Deformations of Lagrangian submanifolds in log-symplectic manifolds
aa r X i v : . [ m a t h . S G ] S e p DEFORMATIONS OF LAGRANGIAN SUBMANIFOLDS INLOG-SYMPLECTIC MANIFOLDS
STEPHANE GEUDENS AND MARCO ZAMBON
Abstract.
This paper is devoted to deformations of Lagrangian submanifolds containedin the singular locus of a log-symplectic manifold. We prove a normal form result for thelog-symplectic structure around such a Lagrangian, which we use to extract algebraic andgeometric information about the Lagrangian deformations. We show that the deformationproblem is governed by a DGLA, we discuss whether the Lagrangian admits deformationsnot contained in the singular locus, and we give precise criteria for unobstructedness of firstorder deformations. We also address equivalences of deformations, showing that the gaugeequivalence relation of the DGLA corresponds with the geometric notion of equivalenceby Hamiltonian isotopies. We discuss the corresponding moduli space, and we prove arigidity statement for the more flexible equivalence relation by Poisson isotopies.
Dedicated to Olga Radko
Contents
Introduction 11. Lagrangian submanifolds in Poisson geometry 52. Poisson vector fields on the cotangent bundle of a foliation 143. Deformations of Lagrangian submanifolds in log-symplectic manifolds: algebraicaspects 214. Deformations of Lagrangian submanifolds in log-symplectic manifolds: geometricaspects 345. Appendix 58References 61
Introduction
Symplectic manifolds are a key concept in modern geometry and physics. A fundamentalrole in symplectic geometry is played by the distinguished class of Lagrangian submanifolds,as emphasized in Weinstein’s symplectic creed [38]:
Everything is a Lagrangian submanifold .The deformation theory of Lagrangian submanifolds is well-behaved: as a consequence ofWeinstein’s Lagrangian neighborhood theorem [37], deformations of a Lagrangian submani-fold L correspond with small closed one-forms on L , and the moduli space under equivalenceby Hamiltonian isotopies can be identified with the first de Rham cohomology group H ( L ) .Poisson manifolds are intimately related with symplectic geometry. The non-degeneratePoisson manifolds are exactly the symplectic ones. If one relaxes the non-degeneracy condi-tion, replacing it with a transverse vanishing condition, one obtains a larger class of Poissonmanifolds, called log-symplectic manifolds: they are symplectic outside of their singularlocus, which is a codimension-one submanifold. Their first appearance occurs in the work of Tsygan-Nest [29]. The study of their geometry was initiated by Radko [32], who classi-fied two-dimensional log-symplectic manifolds (nowadays called Radko surfaces ). Since thesystematic study of their geometry in arbitrary dimension by Guillemin-Miranda-Pires [17],log-symplectic manifolds have attracted lots of attention. One reason for this is that, de-spite the presence of singularities, they behave like symplectic manifolds in many respects.For instance, Mărcuţ-Osorno Torres [27] showed that, on a compact manifold M , the spaceof log-symplectic structures C -close to a given one (modulo C -small diffeomorphisms) issmooth and finite dimensional, parametrized by the second b -cohomology of M .This work originated from the following question: in log-symplectic geometry, is thedeformation theory of Lagrangian submanifolds as nicely behaved as in symplectic geometry? For Lagrangian submanifolds L transverse to the singular locus of the log-symplectic mani-fold, the answer is easily seen to be positive, as shown by Kirchhoff-Lukat [21]: a neighbor-hood of L is equivalent to the b -cotangent bundle of L , and the Lagrangian deformations of L (modulo Hamiltonian isotopy) are parametrized by the first b -cohomology group of L . Inparticular, the moduli space of Lagrangian deformations is smooth and finite dimensionalfor compact Lagrangians L .This paper focuses on the opposite extreme: we assume that the Lagrangian submanifold L n is contained in the singular locus Z of an orientable log-symplectic manifold M n . Notethat the b -calculus developed by Melrose [23], which is one of the main tools in log-symplecticgeometry, does not apply in our setting, due to the complete lack of transversality to Z .The main geometric questions we address are:1) Can L ⊂ Z be deformed smoothly to Lagrangian submanifolds not contained in Z ?2) Can a first order deformation of L be extended to a smooth path of Lagrangiandeformations?3) Is the moduli space of Lagrangian deformations – under the equivalence by Hamil-tonian isotopies – smooth at L ?For “many” Lagrangian submanifolds L , the answer to 1) is positive, ensuring that thedeformation problem we consider does not boil down to the case of symplectic geometry.The answer to 3) is typically negative, in contrast to the symplectic case. The answer to2) is striking, and displays a behaviour that comes close to the symplectic case: first orderdeformations are generally obstructed, but if an obvious quadratic obstruction vanishes,then they can be extended to a smooth path of deformations. Summary of results.
As in many deformation problems in geometry, the first stepconsists in providing a normal form for the log-symplectic structure in a neighborhoodof the Lagrangian L . Notice that as L is contained in the singular locus, it carries acodimension-one foliation F L . Our normal form around L (Cor. 1.18) is constructed in twosteps: we combine a normal form statement around Lagrangian submanifolds transverse tothe symplectic leaves of an arbitrary Poisson manifold (Prop. 1.9) with the normal formaround the singular locus Z of a log-symplectic manifold ( M, Π) due to Guillemin-Miranda-Pires [17], [31]. Since the latter involves the modular class of ( M, Π) , we also need to expressthe first Poisson cohomology of a neighborhood of L in the singular locus Z in terms of L alone (Cor. 2.5). The modular class is then encoded by two objects attached to L :a) A class in H ( F L ) , the first foliated de Rham cohomology.We fix a representative γ ∈ Ω cl ( F L ) .b) An element of X ( L ) F L / Γ( T F L ) ∼ = H ( F L ) .We fix a representative X ∈ X ( L ) F L , a vector field on L that preserves the foliationand is nowhere tangent to it. EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 3
Theorem A.
The log-symplectic structure in a tubular neighborhood of L is isomorphic to (cid:0) U ⊂ T ∗ F L × R , ( V vert + V lift ) ∧ t∂ t + Π can (cid:1) . Here U is a neighborhood of the zero section L , Π can is the canonical Poisson structureon the cotangent bundle T ∗ F L of the foliation F L , and t denotes the coordinate on R .Further, V vert is the vertical fiberwise constant vector field on T ∗ F L which corresponds to γ ∈ Γ( T ∗ F L ) under the natural identification, and V lift is the cotangent lift of X . The above normal form theorem gives an explicit model in which the Lagrangian de-formations of L can be investigated. We can characterize algebraically the Lagrangiandeformations of L , as follows (Thm. 3.2, Cor. 3.9): Theorem B.
Lagrangian deformations C -close to L are exactly the graphs of sections ( α, f ) of the vector bundle T ∗ F L × R → L satisfying the quadratic equation ( d F L α = 0 d F L f + f ( γ − £ X α ) = 0 , where d F L denotes the foliated de Rham differential and γ, X are as above.Further, this equation is the Maurer-Cartan equation of a DGLA. The differential graded Lie algebra mentioned above is the one introduced in greater gen-erality by Cattaneo-Felder [3], and to ensure that it captures the Lagrangian deformationswe need to check that the Poisson structure of Thm. A is fiberwise entire.In turn, Thm. B has several geometric consequences. Before explaining them, we discussbriefly two of the tools we use. First, when L is compact and connected, the followingdichotomy about the foliation F L is well-known [5, Theorem 9.3.13]: either it is the foliationassociated to a fibration L → S , or all leaves are dense. This allows us to prove severalstatements in the compact case by considering the two cases separately.Second, the linear part of the above Maurer-Cartan equation reads d F L α = 0 , d γ F L f = 0 (1)where d γ F L f = d F L f + f γ denotes the foliated de Rham differential twisted by γ . Thecohomology associated to d γ F L is the foliated Morse-Novikov cohomology H • γ ( F L ) . We willcompute it in degree for compact L (Prop. 3.15). The ordinary (untwisted) foliatedcohomology will be denoted by H • ( F L ) .If the modular vector field can be chosen to be tangent to L , i.e. [ γ ] = 0 ∈ H ( F L ) , thenit is easy to see that L can be deformed smoothly to Lagrangian submanifolds outside ofthe singular locus Z . At the opposite end of the spectrum we have (Cor. 4.5, Prop. 4.10): Theorem C.
Assume L is compact and connected.i) Suppose F L is the fiber foliation of a fiber bundle p : L → S . If for every leaf B of F L we have [ γ | B ] = 0 ∈ H ( B ) , then C -small deformations of L stay inside Z . The conversealso holds.ii) Suppose F L has dense leaves, and that H ( F L ) is finite dimensional. If γ ∈ Ω cl ( F L ) isnot exact, then there exists a neighborhood V of in (cid:0) Γ( T ∗ F L × R ) , C ∞ (cid:1) such that C ∞ -smalldeformations of L in V necessarily stay inside Z . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 4
The finite dimensionality assumption in ii) above is necessary: we show this exhibitingan example, in which L is the 2-torus and F L a Kronecker foliation for which the slope λ ∈ R \ Q is a Liouville number. The proof of these statements relies on some functionalanalysis and Fourier analysis.A first order deformation is a solution of eq. (1), the linear part of the Maurer-Cartanequation. The deformation problem is obstructed in general: there are first order defor-mations which do not extend to a (formal or smooth) path of Lagrangian deformations.This is detected by the classical Kuranishi criterium: given a first order deformation ( α, f ) ,where α ∈ Ω ( F L ) and f ∈ C ∞ ( L ) , the class Kr (cid:0) [( α, f )] (cid:1) might not vanish. This class livesin the first foliated Morse-Novikov cohomology group H γ ( F L ) . For a general deformationproblem, the Kuranishi criterium is a necessary – but not sufficient – condition to extend afirst order deformation to a formal curve of deformations. In the case at hand however, wehave the following striking result (Prop. 4.18, Cor. 4.20): Theorem D.
Assume L is compact and connected. The following are equivalent: • A first order deformation ( α, f ) of L is smoothly unobstructed, • Kr (cid:0) [( α, f )] (cid:1) = 0 , • α extends to a closed one-form on L \ Z f , the complement of the zero locus of f . Notice that the third condition is independent of the data ( X, γ ) encoding the modularvector field.Finally, we address moduli spaces. From a geometric point of view, it is natural to identifytwo C -small Lagrangian deformations of L if they are related by a Hamiltonian isotopy ofthe ambient log-symplectic manifold ( M, Π) . We show that this is exactly the equivalencerelation that the DGLA of Thm. B induces on Maurer-Cartan elements (Prop. 4.26). Thusby eq. (1), the resulting moduli space M Ham has formal tangent space at [ L ] given by T [ L ] M Ham = H ( F L ) ⊕ H γ ( F L ) . For most choices of L , this is an infinite dimensional vector space, while the formal tangentspace to M Ham at Lagrangians contained in M \ Z is finite dimensional (at least if L iscompact). Hence, for most choices of L , the moduli space is not smooth at [ L ] . We alsoexhibit some choices of L at which the moduli space is smooth, see §4.3.2.The same phenomenon occurs for the moduli space M P oiss obtained replacing Hamilton-ian isotopies by Poisson isotopies (Prop. 4.30). When L is compact with dense leaves, weshow that L being infinitesimally rigid under Poisson isotopies (i.e. T [ L ] M P oiss = 0 ) impliesthat L is rigid in the following sense: any sufficiently C ∞ -small deformation of L is relatedto L by a Poisson diffeomorphism isotopic to the identity (Prop. 4.34). Organization of the paper.
In §1 and §2 we provide the geometric background andprove the normal form given in Theorem A. In §3 and §4 we address the deformations ofLagrangian submanifolds in log-symplectic manifolds, exhibiting the underlying algebraicstructure and drawing several geometric consequences. We refer to the introductory text ofthe individual sections for more details.
Acknowledgements.
We thank Ioan Mărcuţ for his valuable input during many usefuldiscussions. In particular, we thank him for directing us to [5, Theorem 9.3.13] and [31],and for suggesting the generalization in Remark 3.11 and the proof of Lemma 4.4.
EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 5
We acknowledge partial support by the long term structural funding – Methusalem grantof the Flemish Government, the FWO under EOS project G0H4518N, and the FWO researchproject G083118N (Belgium).1.
Lagrangian submanifolds in Poisson geometry
In this section, we first recall some concepts in Poisson geometry and we introduce the no-tion of Lagrangian submanifold. Then we prove a normal form for Poisson structures aroundLagrangian submanifolds intersecting the symplectic leaves transversely (Prop. 1.9), whichcan be seen as an extension of Weinstein’s Lagrangian neighborhood theorem from sym-plectic geometry. Our main motivation is the study of Lagrangian submanifolds containedin the singular locus of a log-symplectic manifold. In §1.3-§1.4 we use the aforementionedresult to find local and semilocal normal forms around them (Prop. 1.17 and Cor. 1.18).1.1.
Poisson structures.Definition 1.1. A Poisson structure on a manifold M is a bivector field Π ∈ Γ( ∧ T M ) satisfying [Π , Π] = 0 , where [ · , · ] is the Schouten-Nijenhuis bracket of multivector fields.The Schouten-Nijenhuis bracket on Γ ( ∧ • T M ) is a natural extension of the Lie bracketof vector fields, which turns Γ( ∧ • T M )[1] into a graded Lie algebra [12, Section 1.8].The bivector field Π induces a bundle map Π ♯ : T ∗ M → T M , given by contractionof Π with covectors. The rank of Π at a point p ∈ M is defined to be the rank of thelinear map Π ♯p : T ∗ p M → T p M . A Poisson structure is called regular if its rank is the sameat all points. Poisson structures Π ∈ Γ( ∧ T M ) of full rank correspond with symplecticstructures ω ∈ Γ( ∧ T ∗ M ) via ω ↔ − Π − . In general, a Poisson manifold ( M, Π) comeswith an integrable singular distribution Im (Π ♯ ) . Each leaf O of the associated foliation hasan induced symplectic structure, given by ω O = − (Π | O ) − .A map Φ : ( M, Π M ) → ( N, Π N ) between Poisson manifolds is a Poisson map if Π M and Π N are Φ -related, i.e. (cid:0) ∧ d p Φ (cid:1) (Π M ) p = (Π N ) Φ( p ) for all p ∈ M . A vector field X on aPoisson manifold ( M, Π) is called Poisson if its flow consists of Poisson diffeomorphisms,or equivalently, if £ X Π = 0 . Each function f ∈ C ∞ ( M ) determines a Poisson vector field X f := Π ♯ ( df ) , called the Hamiltonian vector field of f . The characteristic distributionIm (Π ♯ ) of a Poisson manifold ( M, Π) is generated by its Hamiltonian vector fields.Thanks to the graded Jacobi identity of the Schouten-Nijenhuis bracket [ · , · ] , the oper-ator [Π , · ] : Γ ( ∧ • T M ) → Γ (cid:0) ∧ • +1 T M (cid:1) squares to zero. The cohomology of the resultingcochain complex (Γ ( ∧ • T M ) , [Π , · ]) is the Poisson cohomology of ( M, Π) , which we denoteby H • Π ( M ) . The cohomology groups in low degrees have geometric interpretations, see forinstance [12, Section 2.1]. We will only encounter the first cohomology group H ( M ) , whichis the quotient of the space of Poisson vector fields by the space of Hamiltonian vector fields.The modular class of ( M, Π) is a distinguished element in H ( M ) which will play a keyrole in this paper. It is defined as follows: upon choosing a volume form µ ∈ Ω top ( M ) , thereis a unique vector field V µmod ∈ X ( M ) such that for all f ∈ C ∞ ( M ) , one has £ X f µ = V µmod ( f ) µ. The vector field V µmod is called the modular vector field associated with µ . One can checkthat V µmod is a Poisson vector field, and that choosing a different volume form µ ′ = gµ changes the modular vector field V µmod by a Hamiltonian vector field: V µ ′ mod = V µmod − X ln | g | . (2) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 6
So the Poisson cohomology class [ V µmod ] ∈ H ( M ) is intrinsically defined; it is called themodular class of ( M, Π) . A Poisson manifold is called unimodular if its modular classvanishes. If M is not orientable, one can still define the modular class using densitiesinstead of volume forms. In this paper, we will only work with modular vector fields onorientable manifolds. For more on the modular class, see [39].We also recall some useful notions from contravariant geometry [6]. The general ideabehind contravariant geometry on Poisson manifolds ( M, Π) is to replace the tangent bundle T M by the cotangent bundle T ∗ M , using the bundle map Π ♯ : T ∗ M → T M . Definition 1.2.
Given a Poisson manifold ( M, Π) , a Poisson spray χ ∈ X ( T ∗ M ) is a vectorfield on T ∗ M that satisfies the following properties:i) p ∗ χ ( ξ ) = Π ♯ ( ξ ) for all ξ ∈ T ∗ M ,ii) m ∗ t χ = tχ for all t > ,where p : T ∗ M → M is the projection and m t : T ∗ M → T ∗ M is multiplication by t .Property ii ) above implies that χ vanishes on M , so that there exists a neighborhood U ⊂ T ∗ M of M where the flow φ χ of χ is defined up to time . The contravariant exponentialmap of χ is defined as exp χ : U ⊂ T ∗ M → M : ξ p ◦ φ χ ( ξ ) . The properties of the Poisson spray imply that exp χ fixes M and that its derivative atpoints x ∈ M is given by d x exp χ : T x M ⊕ T ∗ x M : → T x M : ( v, ξ ) v + Π ♯x ( ξ ) . By property i ) , exp χ maps the fiber U ∩ T ∗ x M into the symplectic leaf through x . Poissonsprays exist for any Poisson manifold ( M, Π) . They proved to be useful in the constructionof symplectic realizations [6] and normal forms [13], for instance.1.2. Lagrangian submanifolds of Poisson manifolds.
Lagrangian submanifolds.
We now introduce Lagrangian submanifolds, which are themain objects of study in this paper. We will use the following definition [36], [15].
Definition 1.3.
A submanifold L of a Poisson manifold ( M, Π) will be called Lagrangian if the following equivalent conditions hold at all points p ∈ L :i) T p L ∩ T p O is a Lagrangian subspace of the symplectic vector space (cid:16) T p O , ( ω O ) p (cid:17) .ii) Π ♯p (cid:0) T p L (cid:1) = T p L ∩ T p O , where T p L ⊂ T ∗ p M denotes the annihilator of T p L .Here ( O , ω O ) denotes the symplectic leaf through the point p .In case ( M, Π) is symplectic, this definition reduces to the usual notion of Lagrangiansubmanifold in symplectic geometry. Another special case of interest is when the manifold L has clean intersection with the leaves of ( M, Π) ; then L is Lagrangian in M exactly when itsintersection with each leaf is Lagrangian inside the leaf, in the sense of symplectic geometry.Coisotropic submanifolds of a Poisson manifold ( M, Π) are defined similarly, replacing“Lagrangian" by “coisotropic" in i) and replacing equality by the inclusion ⊂ in ii). Whilecoisotropic submanifolds have received lots of attention, Lagrangian submanifolds only ap-pear rarely in the context of Poisson geometry. In this regard, there seems to be no standarddefinition for Lagrangian submanifolds L ⊂ ( M, Π) . Another definition that appears in theliterature uses the condition Π ♯ ( T L ) = T L (e.g. [10]). Notice that the latter definition is
EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 7 more restrictive than our Definition 1.3, since it imposes that connected components of L are contained in symplectic leaves and are Lagrangian therein. Examples 1.4. a) The symplectic foliation associated with the Lie-Poisson structure on so ∗ ∼ = R consists of concentric spheres of radius r ≥ . So a plane in so ∗ is Lagrangianexactly when it passes through the origin.b) Let ( M, Π) be a regular Poisson manifold of rank k , and let Φ : ( M, Π) → ( N, be a proper surjective Poisson submersion of maximal rank, i.e. dim N = dim M − k .Assuming that the fibers of Φ are connected, they are Lagrangian tori contained in thesymplectic leaves of ( M, Π) [11, Theorem 2.6].c) Let G be a Lie group acting on a Poisson manifold ( M, Π) with equivariant moment map J : M → g ∗ . Assume the action is free on J − (0) . Then J − (0) ⊂ ( M, Π) is coisotropicand transverse to the symplectic leaves [14, Lemma 3.8]. If the leaves it meets havedimension equal to g , then J − (0) is Lagrangian.1.2.2. Normal forms.
We will prove a normal form theorem around Lagrangian submani-folds L ⊂ ( M, Π) that are transverse to the symplectic leaves, extending Weinstein’s La-grangian neighborhood theorem [37] from symplectic geometry. This is done in Proposition1.9 below. The following lemma reduces the problem to Lagrangian submanifolds of regularPoisson manifolds. Lemma 1.5.
Let ( M, Π) be a Poisson manifold, and L ⊂ ( M, Π) a Lagrangian submanifoldtransverse to the symplectic leaves. Then there exists a neighborhood U of L such that Π | U is regular.Proof. The conditions that L be Lagrangian and transverse to the leaves of ( M, Π) determinethe dimension of the leaves that L meets. Indeed, if p ∈ L and O is the leaf through p , then dim( T p L ) + dim( T p O ) = dim( T p L + T p O ) + dim( T p L ∩ T p O )= dim( T p M ) + 12 dim( T p O ) , so that dim( O ) = 2(dim( M ) − dim( L )) . It now suffices to show that there is an openneighborhood U of L that is contained in the saturation of L (i.e. the union of the leavesthat intersect L ).To construct such a neighborhood, fix a Poisson spray χ ∈ X ( T ∗ M ) . Let E := Π ♯ ( T L ) ,which is a vector bundle of rank dim( M ) − dim( L ) because of the transversality requirement.Choosing a complement to E in T M | L , we get a direct sum decomposition T ∗ M | L = E ∗ ⊕ E . (3)We claim that the contravariant exponential map exp χ : E ∗ → M maps a neighborhood V ⊂ E ∗ of L diffeomorphically onto a neighborhood U ⊂ M of L . Byproperty i ) in Definition 1.2, this neighborhood U is then automatically contained in thesaturation of L . To prove the claim, it suffices to show injectivity of the derivative of exp χ along the zero section d x exp χ : T x L ⊕ E ∗ x → T x M : ( v, ξ ) v + Π ♯x ( ξ ) . (4)To do so, note that if Π ♯x ( ξ ) = − v ∈ T x L , then ξ ∈ (cid:16) Π ♯x (cid:17) − ( T x L ) = E x . But also ξ ∈ E ∗ x ,so that ξ = 0 because of the direct sum (3). This then implies that also v = 0 , which provesinjectivity of the map (4). This finishes the proof. (cid:3) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 8
So in the following, we may assume that L is Lagrangian in a regular Poisson manifold ( M, Π) . In the next lemma, we put the foliation of ( M, Π) in normal form around L , andwe construct the local model for the Poisson structure Π . Lemma 1.6.
