Poisson geometry of b-manifolds. Eva Miranda

Transcription

1 Poisson geometry of b-manifolds Eva Miranda UPC-Barcelona Rio de Janeiro, July 26, 2010 Eva Miranda (UPC) Poisson 2010 July 26, / 45

2 Outline 1 Motivation 2 Classification of b-poisson manifolds in dimension 2 3 Examples of b-manifolds in dimension greater than 2 4 Poisson geometry of the critical hypersurface 5 Extension theorem 6 b-cohomology, Poisson cohomology and Moser theorem 7 Additional structures: Integrable systems 8 Related projects Eva Miranda (UPC) Poisson 2010 July 26, / 45

3 Motivation: b-objects in dimension 2 We consider pairs (M, Z) where M is a compact surface and Z a union of embedded smooth curves: Eva Miranda (UPC) Poisson 2010 July 26, / 45

4 Symplectic structures in dimension 2: Global classification Given an oriented surface, a volume form Ω on it defines a symplectic structure. Theorem (Moser) Two symplectic structures ω 0 and ω 1 on a compact symplectic surface with [ω 0 ] = [ω 1 ] are symplectically equivalent. Idea behind The linear path ω t = (1 t)ω 0 + tω 1 is a path of symplectic structures (Moser s trick) integration of the flow of X t given by the path method gives the diffeomorphism. Eva Miranda (UPC) Poisson 2010 July 26, / 45

5 Motivation: b-structures in dimension 2 Given an oriented surface S with a distinguished union of curves Z, we want to modify the volume form on S by making it explode when we get close to Z. We want this blow up process to be controlled. Eva Miranda (UPC) Poisson 2010 July 26, / 45

6 Motivation: b-structures in dimension 2 What does controlled mean here? Consider the Lie algebra g to be the Lie algebra of the affine group in dimension 2. It is a model for noncommutative Lie algebras in dimension 2 and in a basis e 1, e 2 the brackets are, [e 1, e 2 ] = e 2 We can naturally write this Lie algebra structure (bilinear) as the Poisson structure Π = y x y Eva Miranda (UPC) Poisson 2010 July 26, / 45

7 Motivation: b-structures in dimension 2 This Poisson structure is dual to the 2-form In this example Z is the x-axis: ω = 1 dx dy y In this example Z is formed by symplectic leaves of dimension 0 (points on the line). The upper and lower half-planes are symplectic leaves of dimension 2. Eva Miranda (UPC) Poisson 2010 July 26, / 45

8 Examples in higher dimensions: b-tangent bundle Consider a surface with boundary: Eva Miranda (UPC) Poisson 2010 July 26, / 45

9 Examples in dimension 4: b-tangent bundles A vector field in a point of the boundary has to be tangent to the boundary. Observe that for any point p (H+ n ), the set of vector fields at p tangent to (Hn + ) is generated by T p(h+ n ) = x 1 x,... 1 p x 2 p xn. p Melrose proved that there exists a vector bundle (the b-tangent bundle, b T (M)) whose sections are the set of vector fields tangent to Z. Melrose With this idea Melrose constructed the b-cotangent bundle of this surface. This was the starting point of b-calculus for differential calculus on manifolds with boundary. Eva Miranda (UPC) Poisson 2010 July 26, / 45

10 b-poisson structure on b-tangent bundles Consider the Poisson structure Π = y 1 + x 1 y 1 n i=2 x i y i It can also be interpreted as a section of Λ 2 ( b T (M)). We also have a Liouville one-form interpretation for this Poisson structure. Darboux theorem for our manifolds We can prove a b-darboux theorem which tells us that locally all Poisson b-manifolds look like this model. Notice that for this model the set Z is a smooth hypersurface. Poisson b-manifolds are Poisson manifolds of dimension 2n which have dense symplectic leaves of dimension 2n. We will require that Z is a union of hypersurfaces such that Π n vanishes on Z in a linear way....let s be more precise... Eva Miranda (UPC) Poisson 2010 July 26, / 45

11 Definition of b-poisson manifolds Definition Let (M 2n, Π) be an oriented Poisson manifold such that the map p M (Π(p)) n Λ 2n (T M) is transverse to the zero section, then Z = {p M (Π(p)) n = 0} is a hypersurface and we say that Π is a b-poisson structure on (M, Z). Eva Miranda (UPC) Poisson 2010 July 26, / 45

