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Holomorphic Anosov Systems, Foliations and Fibering

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Holomorphic Anosov Systems, Foliations and Fibering Holomorphic Anosov systems, foliations and fibering com- plex 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 codimen- sion 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 't 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 ' 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 't : 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 't ; 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) ½ 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 't. 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 ' 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 ½ 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 sub- mersion f : M ! R onto a complex manifold R. In this case, according to Ehresmann Theorem [8] M is a locally trivial C1-fibre bundle over R, whose fibers are the connected components of the levels f ¡1(y); y 2 R of f. In particular, these levels are C1-diffeomorphic; usually they are not holomorphically equivalent. In general a compact manifold M equipped with a non-singular closed 1-form admits a C1-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 K¨ahlersurface or a compact homogeneous manifold; whose basic framework is introduced in x2. We prove for instance that (cf. Theorem 2) if M n+1 is a compact K¨ahlermanifold 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 !1; :::; !n 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 xx2 and 3 apply to the study of jointly-integrable holomorphic Anosov flows and codimension one holomorphic Anosov diffeomorphisms. This is done in x4. After recalling some basic examples and introducing some features from [6] we prove that if a compact K¨ahlermanifold M supports a codimension one transitive holomorphic Anosov diffeo- morphism f : M ! M then M is (holomorphically conjugate to) a complex torus C=Λ, for some lattice Λ ½ Cn, and f is (holomorphically conjugate to) a linear automorphism of Anosov type (Theorem 3 x4). The study of jointly integrable holomorphic Anosov flows is carried out in xx4.2 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 x4.3). The questions addressed in the previous sections (xx2,3 and 4) are also interesting in the non- compact case which is considered in x5. 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 codi- mension at least 2 in M (recall that sing(F) ½ M is always an analytic subset). Using Hartogs’ Extension Theorem [10] we first observe that if F invariant under some holomorphic transverse flow on Mn sing(F) then indeed F is non-singular (sing(F) = ;) and we are in the same situation of Lemma 1 of x2 (see Lemma 8 x5.1). Motivated by this and by some concrete examples as, for instance, linear foliations on complex projective spaces (cf. Example 6 in x5.1) we are led to con- sider the following situation: F is a codimension one foliation with singularities on M, there exist an analytic codimension one subset Γ ½ M and a holomorphic vector field Z in M such that Z is transverse to F in MnΓ and FjMnΓ is invariant by the (transverse) flow 't of Z in MnΓ. The non-trivial case arises when Γ is simultaneously F and Z invariant (Remark 11 x5.1). 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)1 = Γ and Ω ¢ Z = 1 in M (Lemma 9). Let also M be compact and L0 ½ MnΓ any leaf of F. We prove that (analogously to Lemmas 2 and 3 of x2) the dynamics of FjMnΓ is classified by the structure of the group of periods of the 0 restriction ΩjMnΓ : there exists a holomorphic covering L0£C ! MnΓ whose group A is (isomorphic to) the group of periods Per(ΩjMnΓ). In particular, if H1(MnΓ; R) = 0 then Per(ΩMnΓ) = 0 and F is a singular compact foliation given by a holomorphic map f : M ! CP (1) (Proposition 5). 1 The leaf space of FjMnΓ is C -diffeomorphic to a complex torus if and only if Per(ΩjMnΓ) is a lattice in C, in which case FjMnΓ has closed leaves on MnΓ. These leaves are also closed in M if rank Per(ΩjMnΓ) = 1. In any other case the leaves of F are not closed in MnΓ. 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 ': C£M !¯M be a holomorphic flow on M. We say that ' is a flow transverse ¯ @' ¯ to F if the vector field Z = @t ¯ (where t 2 C is the complex time) is transverse to (the leaves t=0 of) F.
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