Let ( M, Π) be a regular Poisson manifold with associated symplectic foliation ( F , ω ) . Let L ⊂ ( M, Π) be a Lagrangian submanifold transverse to the leaves of F , anddenote by F L the induced foliation on L . We then have the following:a) There is a foliated diffeomorphism φ between a neighborhood of L in ( M, F ) and a neigh-borhood of L in ( T ∗ F L , p ∗ F L ) , with φ | L = Id. Here T ∗ F L denotes the union of thecotangent bundles of the leaves of F L , and p ∗ F L is the pullback foliation of F L by thebundle projection p : T ∗ F L → L .b) There is a canonical Poisson structure Π can on the total space T ∗ F L which gives rise tothe foliation p ∗ F L .Proof. a) By definition, T F L is a Lagrangian subbundle of the symplectic vector bundle ( T F | L , ω | L ) . Let V be a Lagrangian complement, i.e. T F | L = T F L ⊕ V . The leafwisesymplectic form ω gives an isomorphism of vector bundles − ω ♭ : V → T ∗ F L . (5)Next, by choosing a fiber metric g on the vector bundle T F , we obtain a foliated expo-nential map exp F : U ⊂ T F → M [2, Example 3.3.9]. For each leaf O of F , we havethat exp F : U ∩ T O → O is the usual exponential map of ( O , g | T O ) . Since V ⊂ T F | L is a complement to T L in T M | L , the map exp F gives a local diffeomorphism betweenneighborhoods of L exp F : V → M. (6)Composing (5) and (6) now gives a local diffeomorphism that matches the leaves of F with those of p ∗ F L . Clearly, this map restricts to the identity on L .b) We claim that the canonical Poisson structure Π T ∗ L on T ∗ L pushes forward under therestriction map r : T ∗ L → T ∗ F L , and that Π can := r ∗ (Π T ∗ L ) satisfies the requirement.This is readily checked in coordinates. Take a foliated chart ( x , . . . , x k , x k +1 . . . , x n ) on L such that plaques of F L are level sets of ( x k +1 , . . . , x n ) , and let ( y , . . . , y n ) be theassociated fiber coordinates on T ∗ L . Then the restriction map r : T ∗ L → T ∗ F L is justthe projection onto the first n + k coordinates, which implies that Π T ∗ L = P ni =1 ∂ x i ∧ ∂ y i pushes forward to a Poisson structure r ∗ (Π T ∗ L ) = k X i =1 ∂ x i ∧ ∂ y i . Clearly, the Poisson manifold ( T ∗ F L , Π can ) decomposes into symplectic leaves as follows: ( T ∗ F L , Π can ) = a O∈F L ( T ∗ O , ω T ∗ O ) , (7)where ω T ∗ O denotes the canonical symplectic form on T ∗ O . This finishes the proof. (cid:3) We can now show that ( M, Π) and ( T ∗ F L , Π can ) are Poisson diffeomorphic near L . If φ : ( M, F ) → ( T ∗ F L , p ∗ F L ) denotes the diffeomorphism constructed in Lemma 1.6 (definedon a neighborhood of L ), then we have that ( φ ∗ Π) | L = Π can | L . (8) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 9
This can be checked by direct computation, but instead we refer to the proof of Weinstein’sLagrangian neighborhood theorem in [37], as we are just applying Weinstein’s constructionleaf by leaf. In some detail, we consider the restriction φ : ( S , ω S ) → (cid:0) T ∗ ( L ∩ S ) , ω T ∗ ( L ∩S ) (cid:1) for each leaf S ∈ F , and the usual argument of the Lagrangian neighborhood theorem showsthat φ ∗ ω T ∗ ( L ∩S ) and ω S agree along L ∩ S . This immediately implies the equality (8).Having established (8), we need an appropriate version of Moser’s theorem in order to con-struct a Poisson diffeomorphism between neighborhoods of L in ( M, Π) and ( T ∗ F L , Π can ) .This in turn requires a foliated version of the relative Poincaré lemma. Both statementsalready appeared in the literature; we state them here for the reader’s convenience. Lemma 1.7. [9, Proposition 3.3]
Let ( N, F ) be a foliated manifold, and let p : M → N bea vector bundle over N . Denote by F ′ := p ∗ ( F ) the pullback foliation of F . Suppose that α ∈ Γ (cid:0) ∧ k T ∗ F ′ (cid:1) is a closed foliated k -form whose pullback i ∗ α to ( N, F ) vanishes. Thenthere exists a foliated ( k − -form β ∈ Γ (cid:0) ∧ k − T ∗ F ′ (cid:1) such that d F ′ β = α and β | N = 0 . Lemma 1.8. [7, Lemma 5]
Let ( M, F , ω ) be a symplectic foliation. Consider a foliated -form α ∈ Ω ( F ) satisfying α | N = ( d F α ) | N = 0 for some submanifold N ⊂ M . Then ω + d F α is non-degenerate in a neighborhood U of N , and the resulting symplectic foliation ( U, F | U , ω | U + ( d F α ) | U ) is isomorphic around N to ( M, F , ω ) by a foliated diffeomorphismthat is the identity on N . Altogether, we obtain the following normal form around Lagrangian submanifolds trans-verse to the symplectic leaves of a Poisson manifold.
Proposition 1.9 (Local model around Lagrangians transverse to symplectic leaves) . Givena Poisson manifold ( M, Π) , let L ⊂ ( M, Π) be a Lagrangian submanifold transverse to thesymplectic leaves. Denote by F L the induced foliation on L . Then a neighborhood of L in ( M, Π) is Poisson diffeomorphic with a neighborhood of L in ( T ∗ F L , Π can ) , through adiffeomorphism that restricts to the identity on L .Proof. By Lemma 1.5, we can assume that ( M, Π) is regular, with underlying foliation F .By Lemma 1.6 and (8), there exists a foliated diffeomorphism between neighborhoods of L , φ : U ⊂ ( M, F ) → V ⊂ ( T ∗ F L , p ∗ F L ) , satisfying ( φ ∗ Π) | L = Π can | L and φ | L = Id . Denote by ω, ˜ ω ∈ Ω ( p ∗ F L | V ) the leafwise symplectic forms on V ⊂ T ∗ F L correspondingwith the Poisson structures Π can and φ ∗ Π , respectively. Since ˜ ω − ω is closed and therestriction (˜ ω − ω ) | L vanishes, we can apply Lemma 1.7: shrinking V if necessary, we obtainthat ˜ ω − ω = d p ∗ F L β for some β ∈ Ω ( p ∗ F L | V ) satisfying β | L = 0 . Lemma 1.8 gives anisomorphism of symplectic foliations ψ : ( V, p ∗ F L | V , ˜ ω | V ) → (cid:0) ψ ( V ) , p ∗ F L | ψ ( V ) , ω | ψ ( V ) (cid:1) suchthat ψ | L = Id, again shrinking V if necessary. The map ψ ◦ φ : ( U, Π | U ) → (cid:0) ψ ( V ) , Π can | ψ ( V ) (cid:1) now satisfies the criteria. (cid:3) Remark . One can also obtain Proposition 1.9 by applying some more general resultsthat appeared in [4]. There one shows the following: • [4, Theorem 8.1] Let ( M, D ) be a smooth Dirac manifold. If D ∩ T M has constantrank, then ( M, D ) can be embedded coisotropically into a Poisson manifold ( P, Π) .Explicitly, denote E := D ∩ T M and define P to be the total space of the vectorbundle π : E ∗ → M . Choosing a complement to E inside T M gives an embedding i : E ∗ ֒ → T ∗ M . Then the Dirac structure e i ∗ ω T ∗ M ( π ∗ D ) , obtained by pulling back D along π and applying the gauge transformation by i ∗ ω T ∗ M , defines a Poisson EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 10 structure Π on a neighborhood of M in E ∗ . It has the desired properties: M ⊂ ( P, Π) is coisotropic and the Dirac structure D Π pulls back to D on M . • [4, Proposition 9.4] Suppose we are given a Dirac manifold ( M, D ) for which D ∩ T M has constant rank k , and let ( P , Π ) and ( P , Π ) be Poisson manifolds of dimension dim( M ) + k in which ( M, D ) embeds coisotropically. Assume moreover that thepresymplectic leaves of ( M, D ) have constant dimension. Then ( P , Π ) and ( P , Π ) are Poisson diffeomorphic around M .In our situation, we have a Lagrangian submanifold i : L ֒ → ( M, Π) transverse to thesymplectic leaves of ( M, Π) , so the pullback i ∗ L Π is a smooth Dirac structure on L . More-over, i ∗ L Π ∩ T L has constant rank since it is given by Π ♯ ( T L ) = T F L . The procedure indescribed in the first bullet point above then yields exactly the local model ( T ∗ F L , Π can ) .Now ( L, i ∗ L Π ) is embedded coisotropically in ( M, Π) and in ( T ∗ F L , Π can ) , both of whichhave dimension equal to dim( L ) + rk ( T F L ) . The presymplectic leaves of ( L, i ∗ L Π ) haveconstant dimension, since they are just the leaves of F L . Applying the second bullet pointabove then shows that ( M, Π) and ( T ∗ F L , Π can ) are Poisson diffeomorphic around L . L F L T ∗ F L F L F L Figure 1.
The foliation F L and vector bundle T ∗ F L .Proposition 1.9 implies that C -small deformations of a Lagrangian L ⊂ ( M, Π) transverseto the leaves correspond with Lagrangian sections of ( T ∗ F L , Π can ) . Thanks to the decom-position (7), these can be studied using well-known results from symplectic geometry aboutLagrangian sections in cotangent bundles. We obtain that deformations of L ⊂ ( M, Π) areclassified by the first foliated cohomology group H ( F L ) . Corollary 1.11.
Given a Poisson manifold ( M, Π) , let L ⊂ ( M, Π) be a Lagrangian sub-manifold transverse to the symplectic leaves. Denote by F L the induced foliation on L . • The graph of α ∈ Γ ( T ∗ F L ) is Lagrangian in ( T ∗ F L , Π can ) exactly when d F L α = 0 . • The graphs of closed foliated one-forms α, β ∈ Γ ( T ∗ F L ) are related by a Hamiltoniandiffeomorphism exactly when [ α ] = [ β ] in H ( F L ) . Log-symplectic structures.
The rest of this paper is devoted to a specific class of Poisson structures, called log-symplectic structures. These are generically symplectic, except at some singularities wherethe bivector drops rank in a controlled way.
EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 11
Definition 1.12.
A Poisson structure Π on a manifold M n is called log-symplectic if ∧ n Π is transverse to the zero section of the line bundle ∧ n T M .A log-symplectic structure Π is symplectic everywhere, except at points lying in the set Z := ( ∧ n Π) − (0) , called the singular locus of ( M, Π) . If Z is nonempty, then it is a smoothhypersurface by the transversality condition. In that case, Z is a Poisson submanifold of ( M, Π) with an induced Poisson structure that is regular of corank-one.The geometry of the singular locus ( Z, Π | Z ) has some nice features. The foliation of Π | Z is unimodular, i.e. defined by a closed one-form θ ∈ Ω ( Z ) , and the leafwise symplectic formextends to a closed two-form ω ∈ Ω ( Z ) . The pair ( θ, ω ) defines a cosymplectic structureon Z . The existence of such a pair is equivalent with the existence of a Poisson vector fieldon Z that is transverse to the leaves of Π | Z [16]. One can obtain such a vector field byrestricting a modular vector field on ( M, Π) to Z [17]. Example 1.13.
The standard example of a log-symplectic manifold is R n with coordinates ( x , y , . . . , x n , y n ) and Poisson structure Π = ∂ x ∧ y ∂ y + P ni =2 ∂ x i ∧ ∂ y i . It followsfrom Weinstein’s splitting theorem that any log-symplectic structure looks like this near apoint in its singular locus. In this example, the vector field ∂ x is the modular vector fieldcorresponding with the volume form P ni =1 dx i ∧ dy i . It is indeed transverse to the symplecticleaves of Z = { y = 0 } , which are the level sets of x .The importance of modular vector fields is apparent in the following normal form result,which describes the log-symplectic structure in a neighborhood of its singular locus [17],[31, Prop. 4.1.2]. Proposition 1.14 (Local form around singular locus) . Let Π be a log-symplectic structureon an orientable manifold M , with singular locus ( Z, Π | Z ) . Let V mod ∈ X ( M ) be a modularvector field on M . Then there is a tubular neighborhood U ⊂ Z × R of Z , in which Z corresponds to t = 0 , such that Π | U = V mod | Z ∧ t∂ t + Π | Z . Log-symplectic structures can alternatively be viewed as symplectic forms on a suitableLie algebroid. To any b -manifold ( M, Z ) consisting of a manifold M and a hypersurface Z ⊂ M , one can associate a Lie algebroid b T M whose sections are the vector fields on M that are tangent to Z . Lie algebroid -forms ω ∈ Γ (cid:0) ∧ (cid:0) b T ∗ M (cid:1)(cid:1) that are closed andnon-degenerate are called b -symplectic forms. Having a log-symplectic structure Π on M with singular locus Z is equivalent to having a b -symplectic form on ( M, Z ) [17]. This pointof view allows one to study log-symplectic structures using symplectic techniques.1.4. Lagrangian submanifolds of log-symplectic manifolds.
We now focus on Lagrangian submanifolds L of log-symplectic manifolds ( M, Z, Π) . La-grangians transverse to the degeneracy locus Z can be treated using the b -geometry pointof view, which reduces their study to symplectic geometry. Indeed, the submanifold L isnaturally a b -manifold ( L, L ∩ Z ) and the condition that L be Lagrangian (in the sense ofDef. 1.3) is equivalent with the requirements ( b i ∗ ω = 0dim( L ) = dim( M ) , where ω is the b -symplectic form defined by Π and i : ( L, L ∩ Z ) ֒ → ( M, Z ) is the inclusion.In [21], one shows that a neighborhood of L in ( M, ω ) is b -symplectomorphic with a neigh-borhood of L in its b -cotangent bundle b T ∗ L , endowed with the canonical b -symplectic form. EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 12
As a consequence, the moduli space of Lagrangian deformations of L under Hamiltonianequivalence can be identified with the first b -cohomology group b H ( L ) . All of this is incomplete analogy with what happens in symplectic geometry.We will consider Lagrangians at the other extreme, i.e. those that are contained in thesingular locus of a log-symplectic manifold ( M n , Z, Π) . If L is such a Lagrangian and O n − is the leaf through p ∈ L , then we have dim( T p L ) = dim( T p L + T p O ) − n + 1 , where n − ≤ dim( T p L + T p O ) ≤ n − . So either dim( L ) = n − and connectedcomponents of L lie inside symplectic leaves, or dim( L ) = n and L is transverse to theleaves in Z . In the rest of this note, we will deal with Lagrangians of the second kind: middle dimensional Lagrangian submanifolds contained in the singular locus.Remark . More generally, instead of middle dimensional Lagrangian submanifolds, onecould consider middle dimensional coisotropic submanifolds C ⊂ ( M, Z, Π) . Although thesetwo notions coincide for submanifolds transverse to the degeneracy locus Z , they are notequivalent in general – in particular, they are not equivalent in the setup we consider.An example of middle dimensional coisotropic C contained in Z which is not Lagrangian,is the following. Take M = R and Π = ∂ x ∧ y ∂ y + ∂ x ∧ ∂ y , take C given by theconstraints x − y = 0 and y = 0 . It is coisotropic because the Poisson bracket of theseconstraints is y , thus again a constraint. It is not Lagrangian because T p C = T p O at points p of C where y vanishes, where O denotes the (2-dimensional) symplectic leaf through p . Example 1.16.
In the local model (cid:0) R n , x , y , . . . , x n , y n (cid:1) with its standard log-symplecticstructure Π = ∂ x ∧ y ∂ y + P ni =2 ∂ x i ∧ ∂ y i , the submanifold L = { y = · · · = y n = 0 } isLagrangian of middle dimension, contained in the singular locus.Example 1.16 is in fact the local model for any Lagrangian L n ⊂ Z ⊂ ( M n , Π) . Proposition 1.17 (Local form around a point) . Let ( M n , Z, Π) be a log-symplectic man-ifold and let L n ⊂ Z be a Lagrangian submanifold. Around any point p ∈ L , there existcoordinates ( x , y , . . . , x n , y n ) such that Z = { y = 0 } Π = ∂ x ∧ y ∂ y + P ni =2 ∂ x i ∧ ∂ y i L = { y = · · · = y n = 0 } . Proof.
Applying Prop. 1.9 and Prop. 1.14 locally around p shows that there exists acoordinate system ( U ; x , t, x , y . . . , x n , y n ) such that Π | U = V mod | U ∩ Z ∧ t∂ t + n X i =2 ∂ x i ∧ ∂ y i . (9)Here V mod is a locally defined modular vector field, L = { t = y = · · · = y n = 0 } and Z = { t = 0 } . If we write V mod | U ∩ Z in coordinates as V mod | U ∩ Z = z ( x, y ) ∂ x + n X i =2 g i ( x, y ) ∂ x i + n X i =2 h i ( x, y ) ∂ y i , EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 13 then requiring that V mod | U ∩ Z is Poisson yields that z ( x, y ) only depends on x . Now V mod | U ∩ Z − z ( x ) ∂ x is a Poisson vector field tangent to the leaves, so it is locally Hamil-tonian. This implies that, changing to a different modular vector field, we may assume V mod | U ∩ Z = z ( x ) ∂ x . Note here that z ( x ) is nowhere zero since V mod | U ∩ Z is transverse to the leaves. This allowsus to define a new coordinate ξ by ξ := Z z ( x ) dx . In the new coordinate system ( ξ, t, x , y , . . . , x n , y n ) , the expression (9) becomes Π = ∂ ξ ∧ t∂ t + n X i =2 ∂ x i ∧ ∂ y i , so these coordinates satisfy the criteria. (cid:3) Given a Lagrangian L n contained in the singular locus Z of a log-symplectic manifold ( M n , Π) , Prop. 1.9 describes a neighborhood of L in ( Z, Π | Z ) and Prop. 1.14 describesa neighborhood of Z in ( M n , Π) . Combining the two propositions, we get the followingnormal form around L n ⊂ ( M n , Π) . Corollary 1.18 (Local form around a Lagrangian in the singular locus) . Let ( M n , Z, Π) be an orientable log-symplectic manifold, and make a choice of modular vector field V mod on M . Let L n ⊂ Z be a Lagrangian submanifold, and denote by F L the induced foliation on L . Then a neighborhood of L in ( M, Π) can be identified with a neighborhood of L in thevector bundle T ∗ F L × R → L , endowed with the log-symplectic structure e Π := V ∧ t∂ t + Π can . (10) Here t is a coordinate on R , and V is the image of V mod | Z under the Poisson diffeomorphism ( Z, Π | Z ) → ( T ∗ F L , Π can ) between neighborhoods of L constructed in Prop. 1.9.Remark . The vector field V in (10) is only defined on a neighborhood W of L in T ∗ F L .Note that there is some freedom in the formula (10), in the sense that there we can replace V by any Poisson vector field representing the Poisson cohomology class [ V ] .To see this, take any representative V − X f of [ V ] , for some function f defined on W .Note that V is a modular vector field of e Π , with respect to the volume form Ω on W × R that is uniquely determined by requiring that h Ω , ∧ n e Π i = t . If ˜ f is an extension of f to W × R , then also V − X ˜ f is a modular vector field of e Π , with respect to the volumeform e ˜ f Ω on W × R . Proposition 1.14 now implies that replacing V by V − X f in (10)gives a log-symplectic structure that is Poisson diffeomorphic to e Π in a neighborhood of W ⊂ W × R .An arbitrary representative of the modular class has little to do with the Lagrangian L ;we will remedy this in the next section. One could hope to find a representative of [ V ] thatis tangent to L . This amounts to finding a modular vector field, defined on a neighborhoodof L in ( M, Π) , that is tangent to L . This can always be done locally near a point, as aconsequence of Prop. 1.17 (namely, the vector field ∂ x in the statement of the propositionis modular and tangent to L ). Globally however, this may fail, as we now show. The leaves of the codimension-one foliation F L are the connected components of the intersections of L with the symplectic leaves of Z . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 14
Example 1.20 (No modular vector field is tangent) . Consider the manifold R × S × T with coordinates ( t, τ, θ , θ ) and log-symplectic structure Π = ( ∂ τ + ∂ θ ) ∧ t∂ t + ∂ θ ∧ ∂ θ . The submanifold L := { t = θ = 0 } ∼ = S × S is Lagrangian inside the singular locus Z = S × T . Note that ∂ τ + ∂ θ is a modular vector field for Π (associated with the volumeform dθ ∧ dθ ∧ dt ∧ dτ ). If there existed a modular vector field tangent to L (defined near L ), then its restriction to Z would look like ∂ τ + ∂ θ + ( ∂ θ ∧ ∂ θ ) ♯ ( df ) = ∂ τ + (cid:18) ∂f∂θ (cid:19) ∂ θ − ∂f∂θ ∂ θ for some f ∈ C ∞ ( Z ) , where (cid:18) ∂f∂θ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 = 0 . (11)But then, fixing any value of τ and denoting by i : { τ } × S × { } ֒ → Z the inclusion, weget Z { τ }× S ×{ } i ∗ df = Z { τ }× S ×{ } ∂f∂θ dθ = − Z { τ }× S ×{ } dθ = − π, using Stokes’ theorem, and (11) in the third equality. So there is no modular vector fieldtangent to L .2. Poisson vector fields on the cotangent bundle of a foliation
Let L be a manifold and F L a foliation on L . Denote by Π can the canonical Poissonstructure on T ∗ F L (as in b ) of Lemma 1.6). This section treats Poisson vector fieldson ( T ∗ F L , Π can ) . We show that every class in the first Poisson cohomology group of ( T ∗ F L , Π can ) admits a convenient representative (Thm. 2.2), and use this to computeexplicitly the first Poisson cohomology group (Cor. 2.5). At the beginning of §3, we applythese results to the modular vector field of a log-symplectic manifold, and find a convenientrepresentative of the class [ V ] in (10).2.1. Convenient representatives.
We denote by X ( L ) F L := (cid:8) W ∈ X ( L ) : [ W, Γ( T F L )] ⊂ Γ( T F L ) (cid:9) the Lie subalgebra of vector fields on L whose flow preserves the foliation F L . Lemma 2.1.
Let W ∈ X ( L ) F L and let r : ( T ∗ L, Π T ∗ L ) → ( T ∗ F L , Π can ) denote the restric-tion. We then have the following:(i) The cotangent lift of W pushes forward via r : T ∗ L → T ∗ F L to a Poisson vector fieldon T ∗ F L , which we denote by f W .(ii) When W lies in Γ( T F L ) , the vector field f W is Hamiltonian.Proof. We denote by p T ∗ F L : T ∗ F L → L and p T ∗ L : T ∗ L → L the vector bundle projections.(i) Let W T ∗ L ∈ X ( T ∗ L ) denote the cotangent lift of W . To show that it pushes forwardvia r , we need to show that its action on functions preserves r ∗ ( C ∞ ( T ∗ F L )) . It sufficesto consider fiberwise constant and fiberwise linear functions on T ∗ F L . The fiberwiseconstant ones are of the form p ∗ T ∗ F L g for g ∈ C ∞ ( L ) . Since p T ∗ L = p T ∗ F L ◦ r , we have W T ∗ L ( r ∗ ( p ∗ T ∗ F L g )) = W T ∗ L ( p ∗ T ∗ L g ) = p ∗ T ∗ L ( W ( g )) = r ∗ ( p ∗ T ∗ F L ( W ( g ))) . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 15
Next, fiberwise linear functions on T ∗ F L look like h X : T ∗ F L → R : ( p, α )
7→ h α, X ( p ) i for X ∈ Γ( T F L ) . Clearly, one has a commutative diagram C ∞ lin ( T ∗ F L ) C ∞ lin ( T ∗ L )Γ( T F L ) Γ( T L ) r ∗ ih • h • . Recall that for the standard symplectic structure on T ∗ L , the Poisson bracket satisfies { h X , h Y } = − h [ X,Y ] , for X, Y ∈ Γ( T L ) . Moreover, the cotangent lift W T ∗ L is minusthe Hamiltonian vector field of h W (see e.g. [8, §2]). So for X ∈ Γ( T F L ) we get W T ∗ L ( r ∗ h X ) = W T ∗ L (cid:0) h i ( X ) (cid:1) = − X h W (cid:0) h i ( X ) (cid:1) = −{ h W , h i ( X ) } = h [ W,i ( X )] . The vector field [ W, i ( X )] lies in Γ( T F L ) by assumption, so that W T ∗ L ( r ∗ h X ) lies in r ∗ ( C ∞ lin ( T ∗ F L )) . This shows that W T ∗ L pushes forward under r .The vector field f W is Poisson since the cotangent lift W T ∗ L is a symplectic vector fieldand r is a Poisson map.(ii) If W lies in Γ( T F L ) , then we have f W = r ∗ (cid:0) i ( W ) (cid:1) T ∗ L = r ∗ (cid:16) − X h i ( W ) (cid:17) = − r ∗ ( X r ∗ h W ) = − X h W , where in the second equality we used the above comment about Hamiltonian vectorfields, and in the third the commutativity of the diagram. (cid:3) The rest of this section is devoted to the following theorem, which provides convenientrepresentatives for first Poisson cohomology classes, and its consequences.
Theorem 2.2.
Let ( L, F L ) be a foliated manifold. Consider the standard Poisson structure Π can on the total space of the vector bundle p : T ∗ F L → L . Fix a class in H can ( T ∗ F L ) .Then there exists a representative Y ∈ X ( T ∗ F L ) such that(i) Y is p -projectable and p ∗ Y ∈ X ( L ) F L ,(ii) the vector field Y − g p ∗ Y is vertical and constant on each fiber of p , and Y − g p ∗ Y isclosed when viewed as a foliated 1-form on ( L, F L ) . Notice that given a class in H can ( T ∗ F L ) , a representative Y as in Theorem 2.2 is by nomeans unique: adding to Y a Hamiltonian vector field of the form f W + X p ∗ g for W ∈ Γ( T F L ) and g ∈ C ∞ ( L ) gives a representative of the same class that still satisfies the requirementsof Theorem 2.2 (see Corollary 2.5 below). Example 2.3.
Consider the plane L = R with coordinates x, y , and the foliation F L givenby the lines { x = const } . Then T ∗ F L is R with coordinates x, y, z , with vector bundleprojection p = ( x, y ) : R → R and Poisson structure Π can = ∂ y ∧ ∂ z . An arbitrary Poissonvector field has the form U = f ( x ) ∂ x + g∂ y + k∂ z , where g, k ∈ C ∞ ( R ) satisfy ∂ y g = − ∂ z k . This vector field is not p -projectable in general,because g might depend on z . However, defining h ( x, y, z ) := R z g ( x, y, t ) dt , we obtain afunction on R such that Y := U + X h = f ( x ) ∂ x + ( k + ∂ y h ) ∂ z The lift g p ∗ Y was defined in Lemma 2.1. I.e., when viewed as a section of p : T ∗ F L → L . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 16 is p -projectable. Notice that p ∗ Y = f ( x ) ∂ x lies in X ( L ) F L . Moreover, since the partialderivative ∂ z ( k + ∂ y h ) vanishes, the vertical vector field V := ( k + ∂ y h ) ∂ z is indeed constanton each fiber of p . Regarding V as a foliated 1-form on ( L, F L ) yields ( k + ∂ y h ) dy , whichis closed due to dimension reasons.To prove Theorem 2.2, we need a few general statements about cotangent bundles. Lemma 2.4.
