A global Weinstein splitting theorem for holomorphic Poisson manifolds
Stéphane Druel, Jorge Vitório Pereira, Brent Pym, Frédéric Touzet
aa r X i v : . [ m a t h . AG ] F e b A global Weinstein splitting theorem forholomorphic Poisson manifolds
St´ephane Druel ∗ Jorge Vit´orio Pereira † Brent Pym ‡ Fr´ed´eric Touzet § February 26, 2021
Abstract
We prove that if a compact K¨ahler Poisson manifold has a symplecticleaf with finite fundamental group, then after passing to a finite ´etalecover, it decomposes as the product of the universal cover of the leafand some other Poisson manifold. As a step in the proof, we establish aspecial case of Beauville’s conjecture on the structure of compact K¨ahlermanifolds with split tangent bundle.
In 1983, Weinstein [14] proved a fundamental fact about Poisson brackets, nowa-days known as his “splitting theorem”: if p is a point in a Poisson manifold X at which the matrix of the Poisson bracket has rank 2 k , then p has a neigh-bourhood that decomposes as a product of a symplectic manifold of dimension2 k , and a Poisson manifold for which the Poisson bracket vanishes at p . Animportant consequence of this splitting is that X admits a canonical (and pos-sibly singular) foliation by symplectic leaves, which is locally induced by theaforementioned product decomposition.It is natural to ask under which conditions this splitting is global , so that X decomposes as a product of Poisson manifolds, having a symplectic leaf as afactor (perhaps after passing to a suitable covering space). If X is compact, anobvious necessary condition is that the symplectic leaf is also compact. However,this condition is not sufficient; it is easy to construct examples in the C ∞ contextwhere both X and its leaf are simply connected, but X does not decompose as aproduct.In contrast, we will show that for holomorphic Poisson structures on compactK¨ahler manifolds, the existence of such splittings is quite a general phenomenon: Theorem 1.1.
Let ( X , π ) be a compact K¨ahler Poisson manifold, and supposethat L ⊂ X is a compact symplectic leaf whose fundamental group is finite. Thenthere exist a compact K¨ahler Poisson manifold Y , and a finite ´etale Poissonmorphism e L × Y → X , where e L is the universal cover of L . ∗ CNRS/Universit´e Claude Bernard Lyon 1, [email protected] † IMPA, [email protected] ‡ McGill University, [email protected] § Universit´e Rennes 1, [email protected] X , π )where X is a complex manifold that admits a K¨ahler metric, and π ∈ H ( X , ∧ T X )is a holomorphic bivector that is Poisson, i.e. the Schouten bracket [ π, π ] = 0.We will not make reference to any specific choice of K¨ahler metric at any pointin the paper.We remark that all hypotheses of Theorem 1.1 are used in an essential wayin the proof. Indeed, in Section 4, we give examples showing that the conclusionmay fail if the Poisson manifold X is non-K¨ahler ( C ∞ or complex analytic), if X is not compact, or if the fundamental group of L is infinite.The proof of Theorem 1.1 is given in Section 4. It rests on the following resultof independent interest, which we establish using Hodge theory in Section 2. Theorem 1.2.
Let ( X , π ) be compact K¨ahler Poisson manifold, and supposethat i : L ֒ → X is the inclusion of a compact symplectic leaf. Then the holo-morphic symplectic form on L extends to a global closed holomorphic two-form σ ∈ H ( X , Ω X ) of constant rank such that the composition θ := π ♯ σ ♭ ∈ E nd ( T X ) is idempotent, i.e. θ = θ . Here π ♯ : Ω X → T X and σ ♭ : T X → Ω X are the usual maps defined byinterior contraction into the corresponding bilinear forms. Following work ofFrejlich–M˘arcut , in the C ∞ setting, we refer to a holomorphic two-form σ as inTheorem 1.2 as a subcalibration of π . The key point about a subcalibration,which they observed, is that it provides a splitting T X = F ⊕ G into a pair ofsubbundles F = img θ and G = ker θ that are orthogonal with respect to π .Moreover F is automatically involutive, and G is involutive if and only if thecomponent of σ lying in ∧ F ∨ is closed. But in the compact K¨ahler setting, thelatter condition is automatic by Hodge theory, and furthermore, any splittingof the tangent bundle into involutive subbundles is expected to arise from asplitting of some covering space of X , according to the following open conjectureof Beauville (2000): Conjecture 1.3 ([3, (2.3)]) . Let X be a compact K¨ahler manifold equipped witha holomorphic decomposition T X = L i ∈ I F i of the tangent bundle such that eachsubbundle L j ∈ J F j , for J ⊂ I , is involutive. Then the universal cover of X isisomorphic to a product Q i ∈ I U i in such a way that the given decomposition of T X corresponds to the natural decomposition T Q i ∈ I U i ∼ = L i ∈ I T U i . Thus, our second key step in the proof of Theorem 1.1 is to prove the fol-lowing theorem, which establishes a special case of Conjecture 1.3, but with astronger conclusion:
Theorem 1.4.
Suppose that X is a compact K¨ahler manifold equipped with asplitting T X = F ⊕G of the tangent bundle into involutive subbundles. If F has acompact leaf L with finite holonomy group, then the splitting of T X is induced bya splitting of the universal cover of X as a product of manifolds. If, in addition, L has finite fundamental group and trivial canonical class, then the splitting ofthe universal cover is induced by a splitting of a finite ´etale cover. F are compact with finite holonomy. We then ap-ply the theory of holonomy groupoids to show that the holonomy covers of theleaves assemble into a bundle of K¨ahler manifolds over X , equipped with a flatEhresmann connection induced by the transverse foliation G . Finally, we ex-ploit Lieberman’s structure theory for automorphism groups of compact K¨ahlermanifolds [8] to analyze the monodromy action of the fundamental group of X on the fibres, and deduce the result.The results above have several interesting consequences for the global struc-ture of compact K¨ahler Poisson manifolds. For instance Theorem 1.1 immedi-ately implies the following statement. Corollary 1.5. If ( X , π ) is a compact K¨ahler Poisson manifold, then all simplyconnected compact symplectic leaves in ( X , π ) are isomorphic. Meanwhile, we have the following immediate corollaries of Theorem 1.2:
Corollary 1.6.
