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Poisson manifolds of compact types (PMCT 1)

Marius Crainic, Rui Loja Fernandes and David Martínez Torres

Abstract

This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.

Funding statement: MC and DMT were partially supported by the ERC Starting Grant no. 279729. RLF was partially supported by NSF grants DMS 13-08472 and DMS 14-05671, and by the CNPq Ciências Sem Fronteiras Grant no. 401817/2013-0.

A Linearization of proper symplectic groupoids

In this appendix we give a proof of the following symplectic version of the Zung–Weinstein linearization theorem, which is needed for the proof of the symplectic groupoid neighborhood equivalence theorem (Theorem 8.2):

Proposition A.1.

Let S be an orbit of a proper symplectic groupoid (G,Ω)M. Then one can choose a 2-metric η on G, a principal connection θ on t:s-1(x)S, and a groupoid automorphism Φ:ν(GS)ν(GS), such that the composition

expηΦ:ν(𝒢S)|V𝒢|U

is a groupoid isomorphism and satisfies

((expηΦ)*Ω)|g=Ωlinθ|gfor all g𝒢S.

We will not go into details on 2-metrics on a groupoid 𝒢M. For our purpose, it suffices to know that it amounts to a metric η(0) on the objects M, a metric η(1) on the arrows 𝒢 and a metric η(2) on the composable arrows 𝒢(2) which are compatible with all structure maps. The details can be found in [17]. We will focus our attention on the metric on the arrows η(1), which we write simply as η. As we mentioned above, the main property of a 2-metric is that the exponential map exp:ν(GS)𝒢 is a groupoid morphism.

Recall that a multiplicative distribution in a groupoid 𝒢M is, by definition, a subgroupoid 𝒟D of the tangent groupoid T𝒢TM. We start by remarking that for a proper groupoid, if one is given a multiplicative distribution 𝒟D in T𝒢S𝒢TSM complementary to T𝒢STS, then we can choose a 2-metric which is adapted to the decomposition T𝒢S𝒢=T𝒢S𝒟.

Lemma A.2.

If S is an orbit of a proper Lie groupoid (G,Ω)M and DD in TGSGTSM is a multiplicative distribution complementary to TGSTS, then we can choose a 2-metric in G such that

𝒟=(T𝒢S),D=(TS).

Proof.

The existence of a 2-metric on a proper groupoid is proved in [17] by an averaging procedure. Let us indicate how choices should be made so that the average procedure yields a 2-metric with the desired property.

First, one needs to fix an Ehresmann connection for the source fibration, i.e., to choose a splitting of the short exact sequence

Such a choice allows one to lift any groupoid action θ:𝒢E to a quasi action Tσθ:𝒢TE. We choose a splitting σ:s*TMT𝒢 as follows: first, we restrict our attention to the subgroupoid 𝒢S and choose an Ehresmann connection H for the source map. Then we observe that H𝒟 yields an Ehresmann connection for the source map over the points of S. Such connection over S always extends to a global one. Its corresponding splitting preserves decompositions along 𝒢S:

σ(s*TS)T𝒢S,σ(s*D)𝒟.

Second, we need to choose a metric η on 𝒢 for which the source map s:𝒢M becomes a Riemannian submersion. We choose η so that along S we have additionally

𝒟=(T𝒢S),D=(TS).

Again, this is possible because ds(T𝒢S)=TS and ds(𝒟)=D.

Then, we proceed as in [17]. On the fiber product

𝒢[n]:={(g1,,gn+1)s(g1)==s(gn)},

we consider the restriction of the product metric ηη. The groupoid 𝒢 acts (on the right) on 𝒢[n] in a proper and free fashion and the quotient is 𝒢(n). If we average the product metric on 𝒢[n], using the lifted tangent action, we obtain a metric that descends to the quotient 𝒢(n). The resulting metrics on M=𝒢(0), 𝒢=𝒢(1) and 𝒢(2) give a 2-metric. The important remark is that because 𝒟 and TGS are multiplicative distributions, averaging does not alter the orthogonal decomposition:

𝒟=(T𝒢S),D=(TS).