Let N be a manifold. Consider its cotangent bundle T ∗ N with the standardsymplectic form ω and bundle projection p T ∗ N .(i) Let Y ∈ X ( T ∗ N ) be a symplectic vector field . Then there is h ∈ C ∞ ( T ∗ N ) such that Y + X h is a vertical vector field.(ii) Let V ∈ X ( T ∗ N ) be a vertical symplectic vector field. Then V must be constant oneach fiber. It is closed when viewed as an element of Γ( T ∗ N ) = Ω ( N ) .Proof. (i) Consider the foliation F fiber of T ∗ N by fibers of the projection p T ∗ N . Denoteby i fiber the inclusion of its tangent distribution into the tangent bundle of T ∗ N . Sincethe 1-form ι Y ω ∈ Ω ( T ∗ N ) is closed, its pullback i ∗ fiber ( ι Y ω ) is closed as a foliated -form. It is foliated exact, as the leaves of F fiber are just fibers of a vector bundle(choosing the primitives on each fiber to vanish on the zero section, they assemble toa smooth function on T ∗ N , c.f. Lemma 1.7). So i ∗ fiber ( ι Y ω ) equals d F fiber h for some h ∈ C ∞ ( T ∗ N ) , which implies that ι Y ω − dh ∈ Ω ( T ∗ N ) pulls back to zero under i fiber .As the fibers are Lagrangian, this means that ω − ( ι Y ω − dh ) = Y + X h is a verticalvector field on T ∗ N .(ii) The 1-form ι V ω is closed because V is a symplectic vector field. For any verticalvector field W we have ι W ι V ω = 0 and L W ( ι V ω ) = 0 , so ι V ω = − p ∗ T ∗ N α for a unique,closed α ∈ Ω ( N ) . Writing in local coordinates α = P i f i ( q ) dq i , in the correspondingcanonical coordinates on T ∗ N we have V = − ( ω − )( p ∗ T ∗ N α ) = X i f i ( q ) ∂ p i , showing that V is constant along the fibers. This formula also shows that V , regardedas an element of Γ( T ∗ N ) = Ω ( N ) , is precisely the closed 1-form α . (cid:3) Proof of Thm. 2.2.
Let U ∈ X ( T ∗ F L ) be any representative of the given class in H can ( T ∗ F L ) .Let U ∈ X ( L ) be given by ( U )( x ) := ( d x p )( U ( x )) at each point x ∈ L . So ( U )( x ) is justthe T x L -component of U ( x ) w.r.t. the canonical splitting T x ( T ∗ F L ) = T x L ⊕ T ∗ x F L .We first show that U ∈ X ( L ) F L . Since this is a local statement, it suffices to consideropen subsets of L whose quotient by the restriction of F L is a smooth manifold and showthat the restriction of U projects to a vector field on the leaf space. By abuse of notation,we denote such an open subset by L . Since the leaves of the symplectic foliation F sympl of T ∗ F L are the preimages under p of the leaves of F L , there is a canonical diffeomorphism ofleaf spaces T ∗ F L / F sympl ∼ = L/ F L , induced by the vector bundle projection p : T ∗ F L → L (or equivalently, by the inclusion ofthe zero section). Since U is a Poisson vector field, it projects under T ∗ F L → T ∗ F L / F sympl to some vector field U quot . Restricting to points of the zero section L , we see that U isprojectable under L → L/ F L (to the same vector field U quot ). Part ( i ) of the lemma holds more generally whenever the pullback of ι Y ω to each fiber of p T ∗ N is closed. EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 17
By Lemma 2.1, U lifts to a Poisson vector field f U on T ∗ F L . The Poisson vector field U − f U is tangent to the symplectic foliation on T ∗ F L . Indeed, since the statement is alocal one, we can again work on suitable open subsets of L and use that both U and f U areprojectable to the same vector field under T ∗ F L → T ∗ F L / F sympl .We now apply Lemma 2.4 i ) smoothly to the leaves of F sympl and the vector field U − f U .More precisely, by the proof of Lemma 2.4 i ) , if ω denotes the leafwise symplectic form on T ∗ F L , then we find a function h ∈ C ∞ ( T ∗ F L ) such that the pullback of ι U − f U ω − d F sympl h to the fibers of p is zero. It follows that − Π ♯can (cid:16) ι U − f U ω − d F sympl h (cid:17) = U − f U + X h is vertical, i.e. tangent to the p -fibers. This has two consequences. First, we can applyLemma 2.4 ii) to conclude that this vector field is constant on each fiber, and is closed whenviewed as a foliated 1-form on ( L, F L ) . Second, U + X h is p -projectable and it projects tothe same vector field as f U , namely U ∈ X ( L ) . Hence Y := U + X h is a representative ofthe class H can ( T ∗ F L ) with the required properties. (cid:3) The first Poisson cohomology.
Using Theorem 2.2, we can compute the first Poisson cohomology group of ( T ∗ F L , Π can ) .In the following, H • ( F L ) denotes the cohomology of the foliated differential forms along theleaves of F L . Corollary 2.5.
Let ( L, F L ) be a foliated manifold and denote by Π can the standard Poissonstructure on the total space of the vector bundle p : T ∗ F L → L . There is a linear isomorphism Φ : H can ( T ∗ F L ) → X ( L ) F L / Γ( T F L ) × H ( F L )[ Y ] (cid:0) [ p ∗ Y ] , [ Y − g p ∗ Y ] (cid:1) , (12) where the representative Y satisfies the properties in Thm. 2.2. Notice that X ( L ) F L / Γ( T F L ) agrees with the space of vector fields on L/ F L , wheneverthe latter quotient is smooth. Proof.
We first show that the map Φ is well-defined. For this, due to Thm. 2.2, we onlyneed to show that the above assignment is independent of the choice of representative.Equivalently, since the expression in (12) depends linearly on Y , we have to show that if Y is a Hamiltonian vector field on T ∗ F L satisfying the properties in Thm. 2.2, then p ∗ Y liesin Γ( T F L ) and Y − g p ∗ Y is exact when viewed as a foliated 1-form on ( L, F L ) .Being Hamiltonian, Y is tangent to the symplectic foliation of T ∗ F L , so p ∗ Y is tangentto the foliation F L . Hence g p ∗ Y is a Hamiltonian vector field, by Lemma 2.1. Being thedifference of two Hamiltonian vector fields, the vertical and fiberwise constant vector field V := Y − g p ∗ Y is Hamiltonian. Denote by F ∈ C ∞ ( T ∗ F L ) a Hamiltonian function for V , sothat for each leaf O of F L we have ι V ω T ∗ O = − d ( F | T ∗ O ) . Regarding the vertical constantvector field V as a foliated 1-form yields α ∈ Ω ( F L ) , determined by ι V ω T ∗ O = − p ∗ T ∗ O ( α | O ) , (13)see the proof of Lemma 2.4 ( ii ) . In particular, F is constant along the fibers of p : T ∗ F L → L ,i.e. F = p ∗ ( F | L ) . Thus α = d F L F , showing that it is foliated exact.We show that Φ is surjective. Let W ∈ X ( L ) F L . Then its lift f W is a Poisson vectorfield on T ∗ F L , by Lemma 2.1 i ) . Let α ∈ Ω ( F L ) be a closed foliated 1-form. Denote by V the corresponding vertical fiberwise constant vector field on T ∗ F L . Then V is a Poisson EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 18 vector field, because it is tangent to the symplectic leaves of T ∗ F L and its restriction toeach symplectic leaf is a symplectic vector field, by eq. (13). Hence f W + V is a Poissonvector field on T ∗ F L . By construction it satisfies the properties of Thm. 2.2, and its Poissoncohomology class maps under Φ to ([ W ] , [ α ]) .We show that Φ is injective. Let Y be a Poisson vector field on T ∗ F L satisfying theproperties in Thm. 2.2, so that p ∗ Y lies in Γ( T F L ) and V := Y − g p ∗ Y is exact when viewedas a foliated 1-form on ( L, F L ) . By Lemma 2.1 ii ) , g p ∗ Y is a Hamiltonian vector field. Let α = d F L f ∈ Ω ( F L ) be the exact foliated 1-form corresponding to V , where f ∈ C ∞ ( L ) .Then eq. (13) implies that V = Π ♯can ( p ∗ ( df )) , showing that V is a Hamiltonian vector field.Hence Y = g p ∗ Y + V is Hamiltonian, so [ Y ] = 0 . (cid:3) We discuss the isomorphism (12) in two particular cases.
Example 2.6. i) Suppose F L is the foliation of L by points. Then T ∗ F L is just L withthe zero Poisson structure, and the map Φ is just the identity on X ( L ) .ii) On the other extreme, suppose F L is the one-leaf foliation of L . Then T ∗ F L is thecotangent bundle T ∗ L with its standard symplectic form, and Φ : H can ( T ∗ L ) → H ( L ) . Since Φ is an isomorphism, every class in H can ( T ∗ L ) admits a representative V whichis a vertical fiberwise constant vector field (c.f. Lemma 2.4). The image of this classunder Φ is [ α ] ∈ H ( L ) , where α is just V regarded as a 1-form. The inverse map Φ − reads [ α ]
7→ − [ ω − ( p ∗ α )] , by eq. (13), i.e. it is the composition of the naturalisomorphism p ∗ : H ( L ) → H ( T ∗ L ) and the isomorphism H ( T ∗ L ) ∼ = H can ( T ∗ L ) from de Rham to Poisson cohomology carried on every symplectic manifold. Remark . In case the foliation F L on L is of codimension-one, we can compare ourCorollary 2.5 with some results that appeared in [31].i) In [31, Prop. 1.4.7], one proves the following: if ( M, Π) is a corank-one Poisson manifoldand ( F , ω ) denotes its symplectic foliation, then there is a long exact sequence · · · → H k − ( F , ν ) d → H k ( F ) Π −→ H k Π ( M ) → H k − ( F , ν ) d → H k +1 ( F ) → · · · (14)Here ν := T M/T F denotes the normal bundle of the foliation, and H • ( F , ν ) is thecohomology of the complex (cid:0) Γ( ∧ • T ∗ F ⊗ ν ) , d ∇ (cid:1) , where the differential d ∇ is inducedby the Bott connection ∇ : Γ( T F ) × Γ( ν ) → Γ( ν ) : ∇ X N = [ X, N ] . The connecting map d is, up to sign, given by the cup product with the leafwise variation var ω ∈ H ( F , ν ∗ ) of ω [31, Def. 1.2.14], which vanishes when ω extends to a globallydefined closed -form on M .Specializing to our situation, assume ( L, F L ) is a codimension-one foliation. Then ( T ∗ F L , Π can ) is a corank-one Poisson manifold with symplectic foliation ( F sympl , ω ) .The leafwise symplectic form ω ∈ Γ( ∧ T ∗ F sympl ) extends to a closed -form on T ∗ F L .Indeed, a closed extension of ω is given by q ∗ ω T ∗ L , where q : T ∗ F L → T ∗ L is anysplitting of the restriction map r : T ∗ L → T ∗ F L and ω T ∗ L is the canonical symplecticform on T ∗ L . So the connecting map d in (14) is zero, which implies in particular that H can ( T ∗ F L ) ∼ = H ( F sympl ) ⊕ H ( F sympl , ν ) . (15) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 19
This is equivalent with our isomorphism in Corollary 2.5. Firstly, H ( F sympl ) ∼ = H ( F L ) by homotopy invariance. Secondly, as H ( F sympl , ν ) = X ( T ∗ F L ) F sympl / Γ( T F sympl ) , wehave an isomorphism X ( L ) F L / Γ( T F L ) → H ( F sympl , ν ) : [ X ] e X, where e X is the lift of X as defined in Lemma 2.1. To see that this map is well-defined,just note that e X ∈ X ( T ∗ F L ) F sympl , being a Poisson vector field. Injectivity is clear,for if e X is tangent to F sympl , then its projection p ∗ e X = X is tangent to F L . As forsurjectivity, if U ∈ H ( F sympl , ν ) then U = f U as in the proof of Theorem 2.2, where U ∈ X ( L ) F L . So the isomorphism (15) is equivalent with the one from Corollary 2.5: H can ( T ∗ F L ) ∼ = H ( F L ) ⊕ X ( L ) F L / Γ( T F L ) . (16)ii) In case ( L, F L ) is a unimodular codimension-one foliation, then we can further simplifythe isomorphism (16). Indeed, if θ ∈ Ω ( L ) is a closed defining one-form for F L , thenwet get an isomorphism X ( L ) F L / Γ( T F L ) → H ( F L ) : [ V ] θ ( V ) . An alternative argument, building on i) above, is the following. Since also F sympl is unimodular, the representation of T F sympl on ν given by the Bott connection isisomorphic with the trivial representation of T F sympl on the trivial R -bundle T ∗ F L × R (see [31, Lemma 1.5.15]). So in (15), we get H ( F sympl , ν ) ∼ = H ( F sympl ) ∼ = H ( F L ) .We will now upgrade Corollary 2.5 to an isomorphism of Lie algebras. Note that the Liebracket on X ( L ) restricts to X ( L ) F L thanks to the Jacobi identity. Since Γ( T F L ) is a Liealgebra ideal of (cid:0) X ( L ) F L , [ · , · ] (cid:1) , the Lie bracket passes to the quotient X ( L ) F L / Γ( T F L ) . Weget a representation of this Lie algebra on the vector space H ( F L ) , namely ρ : X ( L ) F L Γ( T F L ) → End (cid:0) H ( F L ) (cid:1) : [ X ] £ X · . (17)Here the Lie derivative £ X α := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 φ ∗ t α (18)of α ∈ Ω ( F L ) along X makes sense since the flow φ t of X preserves the foliation F L .Clearly, the map (17) is well-defined: for any X ∈ X ( L ) F L , the Lie derivative £ X acts on H ( F L ) since it commutes with the foliated differential d F L . Moreover, if X ∈ Γ( T F L ) , then £ X acts trivially in cohomology thanks to Cartan’s magic formula. The fact that ρ is a Liealgebra morphism is simply the identity £ [ X,Y ] = £ X ◦ £ Y − £ Y ◦ £ X for X, Y ∈ X ( L ) F L . Proposition 2.8.
Let L be a manifold and F L a foliation on L . Let Π can denote thestandard Poisson structure on the total space of the vector bundle p : T ∗ F L → L . The map Φ constructed in Corollary 2.5 becomes an isomorphism of Lie algebras Φ : (cid:0) H can ( T ∗ F L ) , [ · , · ] (cid:1) → (cid:0) X ( L ) F L / Γ( T F L ) ⋉ ρ H ( F L ) , [ · , · ] ρ (cid:1) , where [ · , · ] is the usual the Lie bracket of vector fields and [ · , · ] ρ is the semidirect productbracket induced by the Lie algebra representation ρ defined in (17) . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 20
To prove Prop. 2.8, it is convenient to rewrite the action (17) in terms of vertical fiberwiseconstant vector fields on T ∗ F L instead of foliated one-forms on ( L, F L ) . To do so, we usethe correspondence (cid:0) Ω • ( F L ) , d F L (cid:1) → (cid:0) X • vert.const. ( T ∗ F L ) , [Π can , · ] (cid:1) : α (cid:16) ∧ • Π ♯can (cid:17) ( p ∗ α ) , (19)which is an isomorphism of cochain complexes up to a global sign, i.e. it matches d F L with − [Π can , · ] (see for instance [12, Lemma 2.1.3]). Lemma 2.9.
For every X ∈ X ( L ) F L , the correspondence (19) matches £ X and (cid:2) e X, · (cid:3) ,where e X is the lift as described in Lemma 2.1.Proof. For every foliated differential form α ∈ Ω k ( F L ) we have to show that ( ∧ k Π ♯can ) ( p ∗ ( £ X α )) = (cid:2) e X, ( ∧ k Π ♯can )( p ∗ α ) (cid:3) . (20)The left hand side of this equality, using (18), reads ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( ∧ k Π ♯can ) (( φ t ◦ p ) ∗ α ) . Since p ∗ e X = X , we have that φ t ◦ p = p ◦ ψ t , where ψ t denotes the flow of e X . So ( ∧ k Π ♯can ) ( p ∗ ( £ X α )) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( ∧ k Π ♯can ) ( ψ ∗ t ( p ∗ α ))= ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( ψ − t ) ∗ (cid:16) ( ∧ k Π ♯can ) ( p ∗ α ) (cid:17) = h e X, ( ∧ k Π ♯can )( p ∗ α ) i , using in the second equality that ψ t is a Poisson diffeomorphism of ( T ∗ F L , Π can ) . So theequality (20) holds, and this proves the lemma. (cid:3) Proof of Prop. 2.8.
To avoid confusion with too many brackets, we will denote equivalenceclasses by underlining the representatives. Fix classes
Y , Z ∈ H can ( T ∗ F L ) and assumethat the representatives Y, Z satisfy the properties in Theorem 2.2. Then also their Liebracket [ Y, Z ] satisfies these properties: it is p -projectable, p ∗ [ Y, Z ] ∈ X ( L ) F L and [ Y, Z ] − ^ p ∗ [ Y, Z ] = [
Y, Z ] − [ g p ∗ Y , g p ∗ Z ]= h g p ∗ Y + (cid:16) Y − g p ∗ Y (cid:17) , g p ∗ Z + (cid:16) Z − g p ∗ Z (cid:17)i − [ g p ∗ Y , g p ∗ Z ]= h g p ∗ Y , Z − g p ∗ Z i + h Y − g p ∗ Y , g p ∗ Z i , (21)using in the last equation that Y − g p ∗ Y and Z − g p ∗ Z are vertical and fiberwise constant.Lemma 2.9 shows in particular that both terms in (21) are vertical fiberwise constant Poissonvector fields, hence the same holds for [ Y, Z ] − ^ p ∗ [ Y, Z ] . So [ Y, Z ] meets the criteria of EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 21
Theorem 2.2. We can therefore proceed as follows:
Φ ([
Y , Z ]) = Φ (cid:16) [ Y, Z ] (cid:17) = (cid:16) p ∗ [ Y, Z ] , [ Y, Z ] − ^ p ∗ [ Y, Z ] (cid:17) = (cid:18) [ p ∗ Y, p ∗ Z ] , h g p ∗ Y , Z − g p ∗ Z i + h Y − g p ∗ Y , g p ∗ Z i(cid:19) = h(cid:16) p ∗ Y , Y − g p ∗ Y (cid:17) , (cid:16) p ∗ Z, Z − g p ∗ Z (cid:17)i ρ = [Φ( Y ) , Φ( Z )] ρ , using the equation (21) in the third equality and Lemma 2.9 in the fourth equality. (cid:3) Deformations of Lagrangian submanifolds in log-symplectic manifolds:algebraic aspects
In this section, we address the algebra behind the deformation problem of a Lagrangiansubmanifold L n contained in the singular locus of a log-symplectic manifold ( M n , Z, Π) .In §3.1-§3.2 we show that the deformation problem is governed by a DGLA, and we discussthe corresponding Maurer-Cartan equation (Thm. 3.2 and Cor. 3.9). We also compute thecohomology of the DGLA in degree one, by calculating the zeroth foliated Morse-Novikovcohomology in §3.3 (Thm. 3.15). This result will be used in the next section to extractgeometric information about the deformations.To set up the stage, we revisit Corollary 1.18, which states that a neighborhood of aLagrangian submanifold L n contained in the singular locus of an orientable log-symplecticmanifold ( M n , Z, Π) can be identified with a neighborhood of the zero section in T ∗ F L × R ,endowed with the log-symplectic structure e Π := V ∧ t∂ t + Π can . (22)Here V is defined on a neighborhood of L in T ∗ F L , and only its Poisson cohomology class [ V ] is fixed, see Remark 1.19. We can use Theorem 2.2 to choose a convenient representative V that satisfies V = V vert + V lift , where V lift := g p ∗ V is the cotangent lift of p ∗ V ∈ X ( L ) F L and V vert := V − g p ∗ V is vertical,fiberwise constant and closed as a section of p : T ∗ F L → L . Indeed, although Theorem 2.2 isphrased for Poisson vector fields defined on all of T ∗ F L , it is clear that the proof still worksif those vector fields are only defined on a neighborhood of L in T ∗ F L whose intersectionwith each fiber is contractible. We summarize the setup for the rest of this paper:Given a Lagrangian submanifold L n contained in the singular locus Z of an orientablelog-symplectic manifold ( M n , Z, Π) , denote by F L the induced foliation on L . Fix atubular neighborhood embedding ψ : ( Z, Π | Z ) → ( T ∗ F L , Π can ) of neighborhoods of L ,as in Prop. 1.9. Denote by [ V ] the image of [ V mod | Z ] under this map, and assumethat V is a representative that satisfies the assumptions of Thm. 2.2. The local modelaround L is then (cid:0) U ⊂ T ∗ F L × R , e Π := ( V vert + V lift ) ∧ t∂ t + Π can (cid:1) , EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 22 where U is a neighborhood of the zero section L . We denote by γ ∈ Ω cl ( F L ) the closed one-form defined by considering V vert as a section of p : T ∗ F L → L , and wealso write X := p ∗ V ∈ X ( L ) F L . Slightly abusing notation, we will often denote the local model by (cid:0) T ∗ F L × R , e Π (cid:1) althoughit is only defined on the neighborhood U . Throughout the rest of the paper, U denotes thisfixed neighborhood. We only make reference to it when strictly necessary.3.1. The Maurer-Cartan equation.
Studying C -small deformations of L now amounts to studying Lagrangian sections in (cid:0) T ∗ F L × R , e Π (cid:1) , the vector bundle over L given by the Whitney sum of T ∗ F L and the trivial R -bundle. By the following little lemma, it is equivalent to look at coisotropic sections. Lemma 3.1.
The graph of a section ( α, f ) ∈ Γ( T ∗ F L × R ) is coisotropic iff it is Lagrangian.Proof. We only have to check the forward implication at points ( α ( q ) , inside the singularlocus T ∗ F L × { } . The symplectic leaf of (cid:0) T ∗ F L × R , e Π (cid:1) through ( α ( q ) , is given by p − ( O ) × { } , where O is the leaf of F L through q . By assumption, the subspace T ( α ( q ) , Graph ( α, f ) ∩ T ( α ( q ) , (cid:0) p − ( O ) × { } (cid:1) = { ( d q α )( v ) : v ∈ T q O and ( d q f )( v ) = 0 } (23)is coisotropic in T ( α ( q ) , (cid:0) p − ( O ) × { } (cid:1) , so it is at least ( n − -dimensional. But clearlythe right hand side of (23) is at most ( n − -dimensional, which shows that the subspace(23) is Lagrangian in T ( α ( q ) , (cid:0) p − ( O ) × { } (cid:1) . (cid:3) We now derive the equations that cut out coisotropic sections in (cid:0) T ∗ F L × R , e Π (cid:1) . Theorem 3.2.
The graph of a section ( α, f ) ∈ Γ( T ∗ F L × R ) is coisotropic in (cid:0) T ∗ F L × R , e Π (cid:1) exactly when ( d F L α = 0 d F L f + f ( γ − £ X α ) = 0 (24)Recall that any vector bundle E → L carries natural maps ∧ • P E : X • ( E ) → Γ( ∧ • E ) ,given by restriction to L composed with the vertical projection Γ ( ∧ • T E | L ) → Γ( ∧ • E ) . Inparticular, for E := T ∗ F L × R we have the following map in degree two: ∧ P E : X ( E ) → Γ( ∧ T ∗ F L ) ⊕ Γ( T ∗ F L ) . It is clear that L is coisotropic with respect to e Π if and only if ∧ P E ( e Π) = 0 . Below, wedenote the bundle projections by pr E : E → L and pr T ∗ F L : T ∗ F L → L respectively. Proof of Thm. 3.2.