Let ( X , π ) be a compact connected K¨ahler Poisson manifoldsuch that the tangent bundle T X is irreducible. If L ⊂ X is a compact symplecticleaf, then either L = X or L is a single point. Corollary 1.7.
Let ( X , π ) be a compact K¨ahler Poisson manifold such that theHodge number h , ( X ) is equal to zero. If L ⊂ X is a compact symplectic leaf,then L is a single point. Note that the vanishing h , ( X ) = 0 holds for a wide class of manifolds ofinterest in Poisson geometry, including all Fano manifolds, all rational manifolds,and more generally all rationally connected manifolds. Many natural examplesarising in gauge theory and algebraic geometry fall into this class. Applied tothe case in which X is a projective space, this answers a question posed by thethird author about the existence of projective embeddings that are compatiblewith Poisson structures: Corollary 1.8.
A compact holomorphic symplectic manifold of positive dimen-sion can never be embedded as a Poisson submanifold in a projective space, forany choice of holomorphic Poisson structure on the latter.
Acknowledgements:
We thank Pedro Frejlich and Ioan M˘arcut , for corre-spondence, and in particular for sharing their (currently unpublished) work onsubcalibrations with us. We also thank Henrique Bursztyn, Marco Gualtieriand Ruxandra Moraru for interesting discussions. In particular Corollary 1.5and Corollary 1.6 were pointed out by Bursztyn and Gualtieri, respectively.This project grew out of discussions that took place during the school on“Geometry and Dynamics of Foliations”, which was hosted May 18–22, 2020by the Centre International de Rencontres Math´ematiques (CIRM), as part ofthe second author’s Jean-Morlet Chair. We are grateful to the CIRM for their3upport, and for their remarkable agility in converting the entire event to asuccessful virtual format on short notice, in light of the COVID-19 pandemic.S.D. was supported by the ERC project ALKAGE (ERC grant Nr 670846).S.D., J.V.P. and F.T. were supported by CAPES-COFECUB project Ma932/19.S.D. and F.T. were supported by the ANR project Foliage (ANR grant NrANR-16-CE40-0008-01). J.V.P. was supported by CNPq, FAPERJ, and CIRM.B.P. was supported by a faculty startup grant at McGill University, and bythe Natural Sciences and Engineering Research Council of Canada (NSERC),through Discovery Grant RGPIN-2020-05191. Throughout this section, we fix a connected complex manifold X and a holo-morphic Poisson structure on X , i.e. a holomorphic bivector π ∈ H ( X , ∧ T X )such that the Schouten bracket [ π, π ] vanishes identically. We recall that the anchor map of π is the O X -linear map π ♯ : Ω X → T X given by contraction of forms into π . Its image is an involutive subsheaf, defininga possibly singular foliation of X . If i : L ֒ → X is a leaf of this foliation, then π L := π | L is a nondegenerate Poisson structure on L , so that its inverse η := π − L ∈ H ( L , Ω L )is a holomorphic symplectic form. The pair ( L , η ) is called a symplectic leaf of ( X , π ). The following lemma gives a sufficient condition for the holomorphic symplecticform on a symplectic leaf to extend to all of X . Lemma 2.1.
Let ( X , π ) be a compact K¨ahler Poisson manifold, and supposethat ( L , η ) is a compact symplectic leaf with inclusion i : L ֒ → X . Then thereexists a global closed holomorphic two-form σ ∈ H ( X , Ω X ) such that i ∗ σ = η .Proof. This proof is a variant of the arguments in [5, Proposition 3.1] and [9,Theorem 5.6]. Suppose that dim L = 2 k and let ω ∈ H ( X , Ω X ) ∼ = H , ( X ) beany K¨ahler class. Note that since π is holomorphic, the contraction operator ι π on Ω • X descends to the Dolbeault cohomology H • ( X , Ω • X ). In particular, we havea well-defined class α := ι kπ ω k ∈ H k ( X , O X )We claim that i ∗ α ∈ H k ( L , O L ) is nonzero. Indeed, since π | L = π L , we have the4ollowing commutative diagram: H k ( X , Ω k X ) ι kπ / / i ∗ (cid:15) (cid:15) H k ( X , O X ) i ∗ (cid:15) (cid:15) H k ( L , Ω k L ) ι kπ L / / H k ( L , O L )The bottom arrow is an isomorphism since π k L is a trivialization of the anti-canonical bundle of L . Meanwhile i ∗ ω k ∈ H k ( L , Ω k L ) is nonzero since i ∗ ω is aK¨ahler class on L . It follows that i ∗ α = ι kπ L i ∗ ω k = 0 as claimed.Using the Hodge symmetry H k ( X , O X ) ∼ = H ( X , Ω k X ), the complex conjugateof α gives a holomorphic 2 k -form µ ∈ H ( X , Ω k X )such that i ∗ µ ∈ H ( L , Ω k L ) is nonzero. Since the canonical bundle of L is trivial, i ∗ µ must be a constant multiple of the holomorphic Liouville volume form as-sociated with the holomorphic symplectic structure on L . Hence by rescaling µ ,we may assume without loss of generality i ∗ µ = k ! η k . Now consider the globalholomorphic two-form σ := k − ι k − π µ, which is closed by Hodge theory, since X is compact K¨ahler. We claim that itrestricts to the symplectic form on L . Indeed, i ∗ σ = k − i ∗ ( ι k − π µ ) = k − ι k − π L i ∗ µ = k − k ! ι k − π L η k = η as desired.The lemma only gives the existence of the holomorphic two-form σ , butsays little about its global properties. Nevertheless, it can be used to produce atwo-form with the following property, the ramifications of which are explainedin Section 2.2 below: Definition 2.2 (Frejlich–M˘arcut , ) . A holomorphic two-form σ ∈ H ( X , Ω X ) iscalled a subcalibration of ( X , π ) if it is closed, and the composition θ := π ♯ σ ♭ ∈ E nd ( T X )is idempotent (i.e. θ = θ ), where σ ♭ : T X → Ω X is the map defined by contrac-tion of vector fields into σ .We will be interested in subcalibrations that are compatible with our chosensymplectic leaf L in the following sense: Definition 2.3.