12 A Darboux theorem for b-poisson manifolds Using either Weinstein s splitting theorem or the path method, we can prove, Theorem For all p Z, there exists a Darboux coordinate system x 1, y 1,..., x n, y n centered at p such that Z is defined by y 1 = 0 and The b-category Π = y 1 + x 1 y 1 n i=2 x i y i Thus, the moral is that b-poisson manifolds lie between the symplectic and Poisson world and we can get some interesting results that we do not get for general Poisson manifolds. Eva Miranda (UPC) Poisson 2010 July 26, / 45

13 Classification of b-poisson manifolds in dimension 2 Radko classified b-poisson structures on compact oriented surfaces giving a list of invariants: Geometrical: The topology of S and the curves γ i where Π vanishes. Dynamical: The periods of the modular vector field along γ i. Measure: The regularized Liouville volume of S. Figure: Two admissible vanishing curves for Π; the one in (b ) is not admissible. Eva Miranda (UPC) Poisson 2010 July 26, / 45

14 Modular vector field Recall that, Definition Given a Poisson manifold (M, Π) and a volume form Ω, the modular vector field XΠ Ω associated to the pair (Π, Ω) is the derivation given by the mapping f L u f Ω Ω 1 L X Ω Π (Π) = 0 and L X Ω Π (Ω) = 0. 2 X HΩ = X Ω u log(h). its first cohomology class in Poisson cohomology does not depend on Ω. 3 Examples of unimodular (vanishing modular class) Poisson manifolds: symplectic mflds. 4 In the case of Poisson b-manifolds in dimension 2, {x, y} = y and the modular vector field is x. Eva Miranda (UPC) Poisson 2010 July 26, / 45

15 Modular vector fields As a consequence of Darboux theorem, we obtain the following, Modular vector field for Darboux form The modular vector field of a local b-poisson manifold with local normal form, Π = y 1 x 1 y 1 + n i=2 x i y i with respect to the volume form Ω = i dx i dy i is, X Ω = x 1. As a consequence: The modular vector field of a Poisson b-manifold (M, ω Π ) is tangent to the vanishing set Z of the 2n-vector field Π n and is transverse to the symplectic leaves inside Z. Eva Miranda (UPC) Poisson 2010 July 26, / 45

16 Radko s theorem: classification in dimension 2 1 Consider h a function vanishing linearly on the curves γ 1,..., γ n and not vanishing elsewhere. 2 Let V ɛ h (Π) = h >ɛ ω Π 3 The limit V (h) = lim ɛ 0 V ɛ h (Π) exists and is independent of the function. This limit is known as regularized Liouville volume. Theorem (Radko) The set of curves, modular periods and regularized Liouville volume completely determines, up to Poisson diffeomorphisms, the b-poisson structure on a compact oriented surface S. Eva Miranda (UPC) Poisson 2010 July 26, / 45

17 Other related classification results David Martínez-Torres studied classification of generic Nambu structures of top degree and applied similar techniques to the ones of Olga Radko to obtain the following: Theorem (Martínez-Torres) A generic Nambu structure Λ of top degree is determined, up to orientation preserving diffeomorphism, by the the diffeomorphism type of the oriented pair (M, Z) together with its modular periods and regularized Liouville volume. Eva Miranda (UPC) Poisson 2010 July 26, / 45

18 Higher dimensions: Some compact examples. Let (R, π R ) be a Radko compact surface and let (S, π) be a compact symplectic surface, then (R S, π R + π) is a b-poisson manifold of dimension 4. Other product structures to get higher dimensions. We can perturb this product structure to obtain a non-product one. For instance, S 2 with Poisson structure Π 1 = h h θ and the two torus T 2 with Poisson structure Π 2 = θ 1 θ 2. Consider, ˆΠ = h h ( θ + ) + Π 2. θ 1 Then (S 2 T 2, ˆΠ) is a b-poisson manifold. Moser s ideas again Via a path theorem, we can control perturbations that produce equivalent Poisson structures. Eva Miranda (UPC) Poisson 2010 July 26, / 45

19 More examples. Take (N, π) be a regular Poisson manifold with dimension 2n 1 and rank 2n 2 and let X be a Poisson vector field. Now consider the product S 1 N with the bivector field Π = f(θ) θ X + π. This is a b-poisson manifold as long as, 1 the function f vanishes linearly. 2 The vector field X is transverse to the symplectic leaves of N. We then have as many copies of N as zeroes of f. Eva Miranda (UPC) Poisson 2010 July 26, / 45

20 More examples. M = T 4 and Z = T 3 {0}. Consider on Z the codimension 1 foliation given by θ 3 = aθ 1 + bθ 2 + k, with rationally independent a, b R. Then take h = log(sin θ 4 ), a α = a 2 + b dθ b 1 + a 2 + b dθ 1 2 a 2 + b dθ 3, ω = dθ 1 dθ 2 + b dθ 1 dθ 3 a dθ 2 dθ 3, The 2-form ω Π = dh α + ω defines a b-poisson structure in a neighbourhood of Z. In general given an hypersurface Z with a codimension 1 foliation we have topological obstructions for this structure to extend to the whole manifold. Eva Miranda (UPC) Poisson 2010 July 26, / 45