A section ( α, f ) ∈ Γ( E ) gives rise to a diffeomorphism φ ( − α, − f ) : E → E : ( p, ξ, t ) (cid:0) p, ξ − α ( p ) , t − f ( p ) (cid:1) which maps the graph of ( α, f ) to the zero section L ⊂ E . So it suffices to single out thesections ( α, f ) such that L is coisotropic with respect to φ ( − α, − f ) ∗ e Π . This amounts to askingthat ∧ P E (cid:16) φ ( − α, − f ) ∗ (cid:0) ( V vert + V lift ) ∧ t∂ t + Π can (cid:1)(cid:17) . (25) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 23
We now simplify the expression (25) in two steps, identifying throughout vertical fiberwiseconstant vector fields on T ∗ F L with foliated one-forms on ( L, F L ) via the bijection (19).i) Claim 1: ∧ P E (cid:16) φ ( − α, − f ) ∗ (cid:0) ( V vert + V lift ) ∧ t∂ t (cid:1)(cid:17) = (0 , f γ − f £ X α ) . (*)First, we note that P E (cid:16) φ ( − α, − f ) ∗ V vert (cid:17) = P T ∗ F L ( V vert ) = V vert and P E (cid:16) φ ( − α, − f ) ∗ t∂ t (cid:17) = P E (cid:0) ( t + pr ∗ E f ) ∂ t (cid:1) = ( pr ∗ E f ) ∂ t , which yields the first term on the right in (*). Secondly, we have ∧ P E (cid:16) φ ( − α, − f ) ∗ ( V lift ∧ t∂ t ) (cid:17) = P E (cid:16) φ ( − α, − f ) ∗ V lift (cid:17) ∧ ( pr ∗ E f ) ∂ t = P T ∗ F L (cid:0) φ − α ∗ V lift (cid:1) ∧ ( pr ∗ E f ) ∂ t , so Claim 1 follows if we show that P T ∗ F L (cid:0) φ − α ∗ V lift (cid:1) = − £ X α. (26)To do so, we compute P T ∗ F L (cid:0) φ − α ∗ V lift (cid:1) = (cid:0) φ − α ∗ V lift (cid:1)(cid:12)(cid:12) L − ( pr T ∗ F L ) ∗ ( φ − α ∗ V lift )= (cid:0) φ − α ∗ V lift (cid:1)(cid:12)(cid:12) L − ( pr T ∗ F L ) ∗ ( V lift )= (cid:0) φ − α ∗ V lift (cid:1)(cid:12)(cid:12) L − V lift | L = Z ddt (cid:0) φ − tα ∗ V lift (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) L dt, (27)using that pr T ∗ F L ◦ φ − α = pr T ∗ F L . Now, note that ddt (cid:0) φ − tα ∗ V lift (cid:1) = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 φ − tα ∗ (cid:0) φ − sα ∗ V lift (cid:1) = φ − tα ∗ (cid:18) dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 φ − sα ∗ V lift (cid:19) = φ − tα ∗ ([ α, V lift ])= [ α, V lift ]= − [ V lift , α ] , (28)In the fourth equality, we used that [ α, V lift ] is vertical and fiberwise constant, whichfollows from Lemma 2.9. Therefore, the equality (27) becomes P T ∗ F L (cid:0) φ − α ∗ V lift (cid:1) = − Z [ V lift , α ] | L dt = − [ V lift , α ] | L , which is exactly (26) under the identification (19). This proves Claim 1.ii) Claim 2: ∧ P E (cid:16) φ ( − α, − f ) ∗ Π can (cid:17) = ( d F L α, d F L f ) . (**)Since L ⊂ ( T ∗ F L , Π can ) is Lagrangian, we have ∧ P E (Π can ) = ∧ P T ∗ F L Π can = 0 , sothe left hand side of (**) is equal to ∧ P E (cid:16) φ ( − α, − f ) ∗ Π can − Π can (cid:17) . (29) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 24
We can decompose φ ( − α, − f ) ∗ Π can − Π can = A t ∧ ∂ t + B t (30)for some A t ∈ X ( T ∗ F L ) and B t ∈ X ( T ∗ F L ) depending smoothly on t . We find A t bycontracting with dt : A t = − (cid:16) φ ( − α, − f ) ∗ Π can − Π can (cid:17) ♯ ( dt )= − h φ ( − α, − f ) ∗ ◦ Π ♯can ◦ (cid:16) φ ( − α, − f ) (cid:17) ∗ i ( dt )= − φ ( − α, − f ) ∗ (cid:16) Π ♯can ( d ( t − pr ∗ E f )) (cid:17) = φ ( − α, − f ) ∗ (cid:16) X pr ∗ T ∗F L f (cid:17) = X pr ∗ T ∗F L f , (31)using that X pr ∗ T ∗F L f is vertical and fiberwise constant. Next, since £ ∂ t (cid:16) φ ( − α, − f ) ∗ Π can (cid:17) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 φ (0 , − t ) ∗ (cid:16) φ ( − α, − f ) ∗ Π can (cid:17) = φ ( − α, − f ) ∗ (cid:18) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 φ (0 , − t ) ∗ Π can (cid:19) = φ ( − α, − f ) ∗ ( £ ∂ t Π can )= 0 , it follows that £ ∂ t B t = £ ∂ t (cid:16) φ ( − α, − f ) ∗ Π can − Π can − X pr ∗ T ∗F L f ∧ ∂ t (cid:17) = 0 , i.e. B t = B is independent of t . So B is equal to its pushforward under the projection T ∗ F L × R → T ∗ F L , which yields B = φ − α ∗ Π can − Π can = Z ddt (cid:0) φ − tα ∗ Π can (cid:1) dt = Z £ α Π can dt = £ α Π can . (32)Here the third equality follows from a computation similar to the one that led to (28).Inserting (31) and (32) into (30) gives φ ( − α, − f ) ∗ Π can − Π can = X pr ∗ T ∗F L f ∧ ∂ t + £ α Π can . Applying the identification (19) now yields the conclusion of Claim 2: ∧ P E (cid:16) φ ( − α, − f ) ∗ Π can − Π can (cid:17) = ∧ P E (cid:16) X pr ∗ T ∗F L f ∧ ∂ t + £ α Π can (cid:17) = ( d F L α, d F L f ) . Combining Claim 1 and Claim 2, we see that the requirement (25) is equivalent with theequations (24) in the statement of the theorem. (cid:3)
EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 25
Corollary 3.3.
Any Lagrangian section of (cid:0) T ∗ F L × R , e Π (cid:1) can be connected to L by a smoothpath of Lagrangian sections. In particular, the set of Lagrangian sections of (cid:0) T ∗ F L × R , e Π (cid:1) is path connected when endowed with the compact-open topology.Proof. Let ( α, f ) ∈ Γ( T ∗ F L × R ) be a Lagrangian section. Fix a smooth function Ψ ∈ C ∞ ( R ) satisfying Ψ( s ) = 0 for s ≤ , < Ψ( s ) < for < s < and Ψ( s ) = 1 for s ≥ . Define Φ ∈ C ∞ ( R ) by putting Φ( s ) := Ψ( s − , and notice that Φ · Ψ = Φ . Consequently, thesmooth path s (cid:0) Ψ( s ) α, Φ( s ) f (cid:1) consists of Lagrangian sections, since (cid:0) Ψ( s ) α, Φ( s ) f (cid:1) isa solution to the equations (24) for each value of s ∈ R . Clearly, this path is continuousfor the compact-open topology, it passes through the zero section at s = 0 , and it reaches ( α, f ) at time s = 2 . This proves the statement. (cid:3) Remark . We comment on the Maurer-Cartan equation (24).i) Twisting the foliated de Rham differential with a closed element η ∈ Ω ( F L ) gives adifferential d η F L : Ω k ( F L ) → Ω k +1 ( F L ) : α d F L α + η ∧ α. (33)The associated cohomology groups, which we denote by H kη ( F L ) , will be discussedin more detail later. If F L is the one-leaf foliation on L , then we recover what iscalled the Morse-Novikov cohomology, which appears in the context of locally conformalsymplectic structures [19, Section 1].ii) The Maurer-Cartan equation (24) shows that the problem of deforming L into a nearbyLagrangian Graph ( α, f ) can essentially be done in two steps. Indeed, one can solve thefirst (linear) equation in (24) for α , and then solve the second equation – which forfixed α becomes linear – for f . Geometrically, this amounts to the following. First,one deforms L inside the singular locus along the leafwise closed one-form α , and thenone moves the obtained Lagrangian L ′ = Graph ( α ) ⊂ T ∗ F L in the direction normal to T ∗ F L along the function f ∈ H γ − £ X α ( F L ) . Ω cl ( F L ) H γ ( F L ) H γ − £ X α ( F L )0 α ( α, f ) Figure 2.
Deforming L into Graph ( α, f ) . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 26
So heuristically, it seems like deforming L into Graph ( α ) ⊂ T ∗ F L for closed α ∈ Ω ( F L ) transforms γ into γ − £ X α . We will now make this precise. Proposition 3.5.
Let ψ : U ⊂ ( Z, Π | Z ) → U ⊂ ( T ∗ F L , Π T ∗ F L ) be a fixed tubular neighbor-hood embedding of L into T ∗ F L , where Π T ∗ F L denotes the canonical Poisson structure Π can .Assume that V = V vert + V lift is a representative of the Poisson cohomology class ψ ∗ [ V mod | Z ] satisfying the requirements of Thm. 2.2, with associated data ( X, γ ) ∈ X ( L ) F L × Ω cl ( F L ) .Consider a Lagrangian L ′ = Graph ( α ) ⊂ U ⊂ T ∗ F L for some closed α ∈ Ω ( F L ) . Thenthe following hold:i) There is a canonical diffeomorphism of affine bundles (Φ , φ ) : ( T ∗ F L , Π T ∗ F L ) → (cid:0) T ∗ F L ′ , Π T ∗ F L ′ (cid:1) which is a Poisson diffeomorphism between the total spaces and fixes points of L ′ , sothat Φ ◦ ψ is a tubular neighborhood embedding of ψ − ( L ′ ) into (cid:0) T ∗ F L ′ , Π T ∗ F L ′ (cid:1) .ii) The representative Φ ∗ V satisfies the requirements of Thm. 2.2 too, and its associateddata are ( X ′ , γ ′ ) = (cid:16) φ ∗ X, (cid:0) φ − (cid:1) ∗ ( γ − £ X α ) (cid:17) ∈ X ( L ′ ) F L ′ × Ω cl ( F L ′ ) .Proof. i) Since α is closed, the translation map φ − α : ( T ∗ F L , Π T ∗ F L ) ∼ → ( T ∗ F L , Π T ∗ F L ) : ( p, ξ ) ( p, ξ − α ( p )) is a Poisson diffeomorphism; this follows from the computation (32) and the isomor-phism (19). Its restriction to L ′ , which coincides with the restriction of the vectorbundle projection p L to L ′ , is a foliated diffeomorphism τ : ( L ′ , F L ′ ) ∼ → ( L, F L ) , andthe cotangent lift T ∗ τ of τ descends to a Poisson diffeomorphism T ∗F τ : ( T ∗ F L , Π T ∗ F L ) ∼ → (cid:0) T ∗ F L ′ , Π T ∗ F L ′ (cid:1) . In summary, we have a commutative diagram ( T ∗ L ′ , Π T ∗ L ′ ) ( T ∗ L, Π T ∗ L ) (cid:0) T ∗ F L ′ , Π T ∗ F L ′ (cid:1) ( T ∗ F L , Π T ∗ F L )( L ′ , F L ′ ) ( L, F L ) r L ′ r L T ∗ τp L ′ p L T ∗F ττ . (34)The affine bundle map (Φ , φ ) := (cid:0) T ∗F τ ◦ φ − α , τ − (cid:1) meets the requirements.ii) Clearly Φ ∗ V is p L ′ -projectable, since V is p L -projectable and Φ covers the diffeomor-phism φ . Using that p L ◦ φ − α = p L , we have ( p L ′ ) ∗ (Φ ∗ V ) = (cid:0) p L ′ ◦ T ∗F τ ◦ φ − α (cid:1) ∗ V = (cid:0) τ − ◦ p L ◦ φ − α (cid:1) ∗ V = (cid:0) τ − (cid:1) ∗ X. Since τ − : ( L, F L ) → ( L ′ , F L ′ ) is a foliated diffeomorphism and X ∈ X ( L ) F L , wealso have (cid:0) τ − (cid:1) ∗ X ∈ X ( L ′ ) F L ′ . Moreover, ^ ( τ − ) ∗ X = ( T ∗F τ ) ∗ V lift by functoriality.It remains to show that Φ ∗ V − ^ ( p L ′ ) ∗ (Φ ∗ V ) = ( T ∗F τ ◦ φ − α ) ∗ V − ( T ∗F τ ) ∗ V lift is ver-tical, fiberwise constant, and that it corresponds with the closed foliated one-form EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 27 τ ∗ ( γ − £ X α ) ∈ Ω ( F L ′ ) . We rewrite it as ( T ∗F τ ) ∗ (cid:2)(cid:0) φ − α (cid:1) ∗ ( V − V lift ) (cid:3) + ( T ∗F τ ) ∗ (cid:2)(cid:0) φ − α (cid:1) ∗ V lift − V lift (cid:3) = ( T ∗F τ ) ∗ ( V − V lift ) + ( T ∗F τ ) ∗ (cid:2)(cid:0) φ − α (cid:1) ∗ V lift − V lift (cid:3) , (35)using that V − V lift is vertical and fiberwise constant. The computations done in (27)and (28) show that ( φ − α ) ∗ V lift − V lift = − [ V lift , α ] , so it is vertical fiberwise constantand it corresponds with the closed one-form − £ X α ∈ Ω ( F L ) under the identification(19). Since V − V lift corresponds with γ ∈ Ω ( F L ) , we get that the vertical fiberwiseconstant vector field (35) indeed corresponds with the closed one-form τ ∗ ( γ − £ X α ) . (cid:3) The DGLA behind the deformation problem.
We now show that the equations (24) obtained in Theorem 3.2 represent the Maurer-Cartan equation of a differential graded Lie algebra (DGLA) that governs the deformationsof the Lagrangian L ⊂ (cid:0) T ∗ F L × R , e Π (cid:1) . To this end, recall the following.Suppose E → C is a vector bundle and let Π be a Poisson structure on E such that C is coisotropic. Cattaneo and Felder showed in [3] that the graded vector space Γ ( ∧ • E ) [1] supports a canonical L ∞ [1] -algebra structure whose multibrackets are defined by λ k : Γ ( ∧ • E ) [1] ⊗ k → Γ ( ∧ • E ) [1] : ξ ⊗ · · · ⊗ ξ k
7→ ∧ P ([ . . . [[Π , ξ ] , ξ ] . . . , ξ k ]) . (36)Here the ξ i are interpreted as vertical fiberwise constant multivector fields on E and themap ∧ • P : X • ( E ) → Γ( ∧ • E ) is the restriction to C composed with the vertical projection Γ ( ∧ • T E | L ) → Γ( ∧ • E ) . These structure maps λ k only depend on the ∞ -jet of Π along thesubmanifold C , so the L ∞ [1] -algebra usually does not carry enough information to codify Π in a neighborhood of C . Consequently, this L ∞ [1] -algebra fails to encode coisotropicdeformations of C in general (see [33, Ex. 3.2]).However, if the Poisson structure Π is analytic in the fiber directions, then the L ∞ [1] -algebra of Cattaneo-Felder does govern the smooth coisotropic deformation problem of C .In [34], such bivector fields are called fiberwise entire, and there one proves the following. Theorem 3.6. [34, Thm. 1.12]
Let E → C be a vector bundle and Π a fiberwise entirePoisson structure which is defined on a tubular neighborhood U of C in E . Suppose that C is coisotropic with respect to Π , and consider the L ∞ [1] -algebra associated with C ⊂ ( U, Π) .For any section α ∈ Γ( E ) such that Graph ( − α ) is contained in U , the Maurer-Cartan series M C ( α ) converges. Furthermore, for any such α ∈ Γ( E ) , the following are equivalent:(1) The graph of − α is a coisotropic submanifold of ( U, Π) .(2) The Maurer-Cartan series M C ( α ) converges to zero. In the rest of this section, we show that the L ∞ -algebra of Cattaneo-Felder associatedwith (cid:0) T ∗ F L × R , e Π (cid:1) reduces to a DGLA, and that this DGLA governs the deformationproblem of the Lagrangian L . Lemma 3.7.
The Poisson structure e Π = ( V vert + V lift ) ∧ t∂ t + Π can , defined on a neighbor-hood U of L in T ∗ F L × R , is fiberwise entire.Proof. This is straightforward computation. Choose coordinates ( x , . . . , x n ) on L adaptedto the foliation F L , such that plaques of F L are level sets of x . Let ( y , . . . , y n ) be the EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 28 corresponding fiber coordinates on T ∗ L . Then write Π can = n X i =2 ∂ x i ∧ ∂ y i , V vert = n X j =2 f j ( x ) ∂ y j , p ∗ V = n X j =1 h j ( x ) ∂ x j , where p : T ∗ F L → L is the projection and h ( x ) only depends on x since p ∗ V ∈ X ( L ) F L .We then obtain V lift = n X j =1 h j ( x ) ∂ x j − n X i =2 n X j =2 y j ∂h j ∂x i ( x ) ∂ y i . So the Poisson structure e Π reads e Π = n X j =2 f j ( x ) ∂ y j + n X j =1 h j ( x ) ∂ x j − n X i =2 n X j =2 y j ∂h j ∂x i ( x ) ∂ y i ∧ t∂ t + n X i =2 ∂ x i ∧ ∂ y i , which is clearly a fiberwise entire bivector field. (cid:3) Lemma 3.8.
The L ∞ [1] -algebra (cid:0) Γ( ∧ • ( T ∗ F L × R ))[1] , { λ k } (cid:1) of Cattaneo-Felder associatedwith (cid:0) T ∗ F L × R , e Π (cid:1) corresponds to a DGLA-structure on Γ( ∧ • ( T ∗ F L × R )) .Proof. We will show that the multibrackets λ k defined in (36) vanish for k ≥ . Since the λ k are multiderivations, it is enough to evaluate them on elements of C ∞ ( L ) and Γ( T ∗ F L × R ) .As the λ k have degree one, a degree counting argument shows that they can be non-zeroonly when evaluated on tuples of the form ( σ , . . . , σ k ) , ( h, σ , . . . , σ k − ) and ( h, h ′ , σ , . . . , σ k − ) , where h, h ′ ∈ C ∞ ( L ) and σ , . . . , σ k ∈ Γ ( T ∗ F L × R ) . Now let ( α, f ) , ( β, g ) ∈ Γ ( T ∗ F L × R ) and h ∈ C ∞ ( L ) . Let pr : T ∗ F L × R → L denote the projection. If we show that themultivector fields hh e Π , α + pr ∗ ( f ) ∂ t i , β + pr ∗ ( g ) ∂ t i , hh e Π , α + pr ∗ ( f ) ∂ t i , pr ∗ ( h ) i (37)are vertical and fiberwise constant, then the above observation implies that λ k = 0 whenever k ≥ , since clearly hh e Π , pr ∗ ( h ) i , pr ∗ ( h ′ ) i = 0 . One checks that h e Π , α + pr ∗ ( f ) ∂ t i = (cid:2) ( V vert + V lift ) ∧ t∂ t + Π can , α + pr ∗ ( f ) ∂ t (cid:3) = − pr ∗ ( f ) V vert ∧ ∂ t + [ V lift , α ] ∧ t∂ t − pr ∗ ( f ) V lift ∧ ∂ t + [Π can , α ] + [Π can , pr ∗ f ] ∧ ∂ t , where − pr ∗ ( f ) V vert ∧ ∂ t , [Π can , α ] and [Π can , pr ∗ f ] ∧ ∂ t are vertical and fiberwise constant.So only the second and third summand are relevant to compute the expressions (37), andwe get hhe Π , α + pr ∗ ( f ) ∂ t i , β + pr ∗ ( g ) ∂ t i = (cid:2) [ V lift , α ] ∧ t∂ t − pr ∗ ( f ) V lift ∧ ∂ t , β + pr ∗ ( g ) ∂ t (cid:3) = pr ∗ ( f )[ β, V lift ] ∧ ∂ t + pr ∗ ( g )[ α, V lift ] ∧ ∂ t and hh e Π , α + pr ∗ ( f ) ∂ t i , pr ∗ ( h ) i = (cid:2) [ V lift , α ] ∧ t∂ t − pr ∗ ( f ) V lift ∧ ∂ t , pr ∗ ( h ) (cid:3) = pr ∗ (cid:0) f X ( h ) (cid:1) ∧ ∂ t , EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 29 where X = pr ∗ V lift as before. Using Lemma 2.9 we see that the multivector fields (37) arevertical and fiberwise constant, which implies the statement of the lemma. (cid:3) We now established the existence of a DGLA-structure supported on Γ( ∧ • ( T ∗ F L × R )) which governs the deformations of L as a coisotropic submanifold. Thanks to Lemma 3.1,this DGLA in fact governs the Lagrangian deformation problem of L . We now provide moreexplicit descriptions for the structure maps of the DGLA. Corollary 3.9.
The deformation problem of a Lagrangian submanifold L n contained in thesingular locus of an orientable log-symplectic manifold ( M n , Z, Π) is governed by a DGLAsupported on the graded vector space Γ ( ∧ • ( T ∗ F L × R )) = Γ (cid:0) ∧ • T ∗ F L ⊕ ∧ •− T ∗ F L (cid:1) , whosestructure maps ( d, [[ · , · ]]) are defined by d : Γ (cid:16) ∧ k ( T ∗ F L × R ) (cid:17) → Γ (cid:16) ∧ k +1 ( T ∗ F L × R ) (cid:17) : ( α, β ) ( − d F L α, − d F L β − γ ∧ β ) , [[ · , · ]] : Γ (cid:16) ∧ k ( T ∗ F L × R ) (cid:17) ⊗ Γ (cid:16) ∧ l ( T ∗ F L × R ) (cid:17) → Γ (cid:16) ∧ k + l ( T ∗ F L × R ) (cid:17) :( α, β ) ⊗ ( δ, ǫ ) (cid:16) , £ X α ∧ ǫ − ( − kl £ X δ ∧ β (cid:17) . Proof.
We start by writing down explicitly the structure maps λ , λ of the L ∞ [1] -algebra (cid:0) Γ( ∧ • ( T ∗ F L × R ))[1] , λ , λ (cid:1) , as defined in (36). We then apply the décalage isomorphismsto obtain the associated DGLA (cid:0) Γ( ∧ • ( T ∗ F L × R )) , d, [[ · , · ]] (cid:1) . In the computations below, weagain identify elements of Γ (cid:0) ∧ • T ∗ F L (cid:1) with vertical fiberwise constant multivector fieldson T ∗ F L via the isomorphism (19).Choose homogeneous elements ( α, β ) ∈ Γ (cid:0) ∧ k ( T ∗ F L × R ) (cid:1) and ( δ, ǫ ) ∈ Γ (cid:0) ∧ l ( T ∗ F L × R ) (cid:1) .We then have h e Π , α + β ∧ ∂ t i = (cid:2) ( V vert + V lift ) ∧ t∂ t + Π can , α + β ∧ ∂ t (cid:3) = ( − k − [ V lift , α ] ∧ t∂ t − V ∧ β ∧ ∂ t + [Π can , α ] + [Π can , β ] ∧ ∂ t , (38)which implies that λ (cid:0) ( α, β ) (cid:1) = ( − d F L α, − d F L β − γ ∧ β ) . (39)Next, using the computation (38), we have hh e Π , α + β ∧ ∂ t i , δ + ǫ ∧ ∂ t i = h ( − k − [ V lift , α ] ∧ t∂ t − V ∧ β ∧ ∂ t + [Π can , α ] + [Π can , β ] ∧ ∂ t , δ + ǫ ∧ ∂ t i = h ( − k − [ V lift , α ] ∧ t∂ t − V lift ∧ β ∧ ∂ t , δ + ǫ ∧ ∂ t i = ( − k [ V lift , α ] ∧ ǫ ∧ ∂ t − ( − k ( l − [ V lift , δ ] ∧ β ∧ ∂ t , which implies that λ (cid:0) ( α, β ) ⊗ ( δ, ǫ ) (cid:1) = (cid:16) , ( − k £ X α ∧ ǫ − ( − k ( l − £ X δ ∧ β (cid:17) . (40)The décalage isomorphisms act as ( α, β ) ( α, β )( α, β ) ⊗ ( δ, ǫ ) ( − k ( α, β ) ⊗ ( δ, ǫ ) , and applying them to (39) and (40) yields the expressions stated in the corollary. (cid:3) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 30
In more detail, the fact that this DGLA governs the deformations of L means the follow-ing. For convenience, we assume the neighborhood U of L in T ∗ F L × R where e Π is defined tobe invariant under fiberwise multiplication by − . Then for any section ( α, f ) ∈ Γ( T ∗ F L × R ) whose graph lies inside U , we haveGraph ( α, f ) is Lagrangian ⇔ Graph ( α, f ) is coisotropic ⇔ ( − α, − f ) is a Maurer-Cartan element ofthe L ∞ [1] − algebra (cid:0) Γ( T ∗ F L × R )[1] , λ , λ (cid:1) ⇔ ( α, f ) is a Maurer-Cartan element ofthe DGLA (cid:0) Γ( T ∗ F L × R ) , d, [[ · , · ]] (cid:1) ⇔ ( d F L α = 0 d F L f + f ( γ − £ X α ) = 0 , where the first equivalence is Lemma 3.1 and the second one is Thm. 3.6. So we recoverthe equations (24) that we derived in Thm. 3.2 by direct computation. Remark . We do not know whether the DGLA in Corollary 3.9 is formal,i.e. L ∞ -quasi-isomorphic to its cohomology H • ( F L ) ⊕ H •− γ ( F L ) with the induced gradedLie algebra structure. On one side, such a result would not be so surprising when L is com-pact, because of the following. Any graded Lie algebra ( H, [ · , · ]) has the property that theKuranishi map completely characterizes formal unobstructedess: a first order deformation A is formally unobstructed if and only if Kr ( A ) = 0 . When L is compact, we know thatthe DGLA in Corollary 3.9 satisfies this property, as a consequence of Prop. 4.18. Furtherwe expect the property to be invariant under L ∞ -quasi-isomorphisms satisfying mild as-sumptions. We do not address the formality question any further here. A possible approachis to apply Manetti’s formality criteria in Thm. 3.3 or Thm. 3.4 of [22]. Remark . We comment on the structure of the DGLA (cid:0) Γ( T ∗ F L × R ) , d, [[ · , · ]] (cid:1) introducedin Corollary 3.9.i) One can write down this DGLA in more generality. Let (cid:0) A, ρ, [ · , · ] (cid:1) be a Lie algebroidover a manifold M , and let ∇ be a flat A -connection on a line bundle E → M . Let D ∈ Der ( A ) be a derivation of A . Then there is an induced DGLA-structure ( d, [[ · , · ]]) on the graded vector space Γ ( ∧ • ( A ∗ ⊕ E )) = Γ ( ∧ • A ∗ ) ⊕ Γ (cid:0) ∧ •− A ∗ ⊗ E (cid:1) defined by d ( α, ϕ ) = ( d A α, d ∇ ϕ )[[( α, ϕ ) , ( β, ψ )]] = (cid:16) , £ D α ∧ ψ − ( − kl £ D β ∧ ϕ (cid:17) , (41)for homogeneous elements ( α, ϕ ) ∈ Γ (cid:0) ∧ k ( A ∗ ⊕ E ) (cid:1) and ( β, ψ ) ∈ Γ (cid:0) ∧ l ( A ∗ ⊕ E ) (cid:1) . Herethe Lie derivative £ D is obtained extending the derivation on A ∗ dual to D .ii) We discuss the structure of the DGLA (Γ ( ∧ • ( A ∗ ⊕ E )) , d, [[ · , · ]]) . The underlyingcochain complex is a direct sum of complexes (Γ ( ∧ • A ∗ ) , d A ) ⊕ (Γ ( ∧ • A ∗ ⊗ E ) [ − , d ∇ ) .It can also be described as the cochain complex of differential forms on the Lie algebroid A ⊕ E ∗ , the semidirect product of A by the representation on E ∗ given by the dualconnection ∇ ∗ . The underlying graded Lie algebra structure is the semidirect productof the abelian graded Lie algebras Γ ( ∧ • A ∗ ) and Γ ( ∧ • A ∗ ⊗ E ) [ − with respect to theaction Γ ( ∧ • A ∗ ) → Der (cid:0)
Γ ( ∧ • A ∗ ⊗ E ) [ − (cid:1) : α £ D α ∧ • . This is immediate, since if Kr ( A ) = [ A, A ] vanishes then t tA is a curve of Maurer-Cartan elements. EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 31 iii) We can recover the DGLA (cid:0) Γ( T ∗ F L × R ) , d, [[ · , · ]] (cid:1) described in Corollary 3.9 by makingthe following choices in the general construction of i) above: • Take the Lie algebroid A := ( T F L , − ι, − [ · , · ]) , where ι : T F L ֒ → T L is the inclusionand [ · , · ] is the Lie bracket of vector fields. The Lie algebroid differential d A on Γ ( ∧ • A ∗ ) is then − d F L . • Let D := [ X, · ] be the inner derivation determined by X ∈ X ( L ) F L . • Let E := L × R → L be the trivial line bundle. • Let the representation ∇ of A on E be defined by ∇ Y • = £ − Y • − γ ( Y ) • for Y ∈ Γ( A ) . Since γ is closed, this is indeed a representation, and the induceddifferential d ∇ on Γ ( ∧ • A ∗ ) is given by d ∇ • = − d F L • − γ ∧ • . On foliated Morse-Novikov cohomology.