A subcalibration σ of ( X , π ) is compatible with the sym-plectic leaf i : L ֒ → X if i ∗ img θ = T L ⊂ i ∗ T X .5quivalently, the subcalibration is compatible with L if the projection of π to ∧ img θ is a constant rank bivector that is an extension of the nondegeneratePoisson structure π L on L .The following gives a simple condition that allows an arbitrary extension ofthe two-form on L to be refined to a subcalibration compatible with L . Note thatit applies, in particular, whenever X is a compact connected K¨ahler manifold: Lemma 2.4.
Let ( X , π ) be a holomorphic Poisson manifold and let ( L , η ) be asymplectic leaf of ( X , π ) . Suppose that the following conditions hold:1. h ( X , O X ) = 1 , i.e. every global holomorphic function on X is constant2. H ( X , Ω X ) = H ( X , Ω , cl X ) , i.e. every global holomorphic two-form on X isclosed3. η extends to a global holomorphic two-form on X Then η extends to a subcalibration compatible with L .Proof. Let σ ∈ H ( X , Ω X ) be any extension of the symplectic form on L toa global holomorphic two-form on X , and let θ := π ♯ σ ♭ ∈ E nd ( T X ). Since π L = π | L is inverse to the symplectic form η = i ∗ σ on L , it follows easily that θ | L ∈ E nd ( i ∗ T X ) is idempotent with image T L ⊂ i ∗ T X , giving a splitting i ∗ T X ∼ = T L ⊕ ker θ | L In particular, the characteristic polynomial of θ | L is given by P ( t ) = t n − k ( t − k where n = dim X and k = dim L . But the coefficients of the characteristicpolynomial of θ are holomorphic functions on X , and since h ( X , O X ) = 1, suchfunctions are constant. We conclude that P ( t ) is the characteristic polynomialof θ over all of X .Taking generalized eigenspaces of θ , we obtain a decomposition T X ∼ = F ⊕ G where i ∗ F = T L . Projecting the two-form σ to ∧ F ∨ we therefore obtain a newholomorphic two-form σ such that i ∗ σ = η , which has the additional propertythat G ⊂ ker σ . Then θ = π ♯ σ ♭ also has characteristic polynomial P ( t ), givinga splitting T X = F ′ ⊕ G with respect to which θ takes on the Jordan–Chevalleyblock form θ = (cid:18) φ
00 0 (cid:19) where φ ∈ End ( F ′ ) is nilpotent. Using the identity θ π ♯ = π ♯ θ ∨ as mapsΩ X → T X , one calculates that π must be block diagonal, i.e. equal to the sum ofits projections to ∧ G and ∧ F ′ . The latter projection, say π ′ ∈ ∧ F ′ , is thennondegenerate because θ = π ♯ σ ♭ is invertible on F ′ . We may therefore define6 two-form σ := ( π ′ ) − ∈ ∧ F ′ ⊂ Ω X . By construction, the endomorphism θ := π ♯ σ ♭ preserves the decomposition T X = F ′ ⊕ G , acts as the identity on F ′ , and has G as its kernel. In particular, θ is idempotent and σ restricts tothe symplectic form on L . Moreover, σ is closed by hypothesis. Thus σ is asubcalibration of π compatible with L , as desired.Combining Lemma 2.1 and Lemma 2.4, we immediately obtain the followingstatement, which is a rephrasing of Theorem 1.2 from the introduction: Corollary 2.5. If ( X , π ) is a compact K¨ahler manifold equipped with a holo-morphic Poisson structure and L ⊂ X is a compact symplectic leaf, then thereexists a subcalibration of ( X , π ) compatible with L . Note that if σ is a subcalibration of ( X , π ), then the operator θ := π ♯ σ ♭ gives adecomposition T X = F ⊕ G of the tangent bundle into the complementary subbundles F := img θ, G := ker θ. We may then project π onto the corresponding summands in the exterior powersto obtain global bivectors π F ∈ H ( X , ∧ F ) π G ∈ H ( X ∧ G ) . Similarly, the form σ projects to a pair of sections σ F ∈ H ( X , ∧ F ∨ ) σ G ∈ H ( X , ∧ G ∨ ) , which we may view as global holomorphic two-forms on X via the splitting T ∨ X ∼ = F ∨ ⊕ G ∨ .An elementary linear algebra computation then shows that π = π F + π G , σ = σ F + σ G and π F is inverse to σ F on F , so that F = img π F and G = ker σ F . With this notation in place, we may state the following fundamental resultabout subcalibrations, due to Frejlich–M˘arcut , , which will play a key role inwhat follows: Theorem 2.6 (Frejlich–M˘arcut , ) . Suppose that σ is a subcalibration of ( X , π ) .Then the bivector fields π F , π G are Schouten commuting Poisson structures, i.e. [ π F , π F ] = [ π G , π G ] = [ π F , π G ] = 0 . (1) In particular, F = img π F is involutive. Moreover, G is involutive if and onlyif σ F is closed. roof. The proof of Frejlich–M˘arcut , makes elegant use of the notion of a Diracstructure. For completeness, we present here an essentially equivalent argumentbased on the related notion of a gauge transformation of Poisson structures.We recall from [13, Section 4] that if B is a closed holomorphic two-form suchthat the operator 1 + B ♭ π ♯ ∈ E nd (Ω X ) is invertible, then the gauge transforma-tion of π by B is a new Poisson structure B ⋆ π ∈ H ( X , ∧ T X ) whose underlyingfoliation is the same as that of π , but with the symplectic form on each leafmodified by adding the pullback of B . Equivalently, B ⋆ π is determined by itsanchor map, which is given by the formula (
B ⋆π ) ♯ = π ♯ (1+ B ♭ π ♯ ) − : Ω X → T X .The skew symmetry of B ⋆ π follows from the skew symmetry of π and B , whilethe identity [ B ⋆ π, B ⋆ π ] = 0 is deduced using the closedness of B and theequation [ π, π ] = 0.We apply this construction to the family of two-forms B ( t ) := tσ . Note thatsince θ := π ♯ σ ♭ is idempotent, the operator 1 + B ♭ ( t ) π ♯ = 1 + tθ ∨ ∈ E nd (Ω X ) isinvertible for all t = −
1. We therefore obtain a family of holomorphic Poissonbivectors π ( t ) := B ( t ) ⋆ π, t ∈ C \ {− } , with anchor map π ♯ ( t ) = π ♯ (1 + tθ ∨ ) − . Since θ is idempotent, we have (1 + tθ ∨ ) − = 1 − t t θ ∨ , which implies that π ( t ) = π F + π G − t t π F = t π F + π G for all t = −
1. Since [ π ( t ) , π ( t )] = 0 for all t = −