21 Global constructions for higher dimensions: Going backwards... Induced Poisson structures Given a b-poisson structure Π on M 2n we get an induced Poisson structure on Z (vanishing set) which is a regular Poisson structure with symplectic leaves of codimension 1. We can look for converse results. Given a Poisson manifold Z with codimension 1 symplectic foliation L, we want to answer the following questions: 1 Does (Z, Π L ) extend to a b-poisson structure on a neighbourhood of Z in M? 2 If so to what extent is this structure unique? 3 Global results à la Radko? Eva Miranda (UPC) Poisson 2010 July 26, / 45

22 Global constructions for higher dimensions: Going backwards... Semilocal answer We will thicken our regular Poisson manifold Z and we will consider a tubular neighbourhood construction: ω = p (α Z ) df f + p (ω Z ) Using α Z a defining one-form for the symplectic foliation on Z and ω Z a two form that restricts to the symplectic form on every symplectic leaf. These forms need to satisfy more constraints in order to work. So the answer is: Not always. Eva Miranda (UPC) Poisson 2010 July 26, / 45

23 The L-De Rham complex Consider now a regular Poisson manifold (of dimension 2n 1) with a symplectic foliation of codimension 1 given by a one form α Ω 1 (Z) (i Lα = 0 ). Let ω Ω 2 (Z) be a 2-form such that i L ω = ω L. Notice that dα = α β, β Ω 1 (Z) (1) Therefore we can consider the complex Ω k L = Ω k /αω k 1 Consider Ω 0 = α Ω we get a short exact sequence of complexes i 0 Ω 0 Ω j Ω L 0 By differentiation of 1 we get 0 = d(dα) = dβ α β β α = dβ α, so dβ is in Ω 0, i.e., d(jβ) = 0. First obstruction class We define the obstruction class c 1 (Π L ) H 1 (Ω L ) to be c 1 (Π L ) = [jβ] Eva Miranda (UPC) Poisson 2010 July 26, / 45

24 The L-De Rham complex Characterization of the first invariant c 1 (Π L ) = 0 iff we can find a closed one form for the foliation. Assume now c 1 (Π L ) = 0 then, we obtain dω = α β 2. Second obstruction class We define the obstruction class c 2 (Π L ) H 2 (Ω L ) to be c 2 (Π L ) = [jβ 2 ] Main property c 2 (Π L ) = 0 there exists a closed 2-form, ω, such that i L (ω) = ω L. Eva Miranda (UPC) Poisson 2010 July 26, / 45

25 The role of these invariants Integrability The vanishing of c 2 (Π L ) implies that there exists a leafwise symplectic embedding of the symplectic foliation and this implies integrability of the Poisson structure in the sense of Crainic-Fernandes. The role of these invariants c 1 (Π L ) = c 2 (Π L ) = 0 there exists a Poisson vector field v transversal to every symplectic leaf L. Relation of v, ω and α: 1 i v α = 1. 2 i v ω = 0. The fibration is a symplectic fibration and v defines an Ehresmann connection. Eva Miranda (UPC) Poisson 2010 July 26, / 45

26 Dynamics of codimension-1 foliations on Poisson manifolds with vanishing invariants Let β satisfy: dα = α β, β Ω 1 (Z) Then with respect to the volume form α ω n 1, i vmod ω L = β L Theorem A regular corank 1 Poisson manifold is unimodular iff we can choose a closed defining one-form α for the symplectic foliation (i.e. if and only if c 1 (Π L ) = 0). Corollary Foliations with vanishing c 1 (Π L ) have vanishing Godbillon-Vey class. The b-poisson case The Poisson structure induced on the critical hypersurface of a b-poisson structure manifold has vanishing invariants c 1 (Π L ) = 0 and c 2 (Π L ). Eva Miranda (UPC) Poisson 2010 July 26, / 45

27 Summing up, The foliation induced by a b-poisson structure on its critical hypersurface satifies, we can choose the defining one-form α to be closed symplectic structure on leaves which extends to a closed 2-form ω on M Given a symplectic foliation on a corank 1 regular Poisson manifold α and ω exists if and only if the invariants c 1 (Π L ) and c 2 (Π L ) vanish. Question Is every codimension one regular Poisson manifold with vanishing invariants the critical hypersurface of a b-poisson manifold? We will answer this question later. First let s study the geometry of these manifolds. Eva Miranda (UPC) Poisson 2010 July 26, / 45