This subsection discusses the cohomology of the DGLA (cid:0) Γ( ∧ • ( T ∗ F L × R )) , d, [[ · , · ]] (cid:1) indegree one, which in the notation of Remark 3.4 is given by H ( F L ) ⊕ H γ ( F L ) . We explicitlycompute the second summand of this cohomology group for Lagrangians that are compactand connected. We first collect some foliated analogs of well-known facts about Morse-Novikov cohomology [19, Section 1]. Lemma 3.12.
Let L be a manifold, F L a foliation on L and η ∈ Ω ( F L ) a closed foliatedone-form. As before, denote by H • η ( F L ) the cohomology groups of the differential d η F L definedin (33) . We then have the following:i) If [ η ] = [ η ′ ] ∈ H ( F L ) , then H kη ( F L ) ∼ = H kη ′ ( F L ) . In particular, if [ η ] = 0 in H ( F L ) then H kη ( F L ) ∼ = H k ( F L ) .ii) Assume [ η ] = 0 in H ( F L ) and let f ∈ H η ( F L ) . Then there is a leaf O of F L on which f vanishes identically.Proof. i) If η ′ = η + d F L g for g ∈ C ∞ ( L ) , then the following map is an isomorphism ofcochain complexes: (cid:0) Ω • ( F L ) , d η ′ F L (cid:1) → (cid:0) Ω • ( F L ) , d η F L (cid:1) : β e g β. (42)ii) By assumption we have that d F L f + f η = 0 . (43)If f would be nowhere zero, then we could write η = − d F L log | f | , contradicting that η is not exact. So f must have a zero, say in the leaf O ∈ F L . Consider the vanishing set Z f := { x ∈ O : f ( x ) = 0 } , which is nonempty and closed in O . If we show that Z f isalso open in O , then we reach the conclusion f | O ≡ , since O is connected.To this end, let x ∈ Z f . Since η | O ∈ Ω ( O ) is closed, there exist a neighborhood U of x in O and g ∈ C ∞ ( U ) such that η | U = dg . Using the isomorphism (42) for theone-leaf foliation on U , we obtain that d ( e g f | U ) = 0 . So e g f | U is constant on U , andsince f ( x ) = 0 we must have e g f | U ≡ . Consequently f | U ≡ , which shows that U ⊂ Z f . So Z f is open, and this finishes the proof. (cid:3) Remark . If we replace the hypothesis [ η ] = 0 in ii ) of Lemma 3.12 by the strongerrequirement that η | O ∈ Ω ( O ) be not exact for all leaves O ∈ F L , then, restricting theequality (43) to each leaf O , the above proof shows that H η ( F L ) = 0 . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 32
We now specialize to compact, connected manifolds L endowed with a codimension-onefoliation F L defined by a nowhere vanishing closed one-form. Under these assumptions it iswell-known [5, Theorem 9.3.13] that: • either ( L, F L ) is the fiber foliation of a fiber bundle p : L → S , • or all leaves of F L are dense. ( ⋆ )Recall moreover that in the former case, the k -th cohomology groups of the fibers of p : L → S constitute a vector bundle H k over S : H kq = H k (cid:0) p − ( q ) (cid:1) , and one has H k ( F L ) ∼ → Γ (cid:16) H k (cid:17) : [ α ] (cid:16) σ α : q h α | p − ( q ) i(cid:17) . (44)Using the identification X ( S ) ∼ −→ X ( L ) F L Γ( T F L ) : Y Y , one can define a natural flat connection ∇ on the vector bundle H k by the formula ∇ Y σ α := σ £ Y α , (45)for α ∈ Ω cl ( F L ) and Y ∈ X ( S ) . Note that ∇ is well-defined, because of Cartan’s formula. If F denotes the typical fiber of p : L → S and { [ β ] , . . . , [ β m ] } is a basis of H k ( F ) , then in alocal trivialization U × F , the constant functions [ β ] , . . . , [ β m ] ∈ C ∞ ( U, H k ( F )) ∼ = Γ( H k | U ) constitute a local frame of flat sections.To compute the foliated Morse-Novikov cohomology, we will need the following lemma. Lemma 3.14.
Let L be a compact manifold endowed with a foliation F L that is the fiberfoliation of a fiber bundle p : L → S . Let η ∈ Ω ( F L ) be a closed foliated one-form,denote by σ η ∈ Γ( H ) the section corresponding with [ η ] ∈ H ( F L ) under (44) , and let Z η := σ − η (0) . Then there exists a smooth function g ∈ C ∞ ( L ) such that η | p − ( q ) = d (cid:16) g | p − ( q ) (cid:17) for all q ∈ Z η . Proof.
By [24, Lemma 2.28], we can fix an embedded loop τ : S → L transverse to the leavesof F L which hits each leaf of F L exactly once. Define a function h on p − ( Z η ) by setting h | p − ( q ) to be the unique primitive of η | p − ( q ) that vanishes at the point p − ( q ) ∩ τ ( S ) .We claim that h extends to a smooth function g ∈ C ∞ ( L ) . To prove this, it suffices toshow that around each point x ∈ p − ( Z η ) there exist a neighborhood U ⊂ L and a smoothfunction on U that agrees with h on U ∩ p − ( Z η ) .Let x ∈ p − ( q ) for q ∈ Z η and denote y := p − ( q ) ∩ τ ( S ) . Working in a local trivialization V × p − ( q ) , choose a path γ : ( − ǫ, ǫ ) → p − ( q ) such that γ (0) = x and γ (1) = y , takea tubular neighborhood N of this path in p − ( q ) and define U := V × N . Since N iscontractible, we have for each value of v ∈ V that η v ∈ Ω ( N ) is exact. Since one canchoose primitives varying smoothly in v (see [40]), it follows that η | U is foliated exact.Choose any primitive k ∈ C ∞ ( U ) of η | U . Shrinking V if necessary, we can assume that eachfiber { v }× N intersects the loop τ ( S ) . Define a map φ : U → U ∩ τ ( S ) by setting φ ( z ) to bethe intersection point of τ ( S ) with the fiber through z . Then setting e h := k − φ ∗ (cid:0) k | U ∩ τ ( S ) (cid:1) ,we obtain a primitive of η | U that vanishes along U ∩ τ ( S ) . Note that under these assumptions, L is automatically connected. EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 33
Uniqueness of such primitives implies that e h agrees with h wherever both of them aredefined. This shows that h can be extended to a smooth function g ∈ C ∞ ( L ) . (cid:3) We can now compute the zeroth foliated Morse-Novikov cohomology group.
Theorem 3.15.
Let ( L, F L ) be a compact, connected manifold with codimension-one folia-tion defined by a closed one-form. Let η ∈ Ω ( F L ) be a closed foliated one-form.i) Assume F L is the fiber foliation of a fiber bundle p : L → S . Then we have H η ( F L ) ∼ = { f ∈ C ∞ ( S ) : f · σ η = 0 } , where σ η ∈ Γ( H ) denotes the section corresponding with [ η ] ∈ H ( F L ) under (44) .ii) Assume all leaves of F L are dense. Then H η ( F L ) = ( R if η is foliated exact otherwise . Proof. i) Fix a smooth function g ∈ C ∞ ( L ) as constructed in Lemma 3.14 and define Ψ : H η ( F L ) → { f ∈ C ∞ ( S ) : f · σ η = 0 } : h e g h. We first check that Ψ is well-defined. Choosing h ∈ H η ( F L ) , we must show that e g h is constant along the leaves of F L , and that the induced function on the leaf space S lies in the annihilator ideal of σ η ∈ Γ( H ) . Note that for any q ∈ S , we have d (cid:16) h | p − ( q ) (cid:17) + h | p − ( q ) η | p − ( q ) = 0 . In case σ η ( q ) = 0 , then η | p − ( q ) = d ( g | p − ( q ) ) and the isomorphism (42) implies that ( e g h ) | p − ( q ) ∈ H ( p − ( q )) = R . Next, assume that σ η ( q ) = 0 , i.e. η | p − ( q ) is not exact. Then h | p − ( q ) ≡ by applying ii ) of Lemma 3.12 to the one-leaf foliation on p − ( q ) , and therefore ( e g h ) | p − ( q ) ≡ .Clearly, the map Ψ is linear and injective. For surjectivity, we let f ∈ C ∞ ( S ) besuch that f · σ η = 0 and we have to check that e − g p ∗ f ∈ H η ( F L ) , i.e. d F L (cid:18) p ∗ fe g (cid:19) + p ∗ fe g η = 0 . (46)On fibers p − ( q ) with σ η ( q ) = 0 , the equality (46) is satisfied since p ∗ f vanishes there.On fibers p − ( q ) with σ η ( q ) = 0 , we have η | p − ( q ) = d ( g | p − ( q ) ) , so that the left handside of (46) becomes − f ( q )( e − g | p − ( q ) ) d ( g | p − ( q ) ) + f ( q )( e − g | p − ( q ) ) d ( g | p − ( q ) ) = 0 . ii) This is an immediate consequence of Lemma 3.12. (cid:3) Example 3.16.
Take L = ( S × S , θ , θ ) and let F L be the foliation by fibers of theprojection ( S × S , θ , θ ) → ( S , θ ) . To compute H η ( F L ) for closed η ∈ Ω ( F L ) , we canchoose a convenient representative of [ η ] ∈ H ( F L ) , by i ) of Lemma 3.12. In this respect,notice that every class [ g ( θ , θ ) dθ ] ∈ H ( F L ) has a unique representative of the form h ( θ ) dθ . Namely, setting h ( θ ) := π R S g ( θ , θ ) dθ , we have Z S [ g ( θ , θ ) − h ( θ )] dθ = 0 , EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 34 which implies that there exists k ( θ , θ ) ∈ C ∞ ( S × S ) such that g ( θ , θ ) − h ( θ ) = ∂k∂θ ( θ , θ ) . This implies that g ( θ , θ ) dθ − h ( θ ) dθ = ∂k∂θ ( θ , θ ) dθ = d F L k. Uniqueness of such representatives follows by integrating around circles { θ } × S . Now, fix η = h ( θ ) dθ in Ω ( F L ) and assume that f ∈ H η ( F L ) . Then ∂f∂θ dθ + f · h ( θ ) dθ . (47)For fixed θ , the restriction of f to { θ } × S reaches a maximum M and a minimum m .The equality (47) implies that ( M · h ( θ ) = 0 m · h ( θ ) = 0 . So either h ( θ ) = 0 or f | { θ }× S ≡ . Hence, we get that f · h ( θ ) = 0 , and (47) then impliesthat also ∂f /∂θ = 0 . In conclusion, we get H h ( θ ) dθ ( F L ) = { f ( θ ) : f ( θ ) h ( θ ) dθ = 0 } = { f ( θ ) : f ( θ ) · σ h ( θ ) dθ = 0 } , using in the last equality that σ h ( θ ) dθ ( θ ) = 0 ⇔ h ( θ ) dθ = 0 . So we obtain the resultthat was predicted by i ) of Proposition 3.15. Remark . The example we have in mind throughout this subsection is of course thatof a compact connected Lagrangian submanifold L n contained in the singular locus Z of alog-symplectic manifold ( M n , Z, Π) . The induced foliation F L on L is defined by a nowherevanishing closed one-form, which is obtained by pulling back a closed defining one-form forthe foliation on Z . So ( L, F L ) is either the fiber foliation of a fiber bundle L → S , or allleaves of F L are dense in L .Moreover, the foliation type of F L is stable under small deformations of the Lagrangian L inside Z . To see this, we can work in the local model p : T ∗ F L → L , where the total space T ∗ F L is endowed with the pullback foliation p − ( F L ) . Any Lagrangian deformation L ′ of L is of the form L ′ = Graph ( α ) for some α ∈ Ω cl ( F L ) , and the induced foliation F L ′ is obtainedby intersecting L ′ with the leaves of p − ( F L ) . Therefore, the map p : ( L ′ , F L ′ ) → ( L, F L ) isa foliated diffeomorphism (with inverse α : ( L, F L ) → ( L ′ , F L ′ ) ), which shows that ( L, F L ) and ( L ′ , F L ′ ) are of the same type.4. Deformations of Lagrangian submanifolds in log-symplectic manifolds:geometric aspects
We present some geometric consequences of the algebraic results obtained in the previoussection. We address three different geometric questions, each in a separate subsection, aswe now outline. Throughout, we assume the set-up given at the beginning of §3. §4.1 Deformations constrained to the singular locus:
We investigate when allsufficiently small deformations of the Lagrangian L are constrained to the singularlocus. Prop. 4.2 gives a condition under which this does not happen. On theopposite extreme, in Cor. 4.5 and Prop. 4.10 we obtain positive results assuming EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 35 that L is compact, by considering separately the case that L is the total space ofa fibration and the case that L has a dense leaf. The latter case is subtle, and weshow that the conclusion of Prop. 4.10 fails to hold if we remove a certain finitedimensionality assumption. §4.2 Obstructedness of deformations: We ask when infinitesimal deformations ofthe Lagrangian L can be extended to a smooth curve of Lagrangian deformations. Asufficient criterium is given in Prop. 4.13. (All smoothly unobstructed deformationsarise this way under an additional assumption, see Lemma 4.17). Our main resultshere, under the assumption that L is compact, are the computable “if and only if”criteria of Prop. 4.18 and Cor. 4.20. §4.3 Equivalences and rigidity of deformations: On the set of Lagrangian defor-mations of L there are two natural notions of equivalence: an algebraic one and ageometric one, given by Hamiltonian isotopies. In Prop. 4.26 we show that theycoincide. We also show that there are no Lagrangian submanifolds which are in-finitesimally rigid under Hamiltonian isotopies, so that the moduli space (whichtypically is not smooth) does not have any isolated points. This leads us to considerthe more flexible equivalence relation given by Poisson isotopies. The formal tangentspace of its moduli space is computed in Prop. 4.30. There do exist Lagrangianswhich are rigid under Poisson isotopies, as follows using Prop. 4.34. Remark . We summarize here how our results specializeto the local deformation problem, i.e. to a Lagrangian L as in the local model of Prop. 1.17: • L can be deformed smoothly to a Lagrangian submanifold outside of the singularlocus (Remark 4.3). • all first order deformations of L are smoothly unobstructed (Cor. 4.15). • The space of local Lagrangian deformations modulo Hamiltonian isotopies is notsmooth at [ L ] . Indeed, the formal tangent space at [ L ] is isomorphic to C ∞ ( R ) (seeeq. (74)), while at Lagrangians contained in M \ Z it is the zero vector space. Thesame is true if one replaces Hamiltonian isotopies by Poisson isotopies.4.1. Deformations constrained to the singular locus.
We now investigate whether it is always possible to find deformations of the Lagrangian L that escape from the singular locus. Working in the model (cid:0) U ⊂ T ∗ F L × R , V ∧ t∂ t +Π can (cid:1) , asufficient condition is the existence of a representative of the fixed first Poisson cohomologyclass [ V ] that is tangent to L . Below, we denote by W := U ∩ { t = 0 } ⊂ T ∗ F L theneighborhood of L in T ∗ F L where V is defined. Proposition 4.2.
The Poisson cohomology class [ V ] ∈ H can ( W ) has a representativetangent to L if and only if [ γ ] = 0 ∈ H ( F L ) . If these equivalent conditions hold, then thereis a smooth path of Lagrangian deformations L s starting at L = L which is not containedin the singular locus for s > .Proof. We start by showing that the conditions are equivalent. First assume that V − X g for g ∈ C ∞ ( W ) is a representative of [ V ] that is tangent to L . As before, let P denote themap that restricts vector fields on W to L and then takes their vertical component. By EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 36 assumption, we then have P ( V − X g ) = P ( V vert + V lift − X g )= γ − P ( X g − X p ∗ ( i ∗ g ) + X p ∗ ( i ∗ g ) )= γ − P ( X p ∗ ( i ∗ g ) )= γ − d F L ( i ∗ g ) . Here p : W → L and i : L ֒ → W denote the projection and inclusion, respectively, the fourthequality holds since L is coisotropic, and the last equality holds by the correspondence (19).This shows that γ = d F L ( i ∗ g ) , and therefore [ γ ] = 0 ∈ H ( F L ) . Conversely, if γ = d F L g forsome g ∈ C ∞ ( L ) , then V − X p ∗ g is a representative of [ V ] that is tangent to L .If the equivalent conditions hold, then by Remark 1.19 we can assume that γ = 0 . TheMaurer-Cartan equation (24) then shows that any path of the form s (0 , sf ) for a nonzeroleafwise constant function f ∈ C ∞ ( L ) consists of Lagrangian deformations of L that areno longer contained in the singular locus for s > . Alternatively, if γ = d F L g for some g ∈ C ∞ ( L ) , then for any nonzero leafwise constant function f on L , the path s (0 , sf e − g ) meets the criteria. (cid:3) Remark . For the local deformation problem we have γ = 0 , see Prop. 1.17. Hencelocally every half-dimensional Lagrangian submanifold contained in the singular locus canbe deformed smoothly to one outside of the singular locus, by Prop. 4.2.We will single out some Lagrangians whose deformations are constrained to the singularlocus. We restrict ourselves to Lagrangians L that are compact and connected. Recalling thedichotomy ( ⋆ ) from §3.3, these assumptions imply that either ( L, F L ) is the fiber foliationof a fiber bundle L → S or the leaves of F L are dense.4.1.1. The fibration case.
We need the following lemma about the map which, under theidentification (44), assigns to a closed foliated one-form its cohomology class.
Lemma 4.4.
Let ( L, F L ) be a compact manifold, where F L is the fiber foliation of a fiberbundle p : L → S . Then the following map is continuous for the C -topology: (cid:0) Ω cl ( F L ) , C (cid:1) → (cid:0) Γ( H ) , C (cid:1) : α σ α . (48) Proof.
Let F denote the typical fiber of p : L → S . Choose a basis { [ γ ] , . . . , [ γ n ] } ofthe first homology group H ( F ; Z ) , where the representatives γ i : [0 , → F are smooth -cycles. Via the de Rham isomorphism H dR ( F ) → Hom ( H ( F ; Z ) , R ) : [ ω ] n X i =1 c i [ γ i ] n X i =1 c i Z γ i ω ! , we can pick the dual basis { [ α ] , . . . , [ α n ] } of H dR ( F ) , satisfying Z γ i α j = δ ij . This provides an isomorphism H dR ( F ) ∼ = R n . Choose local trivialisations of p : L → S over open subsets U , . . . , U r covering S , and let V , . . . , V r be open subsets whose compactclosures satisfy V i ⊂ U i , such that V , . . . , V r still cover S . Then locally the map (48) reads Ω cl ( F ) (cid:12)(cid:12) p − ( U i ) → Γ (cid:0) U i × H dR ( F ) (cid:1) ∼ = C ∞ ( U i , R ) n : α θ (cid:18)Z γ α θ , . . . , Z γ n α θ (cid:19) . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 37
Therefore the C -norm of σ α is X ≤ i ≤ r X ≤ j ≤ n sup θ ∈ V i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z γ j α θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which can be made arbitrarily small by shrinking α in C . Since the map (48) is linear, thisproves the lemma. (cid:3) The following corollary states that, under hypotheses that are antipodal to those of Prop.4.2, small deformations of L stay inside the singular locus Z . Corollary 4.5.
Let L n be a compact connected Lagrangian submanifold contained in thesingular locus Z of an orientable log-symplectic manifold ( M n , Z, Π) . Assume that theinduced foliation F L on L is the fiber foliation of a fiber bundle p : L → S , and that thesection σ γ ∈ Γ (cid:0) H (cid:1) is nowhere zero. Then C -small deformations of L stay inside Z .Proof. Clearly, we have a continuous map (cid:0) Ω cl ( F L ) , C (cid:1) → (cid:0) Ω cl ( F L ) , C (cid:1) : α γ − £ X α, so composing with the map (48) gives a continuous map (cid:0) Ω cl ( F L ) , C (cid:1) → (cid:0) Γ( H ) , C (cid:1) : α σ γ − £ X α . Therefore, since σ γ ∈ Γ( H ) is nowhere zero, the same holds for σ γ − £ X α ∈ Γ( H ) providedthat α ∈ Ω cl ( F L ) is sufficiently C -small. By Proposition 3.15, this means that the cohomol-ogy group H γ − £ X α ( F L ) is zero for C -small α . Looking at the Maurer-Cartan equation (24),this implies that C -small deformations of L necessarily lie inside the singular locus. (cid:3) Remark . The assumption in Corollary 4.5 cannot be weakened. Clearly, if the interior of σ − γ (0) is nonempty, then by Prop. 3.15 i) there exists f ∈ H γ ( F L ) which is not identicallyzero, and s (0 , sf ) is a path of Lagrangian sections not inside the singular locus for s > .Even if we ask that the support of σ γ be all of S , then the conclusion of Corollary4.5 does not hold. Indeed, one can construct counterexamples where the vanishing set of σ γ − £ X α has nonempty interior, for arbitrarily C -small α ∈ Ω cl ( F L ) .The following is an example of a Lagrangian L for which small deformations stay insidethe singular locus. However there exist “long” paths of Lagrangian deformations that startat L and end at a Lagrangian that is no longer contained in the singular locus. Example 4.7.