1, the bilinearity of the Schoutenbracket therefore implies the desired identities (1). This implies immediatelythat F = img π F is the tangent sheaf of the symplectic foliation of the Poissonbivector π F ; in particular, it is involutive.It remains to show that G is involutive if and only if σ F is closed. To thisend, observe that if σ F is closed, then its kernel G = ker σ F is involutive byelementary Cartan calculus. Conversely, if G is involutive, then both F ∨ and G ∨ generate differential ideals in Ω • X . This implies that the exterior derivativesof σ F ∈ ∧ F ∨ and σ G ∈ ∧ G ∨ lie in complementary subbundles of Ω X , namelyd σ F ∈ ∧ F ∨ ⊕ ( ∧ F ∨ ⊗ G ∨ ) d σ G ∈ ( ∧ G ∨ ⊗ F ∨ ) ⊕ ∧ G ∨ . Since d σ F + d σ G = d σ = 0, we conclude that d σ F = 0, as desired. The subcalibrations discussed in the previous section give, in particular, a de-composition of the tangent bundle into involutive subbundles. In this section wefix a complex manifold X and give criteria for such a decomposition of T X to arisefrom a decomposition of some covering of X as a product as in Conjecture 1.3.The main objects are therefore the following:8 efinition 3.1. Suppose that F is a regular foliation of X . A foliation com-plementary to F is an involutive holomorphic subbundle G ⊂ T X such that T X = F ⊕ G .Note that the definition is symmetric: if G is complementary to F then F is complementary to G . However, in what follows, the foliations F and G willplay markedly different roles. Definition 3.2.
Let
F ⊂ T X be a regular holomorphic foliation. We say that F is a fibration if there exists a surjective holomorphic submersion f : X → Y whose fibres are the leaves of F . In this case, we call Y the leaf space of F and the map f the quotient map . Remark . The submersion f : X → Y , if it exists, is unique up to isomor-phism, so there is no ambiguity in referring to Y as “the” leaf space of thefoliation F . ♦ Suppose that F is a fibration, and G is a foliation complementary to F .Then G is precisely the data of a flat connection on the submersion f : X → Y ,in the sense of Ehresmann [6]. Recall that such a connection is complete if ithas the path lifting propery, i.e. given any point y ∈ Y , any path γ : [0 , → Y starting at γ (0) = y , and any point x lying in the fibre f − ( y ) ⊂ X , there existsa unique path e γ : [0 , → X that is tangent to the leaves of G and starts atthe point e γ (0) = x . If f is proper, then every flat Ehresmann connection iscomplete in this sense. Definition 3.4.
Suppose that ( F , G ) is a pair consisting of a regular holomor-phic foliation F and a complementary foliation G . We say that the pair ( F , G ) isa suspension if F is a fibration for which G defines a complete flat Ehresmannconnection.Suppose that ( F , G ) is a suspension with underlying fibration f : X → Y ,and y ∈ Y is a point in the leaf space of F . Let L = f − ( y ) be the fibre. The monodromy representation at y is the homomorphism π ( Y , y ) → Aut ( L )defined by declaring that a homotopy class [ γ ] ∈ π ( Y , y ) acts on L by sending x ∈ L to e γ (1) where e γ is the path lifting γ with initial condition e γ (0) = x .Lifting arbitrary paths in Y based at y then gives a canonical isomorphism X ∼ = L × e Y π ( Y , y )where e Y is the universal cover of Y based at y , and π ( Y , y ) acts diagonally onthe product. Moreover the pullbacks of F and G to L × e Y coincide with thetangent bundle of the factors L and e Y , respectively, as in Conjecture 1.3. In thisway, we obtain an equivalence between suspensions ( F , G ) and homomorphisms ρ : π ( Y , y ) → Aut ( L ), where Y and L are complex manifolds and y ∈ Y .9 emark . We will make repeated use of the following observation: if X iscompact K¨ahler, the restriction of any K¨ahler class on X to a leaf L of F gives aK¨ahler class ω ∈ H , ( L ) that is invariant under the monodromy representation,i.e. the monodromy representation is given by a homomorphism ρ : π ( Y , y ) → Aut ω ( L )where Aut ω ( L ) is the group of biholomorphisms of L that fix the class ω . Thestructure of Aut ω ( L ) is well understood thanks to work of Lieberman [8], andthis will allow us to control the behaviour of various suspensions. ♦ In Theorem 3.9 below, we will give criteria for a pair of foliations on a compactK¨ahler manifold to be a suspension. Our key technical tool is the holonomygroupoid
Hol ( F ) of a regular foliation F . We briefly recall the construction andrefer the reader to [11, 15] for details. (Note that in [15], the holonomy groupoidis called the “graph” of F .)If x, y are two points on the same leaf L of the foliation F , and γ is a pathfrom x to y in L , then by lifting γ to nearby leaves one obtains a germ of abiholomorphism from the leaf space of F| U x to the leaf space of F| U y where U x , U y ⊂ X are sufficiently small neighbourhoods of x and y , respectively. Thisgerm is called the holonomy transformation induced by γ . We say that twopaths tangent to F have the same holonomy class if their endpoints are thesame, and they induce the same holonomy transformation.The holonomy groupoid Hol ( F ) is the set of holonomy classes of pathstangent to F . It carries a natural complex manifold structure of dimension equalto dim X + rank F , and comes equipped with a pair of surjective submersions s, t : Hol ( F ) → X that pick out the endpoints of paths. The usual compositionof paths then makes Hol ( F ) into a complex Lie groupoid over X . Remark . In the differentiable setting, the holonomy groupoid may fail tobe Hausdorff, but in the analytic setting we consider here, the Hausdorffness isguaranteed by [15, Corollary of Proposition 2.1]. ♦ Remark . The map ( s, t ) :
Hol ( F ) → X × X is an immersion [15, 0.3]. Hencea K¨ahler structure on X induces a K¨ahler structure on Hol ( F ) by pullback. ♦ Suppose that x ∈ X , and let L ⊂ X be the leaf through X . We denote by Hol ( F ) x ⊂ Hol ( F ) the group of holonomy classes of loops based at x . Since ho-motopic loops induce the same holonomy transformation, Hol ( F ) x is a quotientof the fundamental group π ( L , x ). Moreover, it acts freely on the fibre s − ( x ),and the map t descends to an isomorphism s − ( x ) / Hol ( F ) x ∼ = L . Put differ-ently, the fibration s − ( L ) → L is a fibre bundle equipped with a complete flatEhresmann connection whose horizontal leaves are the fibres t − ( y ) ⊂ s − ( L )where y ∈ L . The holonomy group Hol ( F ) x is the image of the homomorphismhol x : π ( L , x ) → Aut (cid:0) s − ( x ) (cid:1) (2)10btained by taking the monodromy of this flat connection.Now suppose that G is a foliation complementary to F . Then the preimage t − G ⊂ T
Hol ( F ) defines a regular foliation on Hol ( F ). The leaves of this foliationare the submanifolds of the form t − ( W ) where W ⊂ X is a leaf of G . Lemma 3.8.