28 A theorem of Tischler: Foliations given by closed forms Theorem Let M be a compact manifold without boundary that admits a non-vanishing closed 1-form. Then M is a fibration over S 1. The irrational flow Observe that this is NOT telling us that the foliation given by α itself IS a fibration. Eva Miranda (UPC) Poisson 2010 July 26, / 45

29 Two variations of Tischler (Tischler) If dim H 1 (M, R) = 1 then the foliation defined by the closed 1-form is isotopic to the foliation given by the fibration. If one leaf is compact, then we can apply Reeb s stability theorem, α closed L has no holonomy the condition of finiteness of the fundamental group can be replaced by trivial holonomy the foliation is a fibration over S 1. Eva Miranda (UPC) Poisson 2010 July 26, / 45

30 The case of vanishing invariants(compact leaves) Theorem (Guillemin-M-Pires) Let M 2n 1 be an oriented compact regular Poisson manifold of rank 2n with c 1 (Π L ) = c 2 (Π L ) = 0. If the foliation has a compact leaf L then all leaves are compact and symplectomorphic. The foliation is given by an S 1 fibration and M 2n 1 is a mapping torus of L associated to the symplectomorphim f given by the flow of the Poisson vector field v. Eva Miranda (UPC) Poisson 2010 July 26, / 45

31 The case of vanishing invariants Corollary The Poisson structure associated of these manifolds is integrable (in the Crainic-Fernandes sense) and the Weinstein s symplectic groupoid is also a mapping torus. Consider the Weinstein s groupoid of a symplectic leaf (Π 1 (L), Ω). And consider the product, (Π 1 (L) T (R), Ω + dλ liouville ). Let f be the time-1-flow of v. In this product consider the mapping ( x, (t, η)) ( f, (t + 1, η)). It preserves the symplectic groupoid structure. Therefore, by identification it induces a symplectic groupoid structure on the mapping torus. Thanks to David Martínez-Torres for key conversations about this integration. Eva Miranda (UPC) Poisson 2010 July 26, / 45

32 The extension property Theorem (Guillemin-M-Pires) Let (M 2n 1, Π 0 ) be a compact corank-1 regular Poisson manifold with vanishing invariants c 1 (Π L ) and c 2 (Π L ), then there exists an extension of (M 2n 1, Π) to a b-poisson manifold (U, Π). The extension is unique, up to isomorphism, among the extensions such that the class of the transversal Poisson vector field [v] is the image of the modular class under the map: H 1 P oisson(u) H 1 P oisson(m 2n 1 ) Eva Miranda (UPC) Poisson 2010 July 26, / 45

33 Construction Given a Poisson vector field v on (Z, Π L ) with v transverse to L chose α Z Ω 1 (Z) and ω Z ω 2 (Z) such that: 1 i v α Z = 1. 2 i v ω Z = 0. 3 α Z is a defining one form for the symplectic foliation. 4 ω Z restricts to the induced symplectic form on each symplectic leaf. Eva Miranda (UPC) Poisson 2010 July 26, / 45

34 Construction Now consider p : U Z a tubular neighbourhood of Z in U and let ω = p (α Z ) df f + p (ω Z ) with f a defining function for Z. Then one can check that this is a semi-local extension to Poisson b-manifold and that the modular vector field of ω restricted to Z is v. Uniqueness relies strongly on a relative Moser s theorem for our b-poisson manifolds that we will spell out shortly. Eva Miranda (UPC) Poisson 2010 July 26, / 45

35 b-derham k-forms Denote by b Ω k (M) the space of sections of the bundle Λ k ( b T M) The usual space of DeRham k-forms sits inside this space but in a somewhat nontrivial way: given µ Ω k (M), we interpret it as a section of Λ k ( b T M) by the convention µ p Λ k (T p M) = Λ k ( b T p M) at p M Z µ p = (i µ) p Λ k (T p Z) Λ k ( b T p M) at p Z where i : Z M is the inclusion map. Every b-derham k-form can be written as ω = α df f + β, with α Ωk 1 (M) and β Ω k (M) (2) Eva Miranda (UPC) Poisson 2010 July 26, / 45

36 b-derham k-forms We can define the exterior De Rham operator d by setting dω = dα df f + dβ. And we get, 0 b Ω 0 (M) d b Ω 1 (M) d b Ω 2 (M) Define b-cohomology as the cohomology of this complex, We obtain Theorem (Mazzeo-Melrose) The b-cohomology groups of M are computable by b H (M) = H (M) H 1 (Z). d... 0 Eva Miranda (UPC) Poisson 2010 July 26, / 45