Consider the manifold ( T × R , θ , θ , ξ , ξ ) with log-symplectic structure Π := ( ∂ θ − ∂ ξ ) ∧ ξ ∂ ξ + ∂ θ ∧ ∂ ξ and Lagrangian L := T × { (0 , } . Note that the leaves of F L are the fibers of the fibration ( T , θ , θ ) → ( S , θ ) . Considering ( ξ , ξ ) as the fiber coordinates on T ∗ L induced by theframe { dθ , dθ } , we have that T ∗ F L = ( T × R , θ , θ , ξ ) with canonical Poisson structure ∂ θ ∧ ∂ ξ . Therefore, using the notation established in the beginning of this section, we have ( X = ∂ θ γ = − dθ . So the section σ γ ∈ Γ( H ) is nowhere zero, and Corollary 4.5 shows that C -small deforma-tions of the Lagrangian L stay inside the singular locus.It is however possible to construct (large) deformations of L that don’t lie in the singularlocus T ∗ F L × { } ⊂ T ∗ F L × R , first deforming L inside the singular locus by large enough EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 38 α ∈ Ω cl ( F L ) such that H γ − £ X α ( F L ) is no longer zero. To do so explicitly, note that (cid:0) g ( θ , θ ) dθ , f ( θ , θ ) (cid:1) ∈ Ω ( F L ) × C ∞ ( L ) , gives rise to a Lagrangian section of T ∗ F L × R exactly when ∂f∂θ − f = f ∂g∂θ . (49)We construct a solution ( g, f ) to (49) with f not identically zero. For instance, let f ( θ ) be any bump function and let H ( θ ) be another bump function with H | supp ( f ) ≡ − and − ≤ H ( θ ) ≤ . Define C := R S H ( θ ) dθ , so C > − π . Let K := − C/ ( C + 2 π ) and put G ( θ ) := K (cid:0) H ( θ ) (cid:1) + H ( θ ) . Notice that G | supp ( f ) ≡ − , and since Z S G ( θ ) dθ = K (2 π + C ) + C = 0 , there exists a periodic primitive g ( θ ) with ∂g/∂θ = G . We check that ( g, f ) is a solutionto the Maurer-Cartan equation (49): for p / ∈ supp ( f ) both sides of (49) evaluate to zero,whereas for p ∈ supp ( f ) both sides of (49) are equal to − f ( p ) . Clearly, f .So first deforming L along α := g ( θ ) dθ and then moving outside of T ∗ F L × { } along f gives a Lagrangian deformation that is no longer contained in the singular locus. As a sanitycheck, looking at i ) of Proposition 3.15, we notice that H γ − £ X α ( F L ) is indeed nonzero andthat f ∈ H γ − £ X α ( F L ) , since the section σ γ − £ X α vanishes on the support of f .Moreover, the proof of Corollary 3.3 shows that this procedure can be done smoothly,in the sense that one can construct a smooth “long” path of Lagrangians that connects L with Lagrangians that are no longer contained in the singular locus. Concretely, let ( g, f ) be the solution to (49) just constructed, and let Ψ : R → R be any smooth functionsatisfying Ψ( s ) = 0 for s ≤ , < Ψ( s ) < for < s < and Ψ( s ) = 1 for s ≥ . Take Φ : R → R to be defined by Φ( s ) = Ψ( s − . Then the path s (Ψ( s ) gdθ , Φ( s ) f ) consistsof Lagrangian sections, it starts at L for s = 0 , passes through ( α, at time s = 1 , and itreaches Graph ( α, f ) at time s = 2 .The Lagrangian Graph ( α, f ) constructed above does not lie entirely outside of the singularlocus. Interestingly, it is not possible to find such deformations of L . For if we assume bycontradiction that ( g, f ) is a solution to (49) with f nowhere zero, then Z π f (cid:18) ∂f∂θ − f (cid:19) dθ = Z π ∂g∂θ dθ = 0 , so that Z π f ∂f∂θ dθ = 2 π. (50)But then we would get Z T d (cid:0) ln | f | dθ (cid:1) = Z T f ∂f∂θ dθ ∧ dθ = − Z π (cid:18)Z π f ∂f∂θ dθ (cid:19) dθ = − π , using Stokes’ theorem in the first and (50) in the last equality. This contradiction showsthat f must have a zero, i.e. the Lagrangian Graph ( α, f ) intersects the singular locus.Alternatively, if there would exist α = g ( θ , θ ) dθ and a function f ∈ H γ − £ X α ( F L ) thatis nowhere zero, then Proposition 3.15 implies that σ γ − £ X α ≡ . Therefore, γ − £ X α isfoliated exact, which implies that there exists a function k ∈ C ∞ ( T ) such that − − ∂g∂θ = ∂k∂θ . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 39
Integrating this equality against the standard volume form dθ ∧ dθ on the torus T givesa contradiction, since the left hand side integrates to − π and the right hand side to zero.4.1.2. The case with dense leaves.
Corollary 4.5 has no counterpart for Lagrangians whoseinduced foliation F L has dense leaves, at least not without additional assumptions. Indeed,looking at Proposition 3.15 and the Maurer-Cartan equation (24), we would need a positiveanswer to the following question: If γ ∈ Ω cl ( F L ) is not exact, then is γ − £ X α still not exact for small enough α ∈ Ω cl ( F L ) ? Drawing inspiration from [1, Section 4], we construct an explicit counterexample whichanswers this question in the negative. Let L = (cid:0) T , θ , θ (cid:1) be the torus with Kroneckerfoliation T F L = Ker ( dθ − λdθ ) , for λ ∈ R \ Q irrational. A global frame for T ∗ F L is givenby dθ , so that every foliated one-form looks like f ( θ , θ ) dθ , which is automatically closedby dimension reasons. It is exact when there exists g ( θ , θ ) ∈ C ∞ (cid:0) T (cid:1) such that f = λ ∂g∂θ + ∂g∂θ . (51)Expanding f and g in double Fourier series, f ( θ , θ ) = X m,n ∈ Z f m,n e πi ( nθ + mθ ) and g ( θ , θ ) = X m,n ∈ Z g m,n e πi ( nθ + mθ ) , the equality (51) is equivalent with f m,n = 2 πi ( m + λn ) g m,n ∀ m, n ∈ Z , (52)which implies in particular that f , = 0 . Note that the g m,n for ( m, n ) = (0 , are uniquelydetermined by the relation (52) since λ is irrational.Assume moreover that the slope λ ∈ R \ Q is a Liouville number (see Definition 5.1 inthe Appendix). In this case the foliated cohomology group H ( F L ) is infinite dimensional[18], [26, Chapter III], as one can construct smooth functions f ( θ , θ ) in such a way thatthe g m,n defined in (52) are not the Fourier coefficients of a smooth function. We give anexample of such a function f ( θ , θ ) in part i) of the proof below. Lemma 4.8.
Consider the torus L = (cid:0) T , θ , θ (cid:1) endowed with the Kronecker foliation T F L = Ker ( dθ − λdθ ) , for λ ∈ R \ Q a Liouville number. There exist a non-exact foliatedone-form γ ∈ Ω ( F L ) and X ∈ X ( L ) F L such that every C ∞ -open neighborhood of in Ω ( F L ) contains a one-form α for which γ − £ X α is foliated exact.Proof. The proof is divided into two steps. In the first step, we construct γ ∈ Ω ( F L ) . Inthe second step, we fix X ∈ X ( L ) F L and we construct a sequence of foliated one-forms α k such that γ − £ X α k is exact for each value of k , and α k → in the Fréchet C ∞ -topology.i) We first have to find a foliated one-form γ = f ( θ , θ ) dθ that is not exact. Moreover,since we want to approach γ by means of exact one-forms, we need that the coefficient f , = π R T f dθ ∧ dθ is zero. This can be done as follows. By Lemma 5.3 in theAppendix, for each integer p ≥ , there exists a pair of integers ( m p , n p ) such that | m p + λn p | ≤ | m p | + | n p | ) p . (53) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 40
We can moreover assume that ( m p , n p ) = ( m q , n q ) for p = q , and that n p ≥ p (seeRemark 5.4). We now define f ( θ , θ ) by means of its Fourier coefficients f m,n , setting f m,n = ( ( m p + λn p ) n p if ( m, n ) = ( m p , n p )0 else . To see that these coefficients define a smooth function, we estimate for k ∈ N : | f m p ,n p | · k ( m p , n p ) k k = | m p + λn p | · n p · k ( m p , n p ) k k ≤ | m p + λn p | · ( | m p | + | n p | ) · ( | m p | + | n p | ) k = | m p + λn p | · ( | m p | + | n p | ) k +1 ≤ | m p | + | n p | ) p · ( | m p | + | n p | ) k +1 ≤ (cid:18) p (cid:19) p − k − , where the last inequality holds for p ≥ k +1 . This shows that sup ( m,n ) ∈ Z | f m,n |k ( m, n ) k k is finite for each value of k ∈ N , and therefore f ( θ , θ ) is indeed smooth. To see that γ = f ( θ , θ ) dθ is not exact, note that the Fourier coefficients of a primitive g ( θ , θ ) are given by (52): g m,n = f m,n πi ( m + λn ) for ( m, n ) = (0 , . (54)Therefore | g m p ,n p | = 12 π n p ≥ π p, which does not tend to zero for p → ∞ . So the g m,n defined in (54) are not the Fouriercoefficients of a smooth function.ii) We let X := ∂ θ . Notice that X ∈ X ( L ) F L and that X is transverse to the leaves of F L .We now construct a sequence α k ∈ Ω cl ( F L ) such that γ − £ X α k is exact and α k → inthe C ∞ -topology. For each integer k ≥ we define α k = h k ( θ , θ ) dθ , where h k ( θ , θ ) is given by its Fourier coefficients h km,n = ((cid:16) m p + λn p πi (cid:17) · φ k (cid:16) p (cid:17) if ( m, n ) = ( m p , n p )0 else . Here φ k is a bump function on R that is identically equal to on the interval [0 , k ] . Asbefore, we see that h k is a smooth function by the estimate | h km p ,n p | · k ( m p , n p ) k l ≤ π | m p + λn p | · ( | m p | + | n p | ) l ≤ π (cid:18) p (cid:19) p − l , where the last inequality holds for p ≥ l . Note that γ − £ X α k = ( f − ∂ θ h k ) dθ isindeed exact because the Fourier coefficients of f − ∂ θ h k are given by f m,n − πi · n · h km,n = ( ( m p + λn p ) n p (cid:16) − φ k (cid:16) p (cid:17)(cid:17) if ( m, n ) = ( m p , n p )0 else , EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 41 only finitely many of which are nonzero. Finally, by letting k increase, we can make α k as C ∞ -small as desired. Indeed, for each integer l we have k h k k l ≤ X ≤ j ≤ l X p ≥ k | m p + λn p | · k ( m p , n p ) k j · (2 π ) j − ≤ X ≤ j ≤ l X p ≥ k (cid:18) p (cid:19) p − j · (2 π ) j − , (55)where the last inequality holds for k ≥ l . The expression (55) tends to zero for k → ∞ ,since the inner sum is the tail of a convergent series for each value of j ∈ { , . . . , l } . (cid:3) The above construction gives a concrete counterexample to the version of Corollary 4.5for Lagrangians ( L, F L ) with dense leaves. We only have to realize L = ( T , θ , θ ) with T F L = Ker ( dθ − λdθ ) as a Lagrangian submanifold contained in the singular locus of somelog-symplectic manifold. The normal form (22) tells us how to construct this log-symplecticmanifold. If ( ξ , ξ ) are the fiber coordinates on T ∗ L , and ξ is the fiber coordinate on T ∗ F L corresponding with the frame { dθ } , then the restriction map reads r : T ∗ L → T ∗ F L : ( θ , θ , ξ , ξ ) ( θ , θ , λξ + ξ ) , and therefore the canonical Poisson structure on T ∗ F L is Π can = r ∗ ( ∂ θ ∧ ∂ ξ + ∂ θ ∧ ∂ ξ ) = ( λ∂ θ + ∂ θ ) ∧ ∂ ξ . Let V denote the vertical Poisson vector field on T ∗ F L defined by the one-form γ ∈ Γ( T ∗ F L ) constructed in Lemma 4.8, and let X := ∂ θ . Then V + X is a Poisson vector field on ( T ∗ F L , Π can ) transverse to the symplectic leaves, so the following is a log-symplectic struc-ture: (cid:16) T ∗ F L × R , ( V + X ) ∧ t∂ t + ( λ∂ θ + ∂ θ ) ∧ ∂ ξ (cid:17) , and L is Lagrangian inside T ∗ F L with induced foliation F L . The above argument shows that,for each integer k ≥ , there exists arbitrarily C k -small α ∈ Ω cl ( F L ) for which γ − £ X α is exact. By Proposition 3.15, there exists f ∈ H γ − £ X α ( F L ) not identically zero, where f can be made arbitrarily C k -small by rescaling with a nonzero constant. The Maurer-Cartan equation (24) now implies that Graph ( α, f ) is an arbitrarily C k -small Lagrangiandeformation of L that is not completely contained in the singular locus T ∗ F L . Remark . In the above counterexample, it is crucial that the slope λ is a Liouvillenumber. If F L is the Kronecker foliation with generic (i.e. not Liouville) irrational slope λ ,then H ( F L ) = R [ dθ ] . In this case, exactness is detected by integration, for if we denote I : C ∞ ( T ) → R : h ( θ , θ ) Z T h ( θ , θ ) dθ ∧ dθ , then hdθ ∈ Ω ( F L ) being exact is equivalent with h ∈ I − (0) . Since integration is C -continuous, it follows that the space of exact one-forms Im ( d F L ) ⊂ (cid:0) Ω ( F L ) , C (cid:1) is closed.Therefore, if we take γ ∈ Ω ( F L ) not exact, so that H γ ( F L ) = 0 , then also γ − £ X α isnot exact for C -small α , so that still H γ − £ X α ( F L ) = 0 . This shows that, if in the abovecounterexample we take a generic slope λ ∈ R \ Q , then C -small deformations of L do stayinside the singular locus. EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 42
The problem in the above counterexample is that the space of exact one-forms in Ω cl ( F L ) is not closed with respect to the Fréchet C ∞ -topology generated by C k -norms {k · k k } k ≥ .Under the additional assumption that H ( F L ) is finite dimensional, this problem does notoccur, and we obtain the following analog to Corollary 4.5. Proposition 4.10.
Let L be a compact, connected Lagrangian whose induced foliation F L has dense leaves. Assume that H ( F L ) is finite dimensional and that γ ∈ Ω cl ( F L ) is notexact. Then there exists a neighborhood V of in (cid:0) Γ( T ∗ F L × R ) , C ∞ (cid:1) such that if Graph ( α, f ) is Lagrangian for ( α, f ) ∈ V , then f ≡ .Proof. Consider the Fréchet space (cid:0) Ω ( F L ) , C ∞ (cid:1) and notice that the space of closed foliatedone-forms Ω cl ( F L ) ⊂ (cid:0) Ω ( F L ) , C ∞ (cid:1) is closed. To see this, it is enough to note that d F L iscontinuous with respect to the C ∞ -topology and that { } ⊂ (cid:0) Ω ( F L ) , C ∞ (cid:1) is closed sinceFréchet spaces are Hausdorff. Consequently, (cid:0) Ω cl ( F L ) , C ∞ (cid:1) is itself a Fréchet space. More-over, the space of exact foliated one-forms Im ( d F L ) ⊂ (cid:0) Ω cl ( F L ) , C ∞ (cid:1) is a closed subspace.Indeed, by assumption, the range of d F L : (cid:0) C ∞ ( L ) , C ∞ (cid:1) → (cid:0) Ω cl ( F L ) , C ∞ (cid:1) has finite codi-mension, so it must be closed because of the open mapping theorem (see [1, Remark 3.2]).As γ is not foliated exact, there exists a C ∞ -open neighborhood of γ consisting of non-exact one-forms. By continuity of the map (cid:0) Ω cl ( F L ) , C ∞ (cid:1) → (cid:0) Ω cl ( F L ) , C ∞ (cid:1) : α γ − £ X α ,we find a C ∞ -open neighborhood U of in Ω cl ( F L ) such that γ − £ X α is not exact for all α ∈ U . There exists a C ∞ -open subset U ′ ⊂ Ω ( F L ) such that U = U ′ ∩ Ω cl ( F L ) . We nowdefine the C ∞ -neighborhood V of in (Γ( T ∗ F L × R ) , C ∞ ) by V := { ( α, f ) ∈ Γ( T ∗ F L × R ) : α ∈ U ′ } . To see that V satisfies the criteria, let ( α, f ) ∈ V be such that Graph ( α, f ) is Lagrangian in ( T ∗ F L × R , e Π) . Then α ∈ U and f ∈ H γ − £ X α ( F L ) = { } , by ii ) of Proposition 3.15. (cid:3) Since the C ∞ -topology is generated by the increasing family of C k -norms, every C ∞ -openneighborhood contains a C k -open neighborhood for some k ∈ N . So shrinking the neighbor-hood V obtained in the above proposition, one can assume that it is a C k -neighborhood ofthe zero section, for some (unspecified) k ∈ N .4.2. Obstructedness of deformations.
Recall that a deformation problem governed by a DGLA ( W, d, [[ · , · ]]) is formally/smoothlyunobstructed if every closed element α ∈ W – i.e. every first order deformation – can beextended to a formal/smooth curve of Maurer-Cartan elements. A way to detect obstruct-edness is by means of the Kuranishi map Kr : H ( W ) → H ( W ) : [ α ] (cid:2) [[ α, α ]] (cid:3) , for if Kr ([ α ]) is nonzero, then α is formally (hence smoothly) obstructed [30, Theorem 11.4].For the deformation problem of a Lagrangian L n contained in the singular locus of a log-symplectic manifold ( M n , Z, Π) , a first order deformation is a pair ( α , f ) ∈ Γ ( T ∗ F L × R ) such that ( d F L α = 0 d F L f + f γ = 0 . (56)Clearly, first order deformations of the specific form ( α , or (0 , f ) are smoothly unob-structed, since s ( α , and s (0 , f ) satisfy the Maurer-Cartan equation (24) for all s ∈ R . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 43
Obstructedness.
We show that the above deformation problem is formally obstructedin general. The Kuranishi map of the DGLA (cid:0)
Γ ( ∧ • ( T ∗ F L × R )) , d, [[ · , · ]] (cid:1) described inCorollary 3.9 reads Kr : H (Γ ( ∧ • ( T ∗ F L × R ))) → H (Γ ( ∧ • ( T ∗ F L × R ))) : [( α, f )] [(0 , f £ X α )] , (57)and the following example shows that this map need not be identically zero. Example 4.11 (An obstructed example) . Consider the manifold T × R , regarded as atrivial vector bundle over T . Denote its coordinates by ( θ , θ , ξ , ξ ) and endow it with alog-symplectic structure Π given by Π = ∂ θ ∧ ξ ∂ ξ + ∂ θ ∧ ∂ ξ . Note that L := T × { (0 , } is a Lagrangian submanifold contained in the singular locus T × R = { ξ = 0 } . It inherits a codimension-one foliation F L with tangent distribution T F L = Ker ( dθ ) , so the cotangent bundle T ∗ F L has a global frame given by dθ . In thenotation established earlier, we now have ( γ = 0 X = ∂ θ , and the differential d of the DGLA acts as d : Γ ( T ∗ F L × R ) → Γ (cid:0) ∧ ( T ∗ F L × R ) (cid:1) : ( gdθ , k ) (cid:18) , − ∂k∂θ dθ (cid:19) . (58)Since the Kuranishi map (57) is given by Kr (cid:16) (cid:2)(cid:0) gdθ , f (cid:1)(cid:3) (cid:17) = (cid:20)(cid:18) , f ∂g∂θ dθ (cid:19)(cid:21) , it is clear that Kr (cid:16) (cid:2)(cid:0) gdθ , f (cid:1)(cid:3) (cid:17) = 0 ⇔ f ∂g∂θ = ∂k∂θ for some k ∈ C ∞ ( T ) ⇔ Z S f ∂g∂θ dθ = 0 . (59)The equation (59) is a non-trivial obstruction to the prolongation of infinitesimal defor-mations. For instance, the section (cid:0) sin( θ ) dθ , cos( θ ) (cid:1) ∈ Γ ( T ∗ F L × R ) is an infinitesimaldeformation of L since it is closed with respect to the differential (58). But it cannot beprolonged to a path of deformations, since the integral (59) is nonzero.4.2.2. Formally unobstructed deformations.
It is well-known that a deformation problemis formally unobstructed whenever the second cohomology group of the DGLA governingit vanishes [30, Theorem 11.2]. Specializing to our situation, say we have a first orderdeformation ( α , f ) as in eq. (56) and we wish to prolong it to a formal power seriessolution P k ≥ ( α k , f k ) ǫ k of the Maurer-Cartan equation. So we require that d X k ≥ ( α k , f k ) ǫ k + 12 X k ≥ ( α k , f k ) ǫ k , X k ≥ ( α k , f k ) ǫ k = 0 . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 44
Collecting all terms in ǫ n gives d ( α n , f n ) + 12 X k,l ≥ k + l = n [[( α k , f k ) , ( α l , f l )]] = 0 ⇔ ( − d F L α n , − d F L f n − f n γ ) + 12 X k,l ≥ k + l = n (0 , f l £ X α k + f k £ X α l ) = 0 ⇔ d F L α n = 0 d F L f n + f n γ − P k,l ≥ k + l = n ( f l £ X α k + f k £ X α l ) = 0 . We can always construct a formal power series solution if H γ ( F L ) = 0 . Concretely, con-structing ( α k , f k ) inductively, we can set α k = 0 for k ≥ and choose f k such that d F L f k + f k γ = f k − £ X α . (60)A quick proof by induction indeed shows that the right hand side of (60) is closed withrespect to the differential d γ F L for each k ≥ . In conclusion, we have proved the following: Corollary 4.12. If H γ ( F L ) = 0 , then the deformation problem is formally unobstructed. Note that this assumption is weaker than requiring that the second cohomology group ofthe DGLA is zero, since the latter is given by H ( F L ) ⊕ H γ ( F L ) .We will see that the vanishing of H γ ( F L ) in fact ensures that the deformation problemis smoothly unobstructed, at least for Lagrangians that are compact and connected.4.2.3. Smoothly unobstructed deformations: general results.
We give a sufficient conditionfor smooth unobstructedness. When £ X α is foliated exact, we have H γ ( F L ) ∼ = H γ − £ X α ( F L ) .Using this isomorphism, from a solution of the linearized Maurer-Cartan equation (56) wecan construct a solution of the Maurer-Cartan equation (24). This leads to the followingresult, which we prove with a short direct computation. Proposition 4.13. If ( α, f ) ∈ Γ ( T ∗ F L × R ) is a first order deformation such that £ X α isfoliated exact, then ( α, f ) is smoothly unobstructed.Proof. Let £ X α = d F L h for h ∈ C ∞ ( L ) . We claim that the path s ( sα, sf e sh ) (61)is a prolongation of ( α, f ) consisting of Lagrangian sections for all times s . Indeed, we havethe following equivalences: d F L ( sf e sh ) + sf e sh ( γ − £ X sα ) = 0 ⇔ d F L ( sf e sh ) + sf e sh ( γ − d F L sh ) = 0 ⇔ d F L sf + sf γ = 0 . (62)In the last equivalence we use i ) of Lemma 3.12, which says that the following map is anisomorphism of cochain complexes: (cid:16) Ω • ( F L ) , d γ − d F L sh F L (cid:17) → (cid:16) Ω • ( F L ) , d γ F L (cid:17) : β e − sh β. The equality (62) is satisfied, since ( α, f ) is a first order deformation. So, by Theorem 3.2, ( sα, sf e sh ) is indeed a Lagrangian section for each time s . Clearly, the path passes throughthe zero section at s = 0 with velocity ( α, f ) . This proves the claim. (cid:3) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 45
As a consistency check, we note that a first order deformation ( α, f ) as in Prop. 4.13maps to zero under the Kuranishi map: we have Kr ([( α, f )]) = [(0 , f £ X α )] by eq. (57).If £ X α = d F L h for some h ∈ C ∞ ( L ) , then d γ F L ( f · h ) = d γ F L f · h + f · d F L h = f £ X α . Remark . We give a geometric interpretation of Proposition 4.13.i) For a closed foliated one-form α ∈ Ω ( F L ) , exactness of £ X α is equivalent with theexistence of a closed one-form e α ∈ Ω ( L ) that extends α . Indeed, if e α ∈ Ω ( L ) is aclosed extension of α and r : Ω ( L ) → Ω ( F L ) is the restriction map, then r ( ι X d e α ) = r ( £ X e α − dι X e α ) = £ X α − d F L ( ι X e α ) , which shows that £ X α = d F L ( ι X e α ) is exact. Conversely, assume that £ X α = d F L h for some h ∈ C ∞ ( L ) . Let e α ∈ Ω ( L ) be the unique extension of α satisfying e α ( X ) = h .Then e α is closed, since r ( ι X d e α ) = r ( £ X e α − dι X e α ) = £ X α − d F L h = 0 . ii) The smooth path of Lagrangian deformations given by (61) is obtained by applyingcertain Poisson diffeomorphisms of T ∗ F L × R to the smooth path of Lagrangian sections s (0 , sf ) . More precisely, as in item i), assume that £ X α = d F L h , and let e α ∈ Ω ( L ) be the closed one-form extending α determined by e α ( X ) = h . As before, denote by pr : T ∗ F L × R → L and p : T ∗ F L → R the vector bundle projections. Since e α is aclosed one-form, it gives rise to a Poisson vector field on T ∗ F L × R , namely e Π ♯ ( pr ∗ e α ) = ( pr ∗ h ) t∂ t + Π ♯can ( p ∗ e α ) . Notice that this vector field is tangent to the fibers of pr , and that the second summandis the constant vector field on the fibers of T ∗ F L with value α . The flow at time s of e Π ♯ ( pr ∗ e α ) maps graph (0 , sf ) to graph ( sα, sf e sh ) .In case α = d F L g is exact, then we can interpret this construction in terms of theDGLA governing the deformation problem. Indeed, Remark 4.23 shows that the gaugeaction by the degree zero element ( g, takes the Maurer-Cartan element (0 , sf ) to (cid:0) sd F L g, sf e sX ( g ) (cid:1) . This is consistent with the above, since X ( g ) is a primitive of £ X α . Corollary 4.15. If H ( F L ) = 0 , then all first order deformations ( α, f ) ∈ Γ ( T ∗ F L × R ) are smoothly unobstructed.Proof. If ( α, f ) is a first order deformation, then α is closed. Since H ( F L ) = 0 , it is exact.The same then holds for £ X α . So the result follows from Proposition 4.13. (cid:3) Corollary 4.15 shows in particular that obstructedness is a global issue, since the coho-mology group H ( F L ) always vanishes locally.One may wonder if all first order deformations ( α, f ) that are smoothly unobstructedarise as in Prop. 4.13. The answer is negative, but it becomes positive if we restrict to firstorder deformations for which f ∈ C ∞ ( L ) is nowhere vanishing. We spell this out in thefollowing remark and lemma. Remark . First order deformations of the form ( α, , hence d F L α = 0 , are smoothlyunobstructed, but in general £ X α is not foliated exact. For instance, consider the log-symplectic manifold ( T × R , Π) and Lagrangian submanifold L := T × { (0 , } as inExample 4.11, for which the foliation F L is one-dimensional. Any α = g ( θ , θ ) dθ ∈ Ω ( F L ) This path can certainly not be obtained by applying Poisson diffeomorphisms to L itself, since the latterpreserve the Poisson submanifold T ∗ F L . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 46 is foliated closed, but in general the integral of α along the fibers of L → S : ( θ , θ ) → θ is not independent of θ , implying that £ X α is not foliated exact. Lemma 4.17.