Suppose that F is a regular foliation of X such that the map s : Hol ( F ) → X is proper, and that G is a foliation complementary to F . Thenthe following statements hold:1. The foliation t − G ⊂ T
Hol ( F ) defines a complete flat Ehresmann connectionon the fibration s : Hol ( F ) → X .2. If L ⊂ X is a leaf of F , then the t − G -horizontal lifts of L are exactly thefibres t − ( y ) for y ∈ L .3. If x ∈ X and L is the leaf of F through x , then the monodromy rep-resentation ρ x : π ( X , x ) → Aut (cid:0) s − ( x ) (cid:1) of t − G extends the holonomyrepresentation of F at x , i.e. the following diagram commutes: π ( L , x ) hol x (cid:15) (cid:15) (cid:15) (cid:15) / / π ( X , x ) ρ x (cid:15) (cid:15) Hol ( F ) x (cid:31) (cid:127) / / Aut (cid:0) s − ( x ) (cid:1) Proof.
For the first statement, note that rank G = dim X − rank F and the fibresof t have dimension equal to rank F . Therefore t − G has rank equal to dim X .Note that G is identified with the normal bundle of the foliation F and therefore t − G surjects by s onto the normal bundle of every leaf. Meanwhile every t -fibresurjects by s onto the corresponding leaf. Considering the ranks, it follows that t − ( G ) is complementary to the fibres of s , defining a flat Ehresmann connection.Since s is proper, this connection is complete, as desired.For the second statement, note that the horizontal lifts of a leaf L ⊂ X of F are, by definition, given by intersecting the preimage s − ( L ) with the leaves of t − G . But t projects s − ( L ) onto L , which is complementary to G . It follows thatthe intersection of the leaves of t − G with s − ( L ) are the fibres t − ( y ) for y ∈ L ,as claimed. The third statement then follows immediately from the descriptionof the holonomy representation (2) above. We are now in a position to prove our main result on complementary foliations.
Theorem 3.9.
Let F be a regular foliation on a compact K¨ahler manifold, andsuppose that F has a compact leaf L ⊂ X with finite holonomy group. Then thefollowing statements hold:1. For every foliation G complementary to F , there exists a finite ´etale cover φ : e X → X such that ( φ − F , φ − G ) is a suspension. . If, in addition, the fundamental group of L is finite and the universal cover e L admits no nonzero holomorphic vector fields, i.e. h ( e L , T e L ) = 0 , then wecan arrange so that the suspension in statement 1 is trivial, i.e. there existsa compact K¨ahler manifold Y and a finite ´etale cover φ : e L × Y → X such that φ − F and φ − G are identified with the tangent bundles of thefactors e L and Y , respectively. As an immediate corollary of part 1 of Theorem 3.9, we obtain the followingspecial case of Beauville’s conjecture:
Corollary 3.10.
