , ϕt in M , be as in Proposition 5, with M compact. Let Per(L0) = {t ∈ C; ϕt(L0) = L0} and ηt : L0 × C → L0 × C be the map 0 0 ηt(x , s) = (x , s + t). Then:

26 0 0 0 (i) If T : L0 × C → L0 × C is a covering transformation of σ : L0 × C → M , (x , t) 7→ ϕt(x )

then T (L0 × {0}) = ηt0 (L0 × {0}) for some t0 = t0(T ) ∈ C.

0 (ii) The correspondence T 7→ t0(T ) deﬁnes an isomorphism A 7→ Per(L0). ½ ¾ R 0 0 0 (iii) Per(L0) = γ Ω ; γ ∈ π1(M ) is the group of periods of Ω .

0 0 Corollary 8. Let F, ϕt , A , Ω , Γ ⊂ M and M compact be as in Proposition 5 above. Then 0 0 ≤ rank A ≤ rank H1(M\Γ, Z). In particular, if rank H1(M\Γ, R) = 0 then F is a singular compact foliation on M.

Proof. It is enough to remark that rank H1(M\Γ, R) ≥ rank H1(M\Γ, Z). ¥

At this point one could ask whether for the case rank H1(M\Γ, Z) = 2, is it true that F is a singular compact foliation. Example 7 above shows that the answer to this question is no. This same example shows that we may have A0 corresponding to a lattice in C, but with corresponding foliation F having some non-closed leaves in M 0. The following result is a straightforward consequence of Proposition 5 and Lemma 10.

Corollary 9. Let F, Ω, ϕt , A, Per(Ω), Γ ⊂ M, M compact be as in Corollary 8. Then: (i) F is a singular compact foliation if, and only if, either Per(Ω) = 0 or Per(Ω) ⊂ C is a (rank two) lattice.

(ii) In case Per(Ω) 6= 0 and is not a rank 2 lattice, the leaves of F are not closed in M 0 = M\Γ.

Proposition 6. Let M be a compact complex manifold supporting a closed holomorphic 1-form Ω ∈ H0(M;Ω1), with singular set of codimension ≥ 2. Assume that Per(Ω) ⊂ (C, +) is discrete. Then F : Ω = 0 is a singular compact foliation and there exists a singular generalized fibration, i.e., a proper holomorphic map f : M → C/Λ of M into a torus, such that the non-singular fibers of f −1 0 are of the form f (y) = Lx , ∀ x ∈ M with f(x) = y ∈ C/Λ. Proof. Owing to (the proof of) Proposition 5 we know that rank A0 = rank Per(Ω) = 2 and since Per(Ω) is discrete it is either trivial or rank two a lattice. Therefore,R may consider a first integral f 0 : M 0 → C/Λ, Λ = {Per(Ω)} ⊂ C for F in M 0. We have f(x) = x Ω (mod Λ), where x0 0 0 0 x0 ∈ M is some fixed point, and x ∈ M is any point. Therefore it remains to show that f extends holomorphically to M. Indeed, we have df 0 = Ω in M 0. Therefore f 0 extends holomorphically to a map f : M → C/Λ. ¥

27 References

[1] Anosov, D.V., 1969, Geodesic ﬂows on closed Riemannian manifolds with negative curvature, Proceedings of the Steklov Institute of Mathematics (A.M.S. translations) 90.

[2] Barth, W., Peters, C. and Van de Ven, A., 1984, Compact Complex Surfaces (Berlin: Springer- Verlag).

[3] Catanese, F., 1991, Moduli and classification of irregular Kählermanifolds (and algebraic vari- eties) with Albanese general type fibrations. Inv. Math., 104, 263-289.

[4] Brunella, M., 1997, A global stability theorem for codimension one transversely holomorphic foliations. Annals of Global Analysis and Geometry, 15(2), 179-186.

[5] Franks, J., 1968, Anosov diﬀeomorphisms. Proceedings of Symposia in Pure Math. Vol XIV, Berkeley, Calif., 61-93.

[6] Ghys, E., 1995, Holomorphic Anosov Systems. Inv. Math., 119, 585-614.

[7] Ghys, E., 1996, Feuilletages holomorphes de codimension un sur les espaces homog`enescom- plexes. Ann. Fac. Sci. Toulouse, (6) 3, 493-519.

[8] Godbillon, C., 1991, Feuilletages: Etudes´ géométriques (Berlin: Birkhäuser).

[9] Grauert, H. and Remmert, R., 1979, Theory of Stein Spaces (Berlin: Springer-Verlag).

[10] Gunning, R.C., 1990, Introduction to holomorphic functions of several variables; vol. I, Func- tion Theory, (Paciﬁc Grove CA: Wadsworth & Brooks/Cole Advanced Books & Software).

[11] Gunning, R.C., 1990, Introduction to holomorphic functions of several variables, vol. II, Local Theory (Monterey CA: Wadsworth & Brooks/Cole Advanced Books & Software).

[12] Griﬃths, P. and Harris, J., 1978, Principles of Algebraic Geometry (New-York: John Wiley & Sons).

[13] Hirsch, M., 1971, Anosov maps, polycyclic groups and homology. Topology, 10, 177-183.

[14] Hirsch, M. and Pugh, C., 1968, Stable manifolds and hyperbolic sets. Collection: Global Anal- ysis (Proc. Sympos. Pure Math.), Vol. XIV, Berkeley, Calif., 133–163.

[15] Huckleberry, A.T., 1990, Actions of groups of holomorphic transformations. Encyclopaedia of Mathematical Sciences, Several Complex Variables, eds. W. Barth & R. Narashimhan, 6, 143- 196.

28 [16] Newhouse, S., 1970, On codimension one Anosov diﬀeomorphisms. American Journal of Math- ematics, 42, 761-770.

[17] Plante, J. F. and Thurston, W. P., 1972, Polynomial Growth in Holonomy Groups of Foliations. Comment. Math. Helvetici, 39 (51), 567-584.

[18] Plante, J., 1972, Anosov Flows. Amer. J. Math., 94, 729-754.

[19] Spanier, E.H., 1966, Algebraic Topology (New York: Mac Graw-Hill).

[20] Smale, S., 1967, Diﬀerentiable dynamical systems. Bulletin of the American Mathematical Society, 73, 747-817.

[21] Tischler, D., 1970, On ﬁbering certain foliated manifolds over S1. Topology, 9, 153-154.

[22] Wells, R.O., 1973, Diﬀerential Analysis on Complex Manifolds (New Jersey: Prentice Hall).