Let ( α, f ) be a first order deformation such that Kr (cid:0) [( α, f )] (cid:1) = 0 . Assumethat f ∈ C ∞ ( L ) is nowhere vanishing. Then £ X α is foliated exact.Proof. The assumption Kr (cid:0) [( α, f )] (cid:1) = 0 is equivalent to [ f £ X α ] = 0 in H γ ( F L ) by (57),so it implies that there exists g ∈ C ∞ ( L ) such that f £ X α = d F L g + gγ. Since f is nowhere zero, we can divide by f and we obtain £ X α = 1 f d F L g + gf γ = 1 f d F L g − gf d F L f = 1 f d F L g + gd F L (cid:18) f (cid:19) = d F L (cid:18) gf (cid:19) , (63)using in the second equality that d γ F L f = 0 . This shows that £ X α is foliated exact. (cid:3) Smoothly unobstructed deformations: the compact case.
We now show that for com-pact connected Lagrangians ( L, F L ) , the condition H γ ( F L ) = 0 from Corollary 4.12 in factimplies that the deformation problem is smoothly unobstructed. We actually prove more:one only needs that the Kuranishi map (57) is trivial. Proposition 4.18.
Let ( L n , F L ) be a compact connected Lagrangian submanifold that iscontained in the singular locus of a log-symplectic manifold ( M n , Z, Π) . A first order defor-mation ( α, f ) ∈ Γ ( T ∗ F L × R ) of L is smoothly unobstructed if and only if Kr (cid:0) [( α, f )] (cid:1) = 0 .Proof. We only have to prove the backward implication. Let ( α, f ) be a first order defor-mation of L with Kr (cid:0) [( α, f )] (cid:1) = 0 . We know that either the leaves of F L are dense, or ( L, F L ) is the foliation by fibers of a fiber bundle over S .i) First assume that the leaves of F L are dense. • If γ is not exact, then H γ ( F L ) = { } by Proposition 3.15. Since ( α, f ) is a firstorder deformation, we have that f ∈ H γ ( F L ) = { } . Therefore ( α, f ) = ( α, anda path of Lagrangian sections that prolongs ( α, is simply given by s ( sα, . • Now assume that γ = d F L k is exact. Thanks to (the proof of) Lemma 3.12 i ) , weknow that e k f is constant on L . So either f ≡ , in which case we conclude that ( α, f ) is smoothly unobstructed as in the previous bullet point. Or f is nowherezero, in which case we can use Lemma 4.17. There we showed that £ X α is foliatedexact, and Proposition 4.13 then implies that the first order deformation ( α, f ) issmoothly unobstructed.ii) Now assume that F L is the fiber foliation of a fiber bundle p : L → S . The closedfoliated one-form £ X α defines a section σ £ X α of the vector bundle H → S viathe correspondence (44). By Lemma 3.14, we can fix a smooth function h ∈ C ∞ ( L ) satisfying ( £ X α ) | p − ( q ) = d (cid:16) h | p − ( q ) (cid:17) ∀ q ∈ Z £ X α , EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 47 where we denote Z £ X α := σ − £ X α (0) . Mimicking the proof of Proposition 4.13, we claimthat the path s ( sα, sf e sh ) is a prolongation of ( α, f ) by Lagrangian sections. Sowe have to show that d F L ( sf e sh ) + sf e sh ( γ − £ X sα ) = 0 . (64)To do so, we denote Z f := f − (0) ⊂ L . Recall here that f ∈ H γ ( F L ) , so that Z f isa union of fibers of p : L → S (cf. the proof of Proposition 3.15). Clearly, the equality(64) holds on Z f . On the other hand, Lemma 4.17 implies that £ X α is exact on L \ Z f .Therefore, £ X α = d F L h on L \ Z f , and the computation (62) in the proof of Prop. 4.13shows that (64) holds on L \ Z f . (cid:3) Remark . A crucial point in the proof of Prop. 4.18 is that h is a smooth functiondefined on the whole of L . Its existence is guaranteed by Lemma 3.14, a statement aboutfiber bundles over S . Due to this, we do not expect the statement of Prop. 4.18 to hold ifone removes the compactness assumption on L .We give an algorithmic overview of first order deformations and their obstructedness, forLagrangians that are compact and connected.i) Assume ( L, F L ) is the foliation by fibers of a fiber bundle p : L → S . Fix a smoothfunction g ∈ C ∞ ( L ) that is a primitive of γ on Z γ := σ − γ (0) , as constructed in Lemma3.14. Thanks to Prop. 3.15 i) and its proof, we can characterize first order deformations ( α, f ) of L by the requirements ( d F L α = 0 e g f is constant on each p -fiber and vanishes on S \ Z γ . By Prop. 4.18, a first order deformation ( α, f ) of L is smoothly unobstructed exactlywhen Kr (cid:0) [( α, f )] (cid:1) = 0 . We claim that the latter condition is equivalent to the following: [ £ X α ] = 0 ∈ H ( F L ) on L \ Z f . (65)Here Z f denotes the zero locus of f , as in the proof of Prop. 4.18.To see that the two conditions are equivalent, recall that Kr (cid:0) [( α, f )] (cid:1) = 0 implies thecondition (65), by Lemma 4.17. Conversely, assume that the condition (65) holds. Asin the proof of Prop. 4.18, choose a smooth function h ∈ C ∞ ( L ) such that £ X α = d F L h on p − ( Z £ X α ) . In particular, this equality holds on L \ Z f . From this, we concludethat f £ X α = d γ F L ( f h ) . Indeed, on Z f this equation holds because both sides are zero; on the complement L \ Z f it also holds because d γ F L ( f h ) = d γ F L f · h + f · d F L h = f £ X α . This shows that [ f £ X α ] = 0 in H γ ( F L ) , which by (57) is equivalent to Kr (cid:0) [( α, f )] (cid:1) = 0 .ii) In case F L has dense leaves, then we distinguish between two types of first order defor-mations. The first type are the ones of the form ( α, for closed α ∈ Ω ( F L ) . Clearly,these are smoothly unobstructed.First order deformations of the second type, those with nonzero second component,can only occur if γ is foliated exact, by Prop. 3.15. They are characterized as the ( α, f ) for which ( d F L α = 0 e g f is a nonzero constant , EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 48 where g ∈ C ∞ ( L ) is a primitive of γ . Such a first order deformation ( α, f ) is smoothlyunobstructed exactly when [ £ X α ] = 0 in H ( F L ) : the forward implication follows fromLemma 4.17, and the backward implication from Prop. 4.13.Notice that we now showed that the criterion (65) for unobstructedness in the fibra-tion case also holds if F L has dense leaves: the two types of infinitesimal deformationsjust described correspond with the extreme cases L \ Z f = ∅ and L \ Z f = L .In conclusion, we have proved the following. Corollary 4.20.
A first order deformation ( α, f ) ∈ Γ( T ∗ F L × R ) of a compact, connectedLagrangian L is smoothly unobstructed exactly when [ £ X α ] = 0 ∈ H ( F L ) on L \ Z f . (66) Here Z f denotes the zero locus of f . Given a first order deformation ( α, f ) , the condition (66) is equivalent with α extendingto a closed one-form on L \ Z f , by the argument of Remark 4.14 i). Therefore the condition(66) is independent of the data ( X, γ ) coming from the modular vector field. Example 4.21.
Consider the manifold T × R , regarded as a trivial vector bundle over T .Denote its coordinates by ( θ , θ , ξ , ξ ) . Let Z := T × R = { ξ = 0 } and L := T × { (0 , } .i) Any orientable log-symplectic structure with singular locus Z so that L is Lagrangianwith induced foliation T F L = Ker ( dθ ) , up to Poisson diffeomorphism, looks as followsnearby L : Π = V ∧ ξ ∂ ξ + ∂ θ ∧ ∂ ξ , where V = g X ( θ ) ∂ θ + g γ ( θ ) ∂ ξ for some function g γ ∈ C ∞ ( S ) and some nowhere vanishing function g X ∈ C ∞ ( S ) .Here we use Corollary 1.18, Remark 1.19 and Corollary 2.5 along with Remark 2.7 ii).We have γ = g γ ( θ ) dθ , and a function on L satisfying the properties of Lemma 3.14is the constant function zero. Hence first order deformations are given by pairs ( α, f ) ,subject to the condition that f = f ( θ ) and f · g γ = 0 .To see when such a first order deformation is unobstructed, we apply Corollary 4.20.In the case at hand, since the fibers of p : L → S are circles and thanks to Stokes’theorem, the condition (66) can be rephrased as:the function q Z p − ( q ) α is locally constant on p ( L \ Z f ) ⊂ S .For instance, in case g γ = 0 (as in Ex. 4.11), any pair ( α, f ) with f = f ( θ ) is a firstother deformation. Such a pair is unobstructed exactly when, writing α = a ( θ , θ ) dθ ,the expression Z { θ }× S a ( θ , θ ) dθ is constant on connected components of p ( L \ Z f ) .ii) Now let λ ∈ R \ Q be a generic (i.e. not Liouville) irrational number. Any orientablelog-symplectic structure with singular locus Z so that L is Lagrangian with inducedfoliation T F L = Ker ( dθ − λdθ ) is Poisson diffeomorphic around L with Π = ( C∂ θ + K∂ ξ ) ∧ ξ ∂ ξ + ( λ∂ θ + ∂ θ ) ∧ ∂ ξ , (67) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 49 for some
C, K ∈ R with C nonzero. This follows from a similar reasoning as above,now using that X ( L ) F L / Γ( T F L ) ∼ = H ( F L ) = R and H ( F L ) = R [ dθ ] . Note that γ = Kdθ is exact if and only if K = 0 . Therefore, first order deformationsare given by ( α, if K = 0 and ( α, c ) if K = 0 , with c ∈ R . Clearly, the Lie derivativealong X = C∂ θ acts trivially in cohomology, since H ( F L ) = R [ dθ ] . Therefore, allfirst order deformations of L are smoothly unobstructed, by Corollary 4.20.The situation is different when λ ∈ R \ Q is a Liouville number. Disregarding triviallyunobstructed first order deformations of the form ( α, , Prop. 3.15 ii) implies that theones with nonzero second component can only occur for log-symplectic structures thatare isomorphic around L to the following model: Π = C∂ θ ∧ ξ ∂ ξ + ( λ∂ θ + ∂ θ ) ∧ ∂ ξ , where C ∈ R . Notice that H ( F L ) is now infinite dimensional, and that the Liederivative along X = C∂ θ no longer acts trivially in cohomology, which is a directconsequence of (the proof of) Lemma 4.8. This shows that there exist obstructed firstorder deformations.4.3. Equivalences and rigidity of deformations.
We now consider two natural equivalence relations on the space of Lagrangian deforma-tions: equivalence by Hamiltonian diffeomorphisms and equivalence by Poisson isotopies.We show that the action by Hamiltonian diffeomorphisms agrees with the gauge action ofthe DGLA that governs the deformation problem. We also discuss rigidity of Lagrangians,both for Hamiltonian and Poisson equivalence.4.3.1.
Hamiltonian isotopies.
We showed in §3.2 that the graph of ( α, f ) ∈ Γ( T ∗ F L × R ) defines a Lagrangian submanifold of ( U, e Π) exactly when ( α, f ) is a Maurer-Cartan elementof the DGLA (cid:0) Γ( ∧ • ( T ∗ F L × R )) , d, [[ · , · ]] (cid:1) . So if we write for shortDef U ( L ) := (cid:8) ( α, f ) ∈ Γ( U ) : graph ( α, f ) is Lagrangian inside (cid:0) U, e Π (cid:1)(cid:9) andMC U (cid:0) Γ( ∧ • ( T ∗ F L × R )) (cid:1) := (cid:8) ( α, f ) ∈ M C (cid:0) Γ( ∧ • ( T ∗ F L × R )) (cid:1) : graph ( α, f ) ⊂ U (cid:9) , then we have a correspondenceDef U ( L ) ←→ MC U (cid:0) Γ( ∧ • ( T ∗ F L × R )) (cid:1) . (68)We now define equivalence relations on both sides of (68) and we show that they agreeunder this correspondence. We closely follow the exposition in [35]. There one considersequivalences of coisotropic submanifolds in symplectic geometry, but most of their resultsremain valid in the more general setting of fiberwise entire Poisson structures. Definition 4.22. i) Two Lagrangian sections ( α , f ) and ( α , f ) in Def U ( L ) are Hamil-tonian equivalent if they are interpolated by a smooth family ( α s , f s ) of Lagrangiansections in Def U ( L ) that is generated by a (locally defined) Hamiltonian isotopy. Inother words, there exists a time-dependent Hamiltonian vector field X H s on U suchthat the associated isotopy φ s maps graph ( α , f ) to graph ( α s , f s ) , for all s ∈ [0 , . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 50 ii) Two Maurer-Cartan elements ( α , f ) , ( α , f ) ∈ MC U (cid:0) Γ( ∧ • ( T ∗ F L × R )) (cid:1) are gaugeequivalent if they are interpolated by a smooth family { ( α s , f s ) } s ∈ [0 , of sections whosegraph lies inside U , and there exists a smooth family { g s } s ∈ [0 , of functions on L suchthat dds ( α s , f s ) = [[( g s , , ( α s , f s )]] − d ( g s , (cid:0) d F L g s , f s £ X g s (cid:1) . (69) Remark . By solving the flow equation (69), we obtain an explicit description for thegauge action of the DGLA. Namely, a path of degree zero elements ( g s , acts on a Maurer-Cartan element ( α , f ) , which yields a path of Maurer-Cartan elements ( α s , f s ) given by ( α s , f s ) = (cid:18) α + d F L (cid:18)Z s g u du (cid:19) , f exp (cid:18) £ X Z s g u du (cid:19)(cid:19) . (70)We rewrite the gauge equivalence relation in more geometric terms. Lemma 4.24.
Two Maurer-Cartan elements ( α , f ) , ( α , f ) ∈ MC U (cid:0) Γ( ∧ • ( T ∗ F L × R )) (cid:1) are gauge equivalent if and only if they are interpolated by a smooth family { ( α s , f s ) } s ∈ [0 , ofsections whose graph lies inside U , and there exists a smooth family { g s } s ∈ [0 , of functionson L such that dds ( α s , f s ) = X pr ∗ g s | graph ( α s ,f s ) . (71) Here pr : U ⊂ T ∗ F L × R → L denotes the bundle projection, and we see (71) as an equalityof sections of the vertical bundle restricted to graph ( α s , f s ) .Proof. We compute the Hamiltonian vector field X pr ∗ g s . As before, let p : T ∗ F L → L denotethe bundle projection. We obtain X pr ∗ g s = (( V vert + V lift ) ∧ t∂ t + Π can ) ♯ ( dpr ∗ g s )= pr ∗ ( £ X g s ) t∂ t + Π ♯can ( p ∗ dg s ) , (72)and therefore X pr ∗ g s | graph ( α s ,f s ) = pr ∗ ( f s £ X g s ) ∂ t + Π ♯can ( p ∗ dg s ) . The section of T ∗ F L × R corresponding with this vertical fiberwise constant vector field is (cid:0) d F L g s , f s £ X g s (cid:1) ∈ Γ( T ∗ F L × R ) , in agreement with (69). This proves the lemma. (cid:3) We need some technical results that appeared in [35]. We state them here for convenience.
Lemma 4.25.
Let A → M be a vector bundle with vertical bundle V . Let X s be one-parameter family of vector fields on A with flow φ s , and let τ be a section of A .i) If τ s is a one-parameter family of sections of A such that graph ( τ s ) = φ s ( graph ( τ )) holds for all s ∈ [0 , , then τ s satisfies the equation dds τ s = P τ s X s ∀ s ∈ [0 , . Here P τ s means vertical projection with respect to T A | graph ( τ s ) = T graph ( τ s ) ⊕ V | graph ( τ s ) . ii) Conversely, assume that the integral curves of X s starting at points of graph ( τ ) existfor all times s ∈ [0 , , and suppose that τ s is a one-parameter family of sections of A satisfying dds τ s = P τ s X s ∀ s ∈ [0 , . Then the family of submanifolds graph ( τ s ) coincides with φ s ( graph ( τ )) for all s ∈ [0 , . EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 51
Making some minor modifications to the proofs of [35, Proposition 3.18] and [35, Propo-sition 3.19], we can show that Hamiltonian equivalence coincides with gauge equivalence.
Proposition 4.26.
The bijection between Lagrangian sections and Maurer-Cartan elementsDef U ( L ) → MC U (cid:0) Γ( ∧ • ( T ∗ F L × R )) (cid:1) : ( α, f ) ( α, f ) descends to a bijection between Def U ( L ) / ∼ Ham and MC U (cid:0) Γ( ∧ • ( T ∗ F L × R )) (cid:1) / ∼ gauge .Proof. First assume that ( α , f ) , ( α , f ) ∈ Def U ( L ) are Hamiltonian equivalent. Thenthey are interpolated by a smooth family of sections ( α s , f s ) ∈ Def U ( L ) generated by theflow φ s of a time-dependent Hamiltonian vector field X H s ∈ X ( U ) . Part i ) of Lemma 4.25then implies that dds ( α s , f s ) = P ( α s ,f s ) X H s (73)for all s ∈ [0 , . Define g s := H s ◦ ( α s , f s ) ∈ C ∞ ( L ) and observe that H s − pr ∗ g s vanishesalong graph ( α s , f s ) . Because graph ( α s , f s ) is coisotropic, this implies that the Hamiltonianvector field X H s − pr ∗ g s = X H s − X pr ∗ g s is tangent to graph ( α s , f s ) . Consequently, the equality(73) becomes dds ( α s , f s ) = P ( α s ,f s ) X pr ∗ g s = X pr ∗ g s | graph ( α s ,f s ) , where we also used that X pr ∗ g s is vertical (which is clear from the expression (72)). ByLemma 4.24, we conclude that ( α , f ) and ( α , f ) are gauge equivalent.Conversely, assume that ( α , f ) , ( α , f ) ∈ MC U (cid:0) Γ( ∧ • ( T ∗ F L × R )) (cid:1) are gauge equivalent.By Lemma 4.24, this means that they are interpolated by a smooth family of sections ( α s , f s ) inside U , such that dds ( α s , f s ) = X pr ∗ g s | graph ( α s ,f s ) = P ( α s ,f s ) X pr ∗ g s ∀ s ∈ [0 , , for a smooth family of functions g s ∈ C ∞ ( L ) . In particular, the integral curve of X pr ∗ g s starting at a point ( α , f )( p ) ∈ graph ( α , f ) is defined up to time , and is given by ( α s , f s )( p ) for s ∈ [0 , . Part ii ) of Lemma 4.25 gives φ s ( graph ( α , f )) = graph ( α s , f s ) for all s ∈ [0 , , where φ s is the flow of X pr ∗ g s . This shows that ( α , f ) and ( α , f ) areHamiltonian equivalent. (cid:3) Remark . The above proof is almost identical to the one presented in [35]. The maindifference is that in [35], one needs to impose compactness on the coisotropic submanifold toobtain the implication “gauge equivalence ⇒ Hamiltonian equivalence”, as otherwise the flowlines of X pr ∗ g s need not be defined for long enough time. Since in our setting Hamiltonianvector fields of basic functions are vertical, we don’t need this additional assumption.As a consequence, we obtain that the formal tangent space at zero to the moduli space M HamU ( L ) := Def U ( L ) / ∼ Ham can be identified with the first cohomology group of thedifferential graded Lie algebra (cid:0)
Γ ( ∧ • ( T ∗ F L × R )) , d, [[ · , · ]] (cid:1) : T [0] M HamU ( L ) = H ( F L ) ⊕ H γ ( F L ) . (74)Indeed, if ( α s , f s ) is a path of Lagrangian deformations of L , then dds | s =0 ( α s , f s ) is closedwith respect to the differential d of the DGLA. Moreover, if the path ( α s , f s ) is generatedby the flow of a time-dependent Hamiltonian vector field, then ( α s , f s ) is obtained by gaugetransforming the zero section, as we just proved. The expression (70) then shows that dds | s =0 ( α s , f s ) is of the form ( d F L g, for g ∈ C ∞ ( L ) . This proves the assertion (74). EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 52
Smoothness of the moduli space by Hamiltonian isotopies.
In general, the moduli space M HamU ( L ) is by no means smooth, since the formal tangent spaces at different points can bedrastically different. For instance, let us look again at Example 4.11, where we considered ( T × R , θ , θ , ξ , ξ ) with log-symplectic structure Π = ∂ θ ∧ ξ ∂ ξ + ∂ θ ∧ ξ and Lagrangian L = T × { (0 , } . The induced foliation on L is the fiber foliation of ( L, θ , θ ) → ( S , θ ) . Since γ = 0 , we get for any nonzero constant c ∈ R a Lagrangian sec-tion (0 , c ) ∈ Γ( T ∗ F L × R ) whose graph lies outside the singular locus. Hence, by symplecticgeometry, we have T [(0 ,c )] M HamU ( L ) ∼ = H ( graph (0 , c )) ∼ = H ( L ) = R , which is finite dimensional. On the other hand, we have T [0] M HamU ( L ) = H ( F L ) ⊕ H γ ( F L ) ∼ = H ( F L ) ⊕ H ( F L ) ∼ = C ∞ ( S ) ⊕ C ∞ ( S ) , which is infinite dimensional.On the other hand, there are instances in which the moduli space is locally smooth.Suppose a Lagrangian submanifold L n contained in the singular locus Z has the propertythat C -small Lagrangian deformations of L stay inside Z . This means that the C -smalldeformations are precisely the graphs of C -small elements of Ω cl ( F L ) . Then M HamU ( L ) isnaturally isomorphic to an open neighborhood of the origin in H ( F L ) , by Corollary 1.11.In particular, M HamU ( L ) is smooth. We present two classes of examples.i) A class of Lagrangians L as above are those satisfying the assumptions of Corollary4.5. In that case M HamU ( L ) is infinite-dimensional. Indeed, recall that H ( F L ) ∼ =Γ( H ) ; if this was finite-dimensional, then H would be of rank zero, which impliesthat H ( F L ) = 0 . Then γ would be exact, which is impossible under the assumptionsof Corollary 4.5.ii) Another class of Lagrangians L as above are those that are C -rigid under Poissonequivalences (see §4.3.6 later on), since Poisson diffeomorphisms of the ambientlog-symplectic manifold necessarily preserve Z . In that case M HamU ( L ) is finite-dimensional by Lemma 4.31, assuming L is compact and connected. We exhibitconcrete examples of such L in Example 4.36. Notice that Proposition 4.34 as stateddoes not quite provide examples, since it makes a statement only about C ∞ -smalldeformations.4.3.3. Rigidity and Hamiltonian isotopies.
At this point, we would like to address somerigidity phenomena. A Lagrangian L is called infinitesimally rigid under Hamiltonian equiv-alence if the formal tangent space T [0] M HamU ( L ) is zero. We call a Lagrangian L rigid underHamiltonian equivalence if small deformations of L are Hamiltonian equivalent with L . Itturns out however that Hamiltonian equivalence is too restrictive for rigidity purposes:there are no Lagrangians that are infinitesimally rigid. Indeed, if the formal tangent space T [0] M HamU ( L ) = H ( F L ) ⊕ H γ ( F L ) is zero, then the triviality of the first summand impliesthat γ is foliated exact. But then H ( F L ) = H γ ( F L ) = { } by Proposition 3.12 i ) , whichis impossible. This is a motivation to look at a more flexible notion of equivalence. EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 53
Poisson isotopies.
We will use flows of Poisson vector fields instead of Hamiltonianvector fields to obtain a less restrictive equivalence relation on the space of Lagrangiandeformations of L . Definition 4.28.
We call two Lagrangian sections ( α , f ) and ( α , f ) in Def U ( L ) Poissonequivalent if they are interpolated by a smooth family ( α s , f s ) of Lagrangian sections inDef U ( L ) that is generated by a (locally defined) Poisson isotopy. In other words, thereexists a time-dependent Poisson vector field Y s on U such that the associated isotopy φ s maps graph ( α , f ) to graph ( α s , f s ) , for all s ∈ [0 , .We denote the moduli space Def U ( L ) / ∼ P oiss of Lagrangian deformations under Poissonequivalence by M P oissU ( L ) . In order to study rigidity under Poisson equivalence, we want tocompute the formal tangent space T [0] M P oissU ( L ) , as done in (74) for Hamiltonian equiva-lence. We now quotient first order deformations of L by elements of the form dds | s =0 ( α s , f s ) ,where ( α s , f s ) is generated by the flow of a time-dependent Poisson vector field Y s ∈ X ( U ) .Lemma 4.25 i ) implies that dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ( α s , f s ) = P ( Y ) , (75)where P : X ( U ) → Γ( T ∗ F L × R ) is the restriction to L composed with the vertical projectioninduced by the splitting (cid:0) T ( T ∗ F L × R ) (cid:1) | L = T L ⊕ ( T ∗ F L × R ) . So we have to take a closerlook at (vertical components of) Poisson vector fields on U ⊂ × R . Lemma 4.29.