Conjecture 1.3 holds for decompositions of the tangent bundleof the form T X = F ⊕ G where F has a compact leaf with finite holonomy.Proof of Theorem 3.9, part 1. Suppose that X is a compact K¨ahler manifold and F is a regular holomorphic foliation having a compact leaf with finite holon-omy group. Then by the global Reeb stability theorem for compact K¨ahlermanifolds [12, Theorem 1], every leaf of F is compact with finite holonomygroup. As observed in [11, Example 5.28(2)], this implies that the submersions s, t : Hol ( F ) → X are proper maps.If G is any foliation complementary to F , we obtain from Lemma 3.8 a flatEhresmann connection on the fibration s : Hol ( F ) → X whose monodromyrepresentation induces the holonomy representation of every leaf. Let us choosea base point x ∈ X , and consider the monodromy representation ρ : π ( X , x ) → Aut (cid:0) s − ( x ) (cid:1) . Note that by Remark 3.7,
Hol ( F ) is a K¨ahler manifold. If we choose a K¨ahlerclass on Hol ( F ), its restriction to s − ( x ) gives a K¨ahler class ω ∈ H , ( s − ( x ))that is invariant under the monodromy action. Hence ρ factors through thesubgroup Aut ω ( s − ( x )) of biholomorphisms of s − ( x ) that fix the class ω .By Lemma 3.11 below applied to the monodromy action of Π := π ( X , x )on Z := s − ( x ), there exists a finite-index subgroup Γ < π ( X , x ) whose image ρ (Γ) < Aut ω ( s − ( x )) contains no finite subgroups that act freely on s − ( x ). Let φ : e X → X be the covering space corresponding to Γ, with base point e x ∈ e X chosen so that φ ∗ π ( e X , e x ) = Γ. Then φ is a finite ´etale cover since Γ has finiteindex.We claim that the pair ( φ − F , φ − G ) is a suspension. Note that to provethis, it suffices to show that the holonomy groups of the leaves of φ − F aretrivial, for in this case the holonomy groupoid of φ ∗ F embeds as the graph ofan equivalence relation in e X × e X , which in turn implies that φ − F is a fibrationwhose quotient map is proper. Then the complementary foliation φ − G is a flatEhresmann connection, as desired, and this establishes part 1 of the theorem.To see that the holonomy groups of the leaves of φ − F are indeed trivial,suppose that e L ⊂ e X is a leaf of φ − F . Then φ restricts to an ´etale cover e L → L where L is a leaf of F . If we choose a base point e y ∈ e L with image y = φ ( e y ) ∈ L ,12hen the germ of the leaf space of φ − F at e y is identified with the germ ofthe leaf space of F at y , so that the holonomy group of e L at e y is canonicallyidentified with the image of the composition π ( e L , e y ) / / π ( L , y ) / / Hol ( F ) y . Choose a homotopy class of a path e γ from e x to e y , let γ be its projection to ahomotopy class from x to y , and let hol γ : s − ( y ) → s − ( x ) be the isomorphismgiven by parallel transport of the Ehresmann connection on Hol ( F ) → X . Theadjoint actions of e γ, γ and hol γ then fit in a commutative diagram π ( e L , e y ) / / Ad e γ (cid:15) (cid:15) π ( L , y ) / / Ad γ (cid:15) (cid:15) Hol ( F ) y (cid:15) (cid:15) ✤✤✤ (cid:31) (cid:127) / / Aut (cid:0) s − ( y ) (cid:1) Ad hol γ (cid:15) (cid:15) π ( e X , e x ) ∼ = Γ (cid:31) (cid:127) / / π ( X , x ) / / Aut ω ( s − ( x )) (cid:31) (cid:127) / / Aut (cid:0) s − ( x ) (cid:1) Note that since
Hol ( F ) y acts freely on s − ( y ), and the holonomy hol γ is anisomorphism between the fibres, the conjugate of Hol ( F ) y by hol γ acts freely on s − ( x ). We conclude that the dashed arrow embeds the holonomy group of e L asa finite subgroup in Aut ω ( s − ( x )) that acts freely on s − ( x ). Since the diagramis commutative, this subgroup lies in the image of Γ and hence by our choice ofΓ it must be trivial, as claimed. Proof of Theorem 3.9, part 2.
By part 1 of the theorem, we may assume with-out loss of generality that ( F , G ) is a suspension with base Y and typical fibre L ′ , where L ′ is a covering space of L . Moreover the monodromy representationof this suspension takes values in the group Aut ω ′ ( L ′ ) of biholomorphism of L ′ that fix some K¨ahler class ω ′ ∈ H , ( L ′ ).Now note that if the fundamental group of L is finite and the universal coverof L has no nonvanishing vector fields, then H ( L ′ , T L ′ ) = 0 as well. Therefore Aut ω ′ ( L ′ ) is finite by [8, Proposition 2.2]. The kernel of the monodromy repre-sentation therefore gives a finite index subgroup of π ( Y ), and passing to thecorresponding finite ´etale cover of Y , we trivialize the suspension, giving an ´etalemap L ′ × Y → X that implements the desired splitting of the tangent bundle.Then passing to the universal cover of L ′ , we obtain the desired statement. Lemma 3.11.
Let Z be a compact K¨ahler manifold with K¨ahler class ω ∈ H , ( Z ) , let Π be a finitely generated group, and let ρ : Π → Aut ω ( Z ) be ahomomorphism to the group of biholomorphisms of Z that fix the class ω . Thenthere exists a finite-index subgroup Γ < Π whose image ρ (Γ) < Aut ω ( Z ) containsno finite subgroups that act freely on Z .Proof. By taking preimages under ρ , we reduce the problem to the case where ρ is injective, so we may assume without loss of generality that Π < Aut ω ( Z ) is asubgroup. Moreover, by [8, Proposition 2.2], the neutral component Aut ( Z ) ofthe group of all biholomorphisms of Z is a finite index subgroup in Aut ω ( Z ). In13articular, Π ∩ Aut ( Z ) has finite index in Π, and is therefore finitely generatedby Schreier’s lemma. Hence we may assume without loss of generality thatΠ < Aut ( Z ) .By [8, Theorems 3.3, 3.12 and 3.14], there is an exact sequence1 / / N / / Aut ( Z ) / / T / / N is the closed subgroup exponentiating the Lie algebra of holomorphicvector fields with nonempty zero locus, and T is a compact complex torus (afinite connected ´etale cover of the Albanese torus of Z ).Since Π is finitely generated, its image in T is a finitely generated abeliangroup. Therefore, by the classification of finitely generated abelian groups, thereis a finite-index subgroup Γ < Π whose image in T is torsion-free. Suppose that G < Γ is a finite subgroup that acts freely on Z . We claim that G is trivial.Indeed, by construction, the image of G in T is a torsion-free finite group, hencetrivial. We must therefore have that G < N . But by Lemma 3.12 below, everyelement of N has a fixed point, and therefore the only subgroup of N that actsfreely is the trivial subgroup. Lemma 3.12.
Let N be a connected complex Lie group, and let X be a compactcomplex manifold. Let a : N × X → X be an action of N on X by biholomor-phisms. Suppose that the vector fields generating the action all have a non-emptyvanishing locus. Then every element of N has a fixed point in X .Proof. Let U ⊂ N be the set of elements with at least one fixed point. Notethat U = p ( a − (∆)), where ∆ ⊂ X × X is the diagonal and p : N × X → N isthe projection. Since a is holomorphic, a − (∆) ⊂ N × X is a closed analyticsubspace. Since p is proper, the Grauert direct image theorem implies that theimage p ( a − (∆)) = U ⊂ N is a closed analytic subvariety. But we assume thatevery generating vector field for the action has at least one zero; therefore U contains the image of the exponential map of N , and in particular it containsan open neighbourhood of the identity. It follows that dim U = dim N , andtherefore U = N since N is connected. In the particular case when the foliation F or its compact leaf L has trivialcanonical class, we can strengthen the results of the previous section. For ex-ample, we have the following: Corollary 3.13.