37 A dual approach to b-poisson structures Definition We say that a closed b-de Rham 2-form ω is b-symplectic if ω p is of maximal rank as an element of Λ 2 ( b T p M) for all p M. As an application of Mazzeo-Melrose we obtain that for a compact b-symplectic M we have, b H 2 (M) 0. Let Π be the dual b-poisson structure. In total analogy with the symplectic case, non-degeneracy of a b-symplectic form proves, that the De Rham cohomology of the manifold is isomorphic to the cohomology of the complex (Λ k ( b T M), d Π ), we now conjecture that, b H (M) = H Π(M) Eva Miranda (UPC) Poisson 2010 July 26, / 45

38 Poisson cohomology for two dimensional b-poisson We can now reinterpret the following result of Radko concerning computation of Poisson cohomology for the 2-dimensional case through the eyes of Mazzeo-Melrose, Theorem (Radko) The Poisson cohomology of a compact oriented stable Poisson surface of genus g with critical curves γ 1,..., γ n is given by: H 0 Π(M, R) = R H 1 Π(M, R) = R n+2g H 2 Π(M, R) = R n+1 Eva Miranda (UPC) Poisson 2010 July 26, / 45

39 Moser s theorems for b-poisson For b-forms we can prove b-poincaré theorem and get local, relative and global Moser s theorems for b-poisson manifolds, Theorem (Guillemin-M-Pires) Let ω i b Ω 2 (U), i = 0, 1 be b-symplectic forms in a neighbourhood of Z such that ω 0 Z = ω 1 Z. Then ω 0 and ω 1 are b-symplectomorphic in a neighbourhood of Z. Eva Miranda (UPC) Poisson 2010 July 26, / 45

40 Another Moser s theorem (global version) Theorem (Guillemin-M.-Pires) Let ω i b Ω 2 (M), i = 0, 1 be b-symplectic forms on M with modular vector fields, v i. Suppose: 1 v = v 1 Z = v 2 Z (modular vector fields coincide when restricted to Z). 2 [ω 0 ] = [ω 1 ]. 3 ω t = (1 t)ω 0 + tω 1 is b-symplectic for all t [0, 1]. Then ω 0 and ω 1 are b-symplectomorphic. Eva Miranda (UPC) Poisson 2010 July 26, / 45

41 Another application of this global Moser theorem Corollary When M is an oriented compact surface, the conditions in our Moser theorem are equivalent to those given by Radko. Therefore, the set of invariants: 1 Topological: The topology of S and the curves γ i where Π vanishes. 2 Dynamical: The periods of the modular vector field along γ i. 3 Measure: The regularized Liouville volume of S. determines the Poisson structure on M. Eva Miranda (UPC) Poisson 2010 July 26, / 45

42 Another applications of this global Moser theorem Theorem (Guillemin-M.-Pires) Let ω t b Ω 2 (M), with 0 t 1, be a family of b-symplectic forms varying smoothly with t. If ω t Z and the cohomology class [ω t ] b H 2 (M) DR are independent of t, then all the ω t s are symplectomorphic. Eva Miranda (UPC) Poisson 2010 July 26, / 45

43 Integrable systems on b-poisson manifolds In the case of b-poisson manifolds, an integrable system is always splitted (in the sense of Weinstein and the integrable system) in a neighbourhood of Z. We then have, Theorem (Guillemin-M.-Pires) An integrable system on a b-poisson manifold of dimension 2n is equivalent to an integrable system with functions (f 1,..., f n ) where in a neighbourhood of the critical set Z: 1 The function f 1 can be chosen to be a defining function for Z. 2 The remaining first integrals (f 2,..., f n ) are functions on Z defining an integrable system on Z with respect to the restricted Poisson structure Π Z. Eva Miranda (UPC) Poisson 2010 July 26, / 45

44 Integrable systems on b-poisson manifolds Therefore most of the classical normal form results for symplectic manifolds also hold for b-poisson manifolds in a neighbourhood of points in Z. In particular, Theorem (Guillemin-M.-Pires) Given an integrable system with non-degenerate singularities on a b-poisson manifold (M, Z), there exists Eliasson-type normal forms in a neighbourhood of points in Z. Eva Miranda (UPC) Poisson 2010 July 26, / 45

45 Related projects 1 Group actions on b-poisson manifolds. 2 Integrability of b-poisson manifolds: The Lie algebroid T M has injective anchor map on an open dense set and therefore (Debord/Crainic-Fernandes) b-poisson manifolds are integrable. Find explicit models. Eva Miranda (UPC) Poisson 2010 July 26, / 45