Given the Poisson structure e Π = V ∧ t∂ t + Π can on U ⊂ T ∗ F L × R , thefollowing map is an isomorphism: H ( L ) ⊕ H ( L ) → H e Π ( U ) : (cid:0) [ ξ ] , g (cid:1) he Π ♯ ( pr ∗ ξ ) + ( pr ∗ g ) V i , where pr : U → L is the projection. We remark that the existence of the isomorphism follows from known facts: L is adeformation retract of U and of U ∩ T ∗ F L , and both cohomology groups appearing aboveare isomorphic to the first b -cohomology group of the pair ( U, U ∩ T ∗ F L ) , by [23] and [27,Prop. 1] respectively. Proof.
Clearly, the map is well-defined. For injectivity, assume e Π ♯ ( pr ∗ ξ )+( pr ∗ g ) V = e Π ♯ ( dh ) for some h ∈ C ∞ ( U ) . Restricting to W := U ∩ { t = 0 } , this implies that Π ♯can ( p ∗ ξ ) + ( p ∗ g ) V is tangent to the symplectic leaves, where p : W → L is the projection. Since V is transverseto the leaves, we get that p ∗ g = 0 , and therefore g = 0 . This means that e Π ♯ ( pr ∗ ξ ) = e Π ♯ ( dh ) ,and since e Π is invertible away from W ⊂ U , we get that pr ∗ ξ = dh on U \ W . By continuity, pr ∗ ξ = dh on all of U , so that ξ = d ( i ∗ L h ) is exact.To prove surjectivity, we use some b -symplectic geometry. The b -symplectic form ω on U obtained by inverting e Π reads [31, Proposition 4.1.2] ω = − e Π − = q ∗ θ ∧ dtt + q ∗ η, where q : U → W is the projection and ( θ, η ) ∈ Ω ( W ) × Ω ( W ) is the cosymplecticstructure corresponding with the pair (Π can , V ) . If Y ∈ X ( U ) is a Poisson vector field, then Y is tangent to W , so we can evaluate ω ♭ ( Y ) = q ∗ h θ, Y | W i dtt + (cid:20) ( h q ∗ θ, Y i − q ∗ h θ, Y | W i ) dtt − (cid:28) dtt , Y (cid:29) q ∗ θ + ι Y q ∗ η (cid:21) , (76) EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 54 which is a closed b -one form on U . Note indeed that the summand between square bracketsis a smooth de Rham form since q ∗ h θ, Y | W i − h q ∗ θ, Y i vanishes along the hypersurface W ↔ t = 0 and Y is tangent to it. Invoking the Mazzeo-Melrose isomorphism [17], [27] b H ( U ) → H ( U ) ⊕ H ( W ) : (cid:20) q ∗ ( h ) dtt + β (cid:21) ([ β ] , h ) , we know that the one-form β := ( h q ∗ θ, Y i − q ∗ h θ, Y | W i ) dtt − (cid:28) dtt , Y (cid:29) q ∗ θ + ι Y q ∗ η appearing in (76) is closed, and that h := h θ, Y | W i is locally constant. We now have Y = − e Π ♯ ( ω ♭ ( Y ))= q ∗ h θ, Y | W i V + e Π ♯ (cid:18) ( q ∗ h θ, Y | W i − h q ∗ θ, Y i ) dtt + (cid:28) dtt , Y (cid:29) q ∗ θ − ι Y q ∗ η (cid:19) = ( q ∗ h ) V + e Π ♯ ( − β ) (77)We make sure that the neighborhood U is such that the map i L ◦ pr : U → U induces theidentity map in cohomology. This means that q ∗ h = pr ∗ ( i ∗ L q ∗ h ) and β − pr ∗ ( i ∗ L β ) is exact.So if we put ξ := − i ∗ L β and g := i ∗ L q ∗ h , then it follows from (77) that [ Y ] = he Π ♯ ( pr ∗ ξ ) + ( pr ∗ g ) V i ∈ H e Π ( U ) . (cid:3) Proposition 4.30.
The formal tangent space T [0] M P oissU ( L ) is given by T [0] M P oissU ( L ) = Ω cl ( F L ) Im (cid:0) r : Ω cl ( L ) → Ω cl ( F L ) (cid:1) + H ( L ) · γ ⊕ H γ ( F L ) , (78) where the map r is restriction of closed one-forms on L to the leaves of F L .Proof. Throughout, for all vector bundles appearing, we denote by P the map that restrictsvector fields to the zero section, and then takes their vertical component. Because of (75),we have to show that the denominator appearing in (78) is equal to { P ( Y ) : Y s ∈ X ( U ) time-dependent Poisson vector field } . Notice that the above set lies in Ω ( F L ) , since all Poisson vector fields on U are tangent to W := U ∩ { t = 0 } . For one inclusion, let Y be a Poisson vector field on U . Using the factthat Y is tangent to W and Lemma 4.29, we have P ( Y ) = P ( Y | W )= P (cid:16) Π ♯can ( p ∗ ξ ) + ( p ∗ g ) V + Π ♯can ( dh ) (cid:17) (79)for some ξ ∈ Ω cl ( L ) , g ∈ H ( L ) and h ∈ C ∞ ( W ) . Here p : W → L is the projection. Nownote that P (cid:16) Π ♯can ( dh ) (cid:17) = P (cid:16) Π ♯can ( p ∗ di ∗ L h ) (cid:17) . Indeed, since L is coisotropic and h − p ∗ i ∗ L h vanishes along L , we have that Π ♯can ( d ( h − p ∗ i ∗ L h )) is tangent to L . So (79) becomes P ( Y ) = P (cid:16) Π ♯can ( p ∗ ξ ) + ( p ∗ g ) V + Π ♯can ( p ∗ di ∗ L h ) (cid:17) = r ( ξ + di ∗ L h ) + gγ, EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 55 where we used the correspondence (19) to obtain the last equality. This proves one inclusion.For the reverse inclusion, given ξ ∈ Ω cl ( L ) and g ∈ H ( L ) , we get a Poisson vector field e Π ♯ ( pr ∗ ξ ) + ( pr ∗ g ) V ∈ X ( U ) , and its vertical component along L is P (cid:16) e Π ♯ ( pr ∗ ξ ) + ( pr ∗ g ) V (cid:17) = P (cid:16) Π ♯can ( p ∗ ξ ) + ( p ∗ g ) V (cid:17) = r ( ξ ) + gγ. (cid:3) Smoothness of the moduli space by Poisson isotopies.
The moduli space M P oissU ( L ) is not smooth in general, since its formal tangent space can change drastically from point topoint. For instance, let us consider the same example as in §4.3.2, i.e. ( T × R , θ , θ , ξ , ξ ) with log-symplectic structure Π = ∂ θ ∧ ξ ∂ ξ + ∂ θ ∧ ξ and Lagrangian L = T ×{ (0 , } . Consider again a Lagrangian section (0 , c ) ∈ Γ( T ∗ F L × R ) for nonzero c ∈ R ; its graph lies outside the singular locus. By symplectic geometry, [(0 , c )] is an isolated point in the moduli space M P oissU ( L ) and therefore T [(0 ,c )] M P oissU ( L ) = 0 . On the other hand, we have T [0] M P oissU ( L ) = Ω cl ( F L ) Im (cid:0) r : Ω cl ( L ) → Ω cl ( F L ) (cid:1) ⊕ H ( F L ) ∼ = C ∞ ( T ) n f ∈ C ∞ ( T ) : ∂∂θ (cid:0)R S f dθ (cid:1) = 0 o ⊕ C ∞ ( S ) , which is infinite dimensional. Here we used Remark 4.14 i) to compute the first summand.4.3.6. Rigidity and Poisson isotopies.
We now address rigidity of Lagrangians under theequivalence relation by Poisson isotopies. As in the case of Hamiltonian equivalence, we calla Lagrangian L infinitesimally rigid under Poisson equivalence if the formal tangent space T [0] M P oissU ( L ) is zero. A Lagrangian L is called rigid under Poisson equivalence if smalldeformations of L are Poisson equivalent with L . Rigidity is a very restrictive property: sincePoisson diffeomorphisms fix the singular locus of the log-symplectic structure, a Lagrangian L can only be rigid if small deformations of it stay inside the singular locus.We will restrict ourselves to Lagrangians L that are compact and connected. It turns outthat asking for infinitesimal rigidity under Poisson equivalence is only a little weaker thanasking for infinitesimal rigidity under Hamiltonian equivalence, as the next lemma shows. Lemma 4.31.
Let L be a Lagrangian that is compact, connected and infinitesimally rigidunder Poisson equivalence. Then H ( F L ) is finite dimensional.Proof. Since L is compact, we know that H ( L ) is finite dimensional. Choose a basis { [ β ] , . . . , [ β k ] } of H ( L ) . If α ∈ Ω cl ( F L ) is a closed foliated one-form, then infinitesimalrigidity implies that α = r ( e α ) + cγ for some e α ∈ Ω cl ( L ) and c ∈ R . Since e α can be writtenas e α = c β + · · · + c k β k + dh for some c , . . . , c k ∈ R and h ∈ C ∞ ( L ) , we get α = c r ( β ) + · · · + c k r ( β k ) + d F L h + cγ. Therefore H ( F L ) is spanned by { [ r ( β )] , . . . , [ r ( β k )] , [ γ ] } , hence finite dimensional. (cid:3) This implies that Lagrangians L for which F L is the foliation by fibers of a fiber bundleover S are never rigid, not even infinitesimally. EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 56
Corollary 4.32. If L is a compact Lagrangian for which F L is the foliation by fibers of afiber bundle p : L → S , then L is not infinitesimally rigid under Poisson equivalence.Proof. Assume to the contrary that L is infinitesimally rigid. By Lemma 4.31, we knowthat H ( F L ) ∼ = Γ( H ) is finite dimensional. So H has to be of rank zero, which impliesthat H ( F L ) = 0 . Consequently, γ is exact, and then Proposition 3.15 i ) guarantees that H γ ( F L ) is nonzero. This contradicts that the infinitesimal moduli space (78) is zero. So L cannot be infinitesimally rigid. (cid:3) Remark . Alternatively, one can obtain Corollary 4.32 by using the flat connection ∇ on H , which was defined in (45). Assuming that L is infinitesimally rigid, fix an open U ⊂ S and a frame { σ η , . . . , σ η m } for H | U consisting of flat sections. If α ∈ Ω cl ( F L ) ,then infinitesimal rigidity implies that α = r ( e α ) + cγ for some e α ∈ Ω cl ( L ) and c ∈ R . Notethat the section σ r ( e α ) ∈ Γ( H ) is flat, since for all Y ∈ X ( S ) we have ∇ Y σ r ( e α ) = σ r ( £ Y e α ) = σ d F L ι Y e α = 0 , where we used Cartan’s magic formula. It follows that σ α | U = c σ η + · · · + c m σ η m + cσ γ | U for constants c , . . . , c k , c ∈ R . This means that necessarily H ( F L ) = 0 , and we obtain acontradiction as in the proof of Corollary 4.32.So fibrations over S don’t give examples of rigid Lagrangians. However, if the foliation F L on L has dense leaves, then we do obtain an interesting rigidity statement: infinitesimalrigidity implies rigidity with respect to the Fréchet C ∞ -topology. Proposition 4.34.
Let L be a compact, connected Lagrangian whose induced foliation F L has dense leaves. Assume that L is infinitesimally rigid under Poisson equivalence. Thenthere exists a neighborhood V ⊂ (Γ( T ∗ F L × R ) , C ∞ ) of such that if Graph ( α, f ) is La-grangian for ( α, f ) ∈ V , then ( α, f ) is Poisson equivalent with the zero section of T ∗ F L × R .Proof. Infinitesimal rigidity implies that H γ ( F L ) = 0 , so γ is not foliated exact by ii ) ofProposition 3.15. Moreover, H ( F L ) is finite dimensional by Lemma 4.31. By Proposition4.10, we obtain a neighborhood V ⊂ (Γ( T ∗ F L × R ) , C ∞ ) of such that if Graph ( α, f ) isLagrangian for ( α, f ) ∈ V , then f ≡ . To show that V satisfies the criteria, we distinguishbetween two cases.Case 1: γ extends to a closed one-form on L . The assumption of infinitesimal rigidity thenimplies that Ω cl ( F L ) = Im (cid:0) r : Ω cl ( L ) → Ω cl ( F L ) (cid:1) . So if ( α, f ) = ( α, ∈ V is such that thegraph of ( α, ∈ Γ( T ∗ F L × R ) is Lagrangian, then we have α = r ( e α ) for some e α ∈ Ω cl ( L ) .The time 1-flow of the Poisson vector field e Π ♯ ( pr ∗ e α ) then takes L to Graph ( α, .Case 2: γ does not extend to a closed one-form on L . In this case, infinitesimal rigidityimplies that (cid:0) Ω cl ( F L ) , C ∞ (cid:1) splits into an algebraic direct sum Ω cl ( F L ) = Im (cid:0) r : Ω cl ( L ) → Ω cl ( F L ) (cid:1) ⊕ R γ. (80)Since r is C ∞ -continuous, linear and Im ( r ) ⊂ Ω cl ( F L ) is of finite codimension, we get thatIm ( r ) ⊂ (cid:0) Ω cl ( F L ) , C ∞ (cid:1) is closed. This implies that (80) is in fact a topological direct sum: R γ is an algebraic complement to a maximal closed subspace, and therefore a topologicalcomplement [28, Theorem 4.9.5]. So the projection onto the second summand of (80) iscontinuous, and therefore we get a continuous map p : (Ω cl ( F L ) , C ∞ (cid:1) → R : r ( e α ) + cγ c. EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 57
This implies that, shrinking the neighborhood V constructed above if necessary, we canassume that p ( £ X α ) < whenever ( α, f ) = ( α, ∈ V is a Lagrangian section.Now suppose that ( α, f ) = ( α, ∈ V is such that the graph of ( α, ∈ Γ( T ∗ F L × R ) isLagrangian. We decompose α and £ X α in the direct sum (80): ( α = r ( ξ ) + Cγ £ X α = r ( η ) + Kγ (81)for ξ, η ∈ Ω cl ( L ) and C, K ∈ R with K < . We define smooth families ξ s ∈ Ω cl ( L ) and C s ∈ R for s ∈ [0 , by the formulas ξ s := ξ + C − sK sη, C s := C − sK . Note that the denominator − sK occurring in these expressions is never zero for s ∈ [0 , since K < . We claim that the isotopy φ s generated by the time-dependent Poisson vectorfield e Π ♯ ( pr ∗ ξ s ) + C s V takes the zero section of T ∗ F L × R to graph ( α, , or more precisely,that φ s ( L ) = graph ( sα, for s ∈ [0 , . To prove this, by [35, Lemma 3.15] it is enough tocheck that dds ( sα,
0) = P ( sα, (cid:16) e Π ♯ ( pr ∗ ξ s ) + C s V (cid:17) , (82)where P ( sα, denotes the vertical projection induced by the direct sum decomposition of T ( T ∗ F L × R ) | graph ( sα, into T graph ( sα, and the vertical bundle along graph ( sα, .Computing the right hand side of (82) gives P ( sα, (cid:16) e Π ♯ ( pr ∗ ξ s ) + C s V (cid:17) = P sα (cid:16) Π ♯can ( p ∗ ξ s ) + C s V (cid:17) = r ( ξ s ) + C s P sα ( V vert + V lift )= r ( ξ s ) + C s ( γ − £ X ( sα ))= r (cid:18) ξ + C − sK sη (cid:19) + C − sK ( γ − sr ( η ) − sKγ )= r ( ξ ) + Cs − sK r ( η ) + C (1 − sK )1 − sK γ − Cs − sK r ( η )= r ( ξ ) + Cγ = α. Here we used the correspondence (19) in the second equality, Lemma 4.35 below in the thirdequality and the expressions (81) in the fourth equality. This finishes the proof. (cid:3)
By definition of the C ∞ -topology, one can rephrase the above proposition as follows:infinitesimal rigidity of L implies the existence of some k ∈ N such that L is C k -rigid. Lemma 4.35.
Let α ∈ Γ( T ∗ F L ) , and denote by P α the vertical projection induced by thesplitting of T ( T ∗ F L ) | graph ( α ) into T graph ( α ) and the vertical bundle along graph ( α ) . Wethen have P α ( V lift ) = − £ X α. Proof.
Denote by φ − α the translation map φ − α : T ∗ F L → T ∗ F L : ( p, ξ ) ( p, ξ − α ( p )) , EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 58 and let P := P be the vertical projection along the zero section. We then have a commu-tative diagram T ( T ∗ F L ) | graph ( α ) T ( T ∗ F L ) | L Γ( T ∗ F L ) ( φ − α ) ∗ P α P , so the lemma follows from the equality (26). (cid:3) Example 4.36.
Let L = ( T , θ , θ ) with Kronecker foliation T F L = Ker ( dθ − λdθ ) forgeneric (i.e. not Liouville) λ ∈ R \ Q . Let ξ be the fiber coordinate on T ∗ F L correspondingwith the frame { dθ } . As in eq. (67), we take a log-symplectic structure (cid:16) T ∗ F L × R , e Π := ( C∂ θ + K∂ ξ ) ∧ t∂ t + ( λ∂ θ + ∂ θ ) ∧ ∂ ξ (cid:17) , where C, K ∈ R and K is nonzero. Since for generic λ ∈ R \ Q , we have H ( F L ) = R [ dθ ] ,it is clear that every element of Ω cl ( F L ) extends to a closed one-form on L . Moreover,since γ = Kdθ is not exact, we have that H γ ( F L ) = 0 by Proposition 3.15 ii ) . So L isinfinitesimally rigid: T [0] M P oissU ( L ) = Ω cl ( F L ) Im (cid:0) r : Ω cl ( L ) → Ω cl ( F L ) (cid:1) + R γ ⊕ H γ ( F L ) = 0 , and therefore L is C ∞ -rigid, by Proposition 4.34.In this particular example, we in fact know a bit more. We already noted in Remark4.9 that C -small deformations of L stay inside the singular locus, i.e. they are of the form ( α, f ) = ( α, ∈ Γ( T ∗ F L × R ) for α ∈ Ω cl ( F L ) . Along with the fact that foliated closedone-forms extend to closed one-forms on L , this implies that the Lagrangian L is C -rigidunder Poisson equivalence. For if e α ∈ Ω cl ( L ) is a closed extension of α , then the flow of thePoisson vector field e Π ♯ ( pr ∗ e α ) takes L to graph ( α, .If instead we take λ ∈ R \ Q to be a Liouville number, then L is not infinitesimally rigidby Lemma 4.31, since in that case H ( F L ) is infinite dimensional.5. Appendix
This short appendix summarizes some facts about Liouville numbers and Fréchet spaces.5.1.
Liouville numbers.
We collect some facts about Liouville numbers that are used in §4.1.
Definition 5.1.
A Liouville number is a real number α ∈ R with the property that, for allintegers p ≥ , there exist integers m p , n p ∈ Z such that n p > and < (cid:12)(cid:12)(cid:12)(cid:12) α − m p n p (cid:12)(cid:12)(cid:12)(cid:12) < n pp . Liouville numbers are irrational (even transcendental, see [25, Theorem 4.5]).
Remark . For any sequence ( m p , n p ) p ∈ N as in Definition 5.1, the set of denominators { n p : p ∈ N } is unbounded. Indeed, assume to the contrary that this set is bounded bysome constant M . Since n p > , the sequence ( m p /n p ) p ∈ N converges to α . As there areonly finitely many fractions a/b such that < b ≤ M and a/b lies within distance of α , EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 59 the sequence ( m p /n p ) p ∈ N must have a constant subsequence. This subsequence must alsoconverge to α , which implies that α ∈ Q . This contradiction shows that { n p : p ∈ N } isunbounded.The next statement is used in the proof of Lemma 4.8. It appears without proof in [1]. Lemma 5.3. If α is a Liouville number, then for each integer p ≥ , there exists a pair ofintegers ( m p , n p ) ∈ Z such that | m p + αn p | ≤ | m p | + | n p | ) p . Proof.
Since α is Liouville, we can fix a sequence ( M p , N p ) for integers p ≥ , satisfying < (cid:12)(cid:12)(cid:12)(cid:12) α − M p N p (cid:12)(cid:12)(cid:12)(cid:12) < N pp , N p ≥ . The sequence ( M p /N p ) p ∈ N is convergent hence bounded, so there exists an integer k ≥ such that | M p | ≤ k N p , ∀ p ≥ . (83)Notice that (cid:12)(cid:12)(cid:12)(cid:12) α − M ( k +2) p N ( k +2) p (cid:12)(cid:12)(cid:12)(cid:12) < N ( k +2) p ( k +2) p = 1 N p ( k +2) p · N ( k +1) p ( k +2) p ≤ N p ( k +2) p · ( k +1) p . (84)Since the function x x p is convex on (0 , ∞ ) , we have (cid:18) | N ( k +2) p | + | M ( k +2) p | (cid:19) p ≤ | N ( k +2) p | p + | M ( k +2) p | p , and therefore (cid:0) | N ( k +2) p | + | M ( k +2) p | (cid:1) p ≤ p − (cid:0) | N ( k +2) p | p + | M ( k +2) p | p (cid:1) ≤ p max (cid:0) | N ( k +2) p | p , | M ( k +2) p | p (cid:1) ≤ p · kp | N ( k +2) p | p = 2 ( k +1) p | N ( k +2) p | p , (85)using (83) in the third inequality. Combining the inequality (84) with (85) gives (cid:12)(cid:12)(cid:12)(cid:12) α − M ( k +2) p N ( k +2) p (cid:12)(cid:12)(cid:12)(cid:12) < (cid:0) | N ( k +2) p | + | M ( k +2) p | (cid:1) p . Replacing M p by − M p , this implies that (cid:12)(cid:12) M ( k +2) p + αN ( k +2) p (cid:12)(cid:12) < N ( k +2) p (cid:0) | N ( k +2) p | + | M ( k +2) p | (cid:1) p ≤ (cid:0) | N ( k +2) p | + | M ( k +2) p | (cid:1) p − . So if we set ( m p , n p ) := (cid:0) M ( k +2)( p +1) , N ( k +2)( p +1) (cid:1) , then we have | m p + αn p | < | m p | + | n p | ) p . (cid:3) Remark . The proof of Lemma 5.3 shows that we can make the additional assumptions n p ≥ p and ( m p , n p ) = ( m q , n q ) for p = q . Indeed, since the set of denominators { N p : p ∈ N } of the sequence ( M p , N p ) p ∈ N is unbounded, we can ensure that N p ≥ p . For if N p < p , then EFORMATIONS OF LAGRANGIAN SUBMANIFOLDS IN LOG-SYMPLECTIC MANIFOLDS 60 we know that there exists p ′ > p such that the element ( M p ′ , N p ′ ) satisfies N p ′ ≥ p . Wethen have (cid:12)(cid:12)(cid:12)(cid:12) α − M p ′ N p ′ (cid:12)(cid:12)(cid:12)(cid:12) < N p ′ p ′ < N pp ′ , so we can just replace ( M p , N p ) by ( M p ′ , N p ′ ) . It then follows that n p = N ( k +2)( p +1) ≥ ( k + 2)( p + 1) ≥ p. Similarly, we can make sure that N p = N q for q = p , so that also ( m p , n p ) = ( m q , n q ) .5.2. Fréchet spaces.
We recall some basic facts about Fréchet spaces, which are used in §4.1 and §4.3. Formore details, see for instance [20].
Definition 5.5.
A Fréchet space is a topological vector space X that satisfies the followingthree properties:i) X is Hausdorff.ii) The topology on X is induced by a countable family of seminorms {k · k k } k ≥ .iii) X is complete.By item ii), a base of neighborhoods of x ∈ X is given by subsets of the form B k r ( x ) ∩ · · · ∩ B k n r ( x ) for n ∈ N and r > , where B k j r ( x ) denotes the open ball B k j r ( x ) = { y ∈ X : k y − x k k j < r } . A sequence x n converges to x if and only if k x n − x k k converges to zero for each k ≥ . Example 5.6. If L is compact, the space of sections of any vector bundle over L becomesa Fréchet space when endowed with the C ∞ -topology generated by C k -norms k · k k . Werecall the construction of such norms in the situation that is of interest to us. Let ( L, F L ) be a compact manifold endowed with a codimension-one foliation; we will define C k -normson the space Ω • ( F L ) = Γ( ∧ • ( T ∗ F L )) of foliated forms of fixed degree. Fix a finite cover { U , . . . , U m } of L consisting of foliated charts with coordinates ( x , . . . , x n − , x n ) , such thatplaques of F L are level sets of x n . Choose open subsets V i for i = 1 , . . . , m that still cover L and have compact closures satisfying V i ⊂ U i . The k -norm of a foliated form η ∈ Ω l ( F L ) with coordinate representation η | U j = X ≤ i < ···
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