Suppose X is a compact K¨ahler manifold, and F ⊂ T X isa regular foliation having a compact leaf L with finite fundamental group andtrivial canonical class c ( L ) = 0 . If G is any complementary foliation, then thereexists a finite ´etale cover φ : e L × Y → X inducing the splitting of the tangentbundle of X as in Theorem 3.9, part 2.Proof. By Theorem 3.9, part 2, it suffices to show that h ( e L , T e L ) = 0 where e L isthe universal cover of L , but this vanishing is well known. Indeed, in this case,14he canonical bundle of e L is trivial, so that T e L ∼ = Ω n − e L where n = dim L . Wethen have h ( e L , T e L ) = h ( e L , Ω n − e L ) = h n − ( e L , O e L ) = h ( e L , O e L ) = dim H ( e L ; C ) = 0by Hodge symmetry, Serre duality, the Hodge decomposition theorem and thesimple connectivity of e L .This corollary, in turn has consequences for the foliation itself: Corollary 3.14.
Suppose that X is a compact K¨ahler manifold and that F isa (possibly singular) foliation with trivial canonical class c ( F ) = 0 . If F hasa compact leaf L with finite fundamental group, then there exists a finite ´etalecover φ : e L × Y → X such that φ − F = T e L .Proof. By [9, Theorem 5.6], F is automatically regular and admits a comple-mentary foliation, so Corollary 3.13 applies.In the situation where c ( F ) = 0 but the compact leaf L has infinite funda-mental group, the situation becomes more complicated. However, the Beauville–Bogomolov decomposition theorem [2, 4] implies that L has a finite ´etale coverof the form Z × T , where Z is a simply connected compact K¨ahler manifold with c ( Z ) = 0, and T is a compact complex torus. Moreover, amongst all such cov-erings there is a “minimal” one through which all others factor. This minimalsplit covering is unique up to a non-unique isomorphism [1, Proposition 3]. Byexploiting this result we can split the leaves of F in a uniform fashion: Proposition 3.15.
Let X be a compact K¨ahler manifold and let F be a possiblysingular foliation on X with c ( F ) = 0 . If L ⊂ X is a compact leaf whoseholonomy group is finite, then there exists a simply connected compact K¨ahlermanifold Z with c ( Z ) = 0 , a compact complex torus T , a locally trivial fibration f : W → Y between complex K¨ahler manifolds with typical fibre T , and a finite´etale cover φ : Z × W → X such that φ − F is given by the fibres of the natural morphism Z × W → Y induced by f . Moreover if G is any foliation complementary to F , then we maychoose φ so that φ − G is the pullback of a flat Ehresmann connection on f .Proof. By [9, Theorem 5.6], F is regular and there exists a foliation G comple-mentary to F . Then by Theorem 3.9 part 1, we may assume without loss ofgenerality that ( F , G ) is a suspension with base Y and typical fibre L .Let Z × T → L be the minimal split cover of L as in [1, Section 3]. Sinceevery automorphism of Z × T respects the product decomposition [1, Section3, Lemma], the foliations on Z × T given by the tangent bundles of the factorsdescend to canonical foliations on L that split the tangent bundle T L . Then,since f is a locally trivial fibration, we obtain a decomposition F ∼ = F Z ⊕ F T into involutive subbundles with compact leaves. Note that by construction, theleaves of F Z are finite quotients of Z and therefore F Z satisfies the hypotheses15f Corollary 3.13. Hence by passing to an ´etale cover, we may assume that X = Z × W , so that F Z is identified with the tangent bundle of Z , and F T ⊕ G is identified with the tangent bundle of W .This reduces the problem to the case in which F Z = 0, or equivalently L isa finite quotient of a torus T , and F is the suspension of a representation ρ : π ( Y , y ) → Aut ω L ( L )for some K¨ahler class ω L ∈ H , ( L ) and some base point y ∈ Y . Let ω ∈ H , ( T )be the induced K¨ahler class on T . By Lemma 3.16 below, there exists a finiteindex subgroup Γ < π ( Y , y ) such that ρ | Γ lifts to a homomorphism e ρ : Γ → Aut ω ( T )Let e Y → Y be the covering determined by Γ, and let e X → e Y be the suspensiondetermined by e ρ . Then we have a natural ´etale cover e X → X lifting e Y → Y whose restriction to each fibre corresponds to the quotient map T → L , givingthe result. Lemma 3.16.
Let L be a compact K¨ahler manifold equipped with a finite ´etalecover T → L , where T is a compact complex torus. Let ω ∈ H , ( L ) be anyK¨ahler class. If Π is a finitely generated group and ρ : Π → Aut ω ( L ) is ahomomorphism, then there exists a finite-index subgroup Γ < Π whose actionon L lifts to an action on T .Proof. As in the proof of Lemma 3.11 we may assume without loss of generalitythat ρ is the inclusion Π ֒ → Aut ( L ) < Aut ω ( L ) of a subgroup of the neutralcomponent of the full automorphism group. By a theorem of Lichnerowicz [7], Aut ( L ) is a torus. In particular, Π is a finitely generated abelian group, andhence it has a free abelian subgroup Γ < Π of finite index.Let T → L be the minimal split cover of L as in [1, Section 3], and let T → T be a factorization of T → L through T → L . We may assume withoutloss of generality that T → T is a morphism of tori. Let ω T be the pullbackof the class ω to T . Since the minimal split cover is unique up to isomorphism,any automorphism of L that fixes the class ω extends to an automorphism of T that fixes ω T .Let G < Aut ω T ( T ) be the subgroup of automorphisms of T that lift auto-morphisms in Aut ( L ) . Then G is a complex Lie subgroup of Aut ω T ( T ) and thenatural map G → Aut ( L ) is a surjective morphism of complex Lie groups withfinite kernel. This in turn implies that the neutral component G < Aut ( T ) of G is a torus where Aut ( T ) ∼ = T denotes the neutral component of the auto-morphism group of T . In particular, G is abelian. It follows that Γ < Aut ( L ) lifts to an inclusion Γ < G < Aut ( T ) .Let now Aut ( T ) ∼ = T be the neutral component of the automorphism groupof T . Since the map T → T is a morphism of complex Lie groups, we have asurjective natural morphism Aut ( T ) → Aut ( T ) of complex Lie groups. Hencethe inclusion Γ < Aut ( T ) lifts to an inclusion Γ < Aut ( T ) , as desired.16 emark . If one keeps the hypotheses of Proposition 3.15 and further as-sumes that X is projective, then there exists a finite ´etale cover of X isomorphicto a product where the foliation F becomes the relative tangent bundle of theprojection to one of the factors; this follows by combining Proposition 3.15 withthe fact that there exists a fine moduli scheme for polarized abelian varietiesof dimension g with level N structures, provided that N is large enough. Thisgives a simpler proof of [9, Theorem 5.8]. ♦ We now combine the results of the previous sections to establish our main result(Theorem 1.1 from the introduction), whose statement we now recall:
Theorem.
Let ( X , π ) be a compact K¨ahler Poisson manifold, and suppose that L ⊂ X is a compact symplectic leaf whose fundamental group is finite. Thenthere exist a compact K¨ahler Poisson manifold Y , and a finite ´etale Poissonmorphism e L × Y → X , where e L is the universal cover of L .Proof. By Corollary 2.5, there exists a subcalibration σ ∈ H ( X , Ω X ) compatiblewith π . Let T X = F ⊕ G be the corresponding splitting of the tangent bundleas in Section 2.2, and let π = π F + π G and σ = σ F + σ G be the correspondingdecompositions. Note that σ F is closed since it is holomorphic, and X is acompact K¨ahler manifold. Therefore by Theorem 2.6, F and G are involutive.Moreover, since L is a holomorphic symplectic manifold, we have c ( L ) = 0, soby Corollary 3.13 there exists an ´etale cover e L × Y → X that induces the givensplitting of the tangent bundle.It remains to verify that the induced Poisson structure on this covering spaceis the sum of the pullbacks of Poisson structures on the factors. But since e L × Y is compact, this property follows immediately from the K¨unneth decomposition H ( e L × Y , ∧ T e L × Y ) ∼ = H ( e L , ∧ T e L ) ⊕ (cid:16) H ( e L , T e L ) ⊗ H ( Y , T Y ) (cid:17) ⊕ H ( Y , ∧ T Y ) , and the π -orthogonality of the factors, which ensures that the induced bivectorprojects trivially to the middle summand in this decomposition.We conclude the paper by giving some examples which demonstrate that theconclusion of Theorem 1.1 may fail if the hypotheses are weakened. Example . The analogue of Theorem 1.1 fails in the C ∞ or real analyticcontexts, even for Poisson structures of constant rank.For instance, any C ∞ symplectic fibre bundle (as in [10, Chapter 6]) definesa regular Poisson manifold for which the symplectic leaves are the fibres. Suchbundles need not be trivial, even if the base and fibres are simply connected.The simplest nontrivial example is the nontrivial S -bundle over S underlyingthe odd Hirzebruch surfaces, equipped with the C ∞ Poisson structure inducedby a fibrewise K¨ahler form. This four-manifold is simply connected but is notdiffeomorphic to S × S . 17ote that given a symplectic fibre bundle, we may rescale the symplecticform on the fibres by the pullback of an arbitrary nonvanishing function on thebase, to obtain a Poisson manifold whose symplectic leaves are pairwise non-symplectomorphic. In particular, even when the underlying manifold splits asa product, the Poisson structure need not decompose as a product of Poissonstructure on the factors. ♦ Example . The conclusion of Theorem 1.1 can fail if the K¨ahler conditionis dropped. For instance, by taking the mapping torus of an infinite-orderholomorphic symplectic automorphism of a K3 surface, we may construct aholomorphic symplectic fibre bundle whose total space is non-K¨ahler. It splitsonly after passing to the universal cover, which has infinitely many sheets. ♦ Example . The conclusion of Theorem 1.1 can fail if the leaf L has infinitefundamental group. For example, let Z n ∼ = Λ ⊂ C n be a lattice, and let Y be a compact K¨ahler Poisson manifold on which Λ acts by holomorphic Poissonisomorphisms. Equip C n with the standard holomorphic Poisson structure inDarboux form. Then the quotient X := ( C n × Y ) / Λ is a compact holomorphicPoisson manifold, which is a flat fibre bundle over the symplectic base torus L := C n / Λ. If p ∈ Y is a point where the Poisson structure vanishes, then e L := ( C n × Λ · p ) / Λ defines a symplectic leaf of X for which the projection e L → L is a local diffeomorphism of holomorphic Poisson manifolds. Two possibilitiesare of note: i) if p is a Λ-fixed point, then e L ∼ = L , and ii) if the orbit of p isinfinite, then L is non-compact.Note that if both i) and ii) occur for some points p ∈ Y , then X cannotsplit. It is easy to construct examples of this phenomenon, e.g. take Y = P equipped with a Poisson bivector given by an anticanonical section vanishingon the standard toric boundary divisor (a triangle). Let Λ act by irrationalrotations of the torus. Then the vertices of the triangle are fixed, and the smoothpoints of the triangle are symplectic leaves with infinite orbits. Moreover, inthis case X is K¨ahler. ♦ Example . The conclusion of Theorem 1.1 can fail if X is not compact. Let Y be a smooth projective manifold and let f : Y → B be a non-isotrivial fibrationwhose general fibres are K U is a sufficiently small euclidean opensubset of B that does not intersect the critical locus of f , and X = f − ( U ) then X is K¨ahler, simply-connected, and admits a holomorphic Poisson structure withsymplectic leaves given by the fibres of f , but it does not split as a product. ♦ References [1] A. Beauville,
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