Let (𝒢,Ω)M be a proper symplectic Lie groupoid with orbit S. Note that 𝒢S is a coisotropic submanifold and that the restriction of the symplectic form to 𝒢S has kernel

K=Ker(Ω|𝒢S)=Ker(ds|T𝒢S)Ker(dt|T𝒢S).

This is a multiplicative distribution. We will need a version of the standard coisotropic neighborhood theorem which is valid in our multiplicative setting.

Let us start by recalling first the symplectic linear algebra statement, which we adapt here to the case of symplectic vector bundles:

Lemma A.3.

Let (VQ,Ω) be a symplectic vector bundle and assume that there are sub-bundles C,E,LV such that

  1. (i)

    CQ is a coisotropic sub-bundle;

  2. (ii)

    C=KE, where K=Ker(Ω|C);

  3. (iii)

    EΩ=KL, where LV is a Lagrangian sub-bundle.

Then there is a canonical isomorphism of symplectic vector bundles

A:(V,Ω)(EKK*,Ω|E+Ωcan),

where Ωcan denotes the standard symplectic form on KK*.

Note that Ω|E is symplectic. Hence, one has the direct sum decomposition

V=CL=EKL.

The bundle isomorphism A is obtained by combining this direct sum decomposition with the isomorphism IΩ:LK*, v(ivΩ)|K. In other words, it is the unique isomorphism that makes the following diagram commute:

We now turn to multiplicative versions of these results:

Lemma A.4.

Let (G,Ω)M be a proper symplectic Lie groupoid with orbit S. Then one can choose multiplicative distributions E,LTGSG such that

  1. (i)

    T𝒢S=KE, where K=Ker(Ω|GS) and E is a symplectic sub-bundle;

  2. (ii)

    EΩ=KL, where L is a Lagrangian sub-bundle.

Proof.

We can assume that the groupoid has already be linearized. Hence, we look at a multiplicative symplectic form Ω in the groupoid ν(GS)ν(S).

Fixing a point xS, our groupoid can also be identified with

ν(𝒢S)(s-1(x)×s-1(x)×νx(S))/Gx.

In other words, ν(𝒢S)ν(S) is a quotient by a proper and free Gx-action by groupoid automorphisms of the pair groupoid:

s-1(x)×s-1(x)×νx(S)s-1(x)×νx(S).

One now checks easily that a multiplicative form Ω on ν(𝒢S)ν(S) corresponds to a Gx-basic multiplicative closed 2-form

Ω~=pr1*ω¯-pr2*ω¯Ω2(s-1(x)×s-1(x)×νx(S)),

where ω¯Ω2(s-1(x)×νx(S)) is a Gx-basic form on s-1(x)×νx(S).

In general, it is difficult to produce multiplicative distributions in a groupoid. But, in our case, it is easy to check that any choice of a Gx-invariant distribution D¯ in s-1(x)×νx(S) gives rise to a multiplicative Gx-invariant distribution D~ in s-1(x)×s-1(x)×νx(S), by setting

D~:={(v1,v2,w)T(s-1(x)×s-1(x)×νx(S))(v1,w),(v2,w)D}.

The quotient D:=D~/G is a multiplicative distribution on ν(𝒢S)ν(S). Two instances of this are as follows:

  1. (i)

    The restriction of the symplectic form Ω to 𝒢S has kernel the multiplicative distribution:

    K=Ker(Ω|𝒢S)=Ker(ds|T𝒢S)Ker(dt|T𝒢S).

    This multiplicative distribution corresponds to the Gx-invariant distribution in

    s-1(x)s-1(x)×{0}

    defined by the tangent spaces to the Gx-orbits: K¯=Kerdt, where t:s-1(x)S.

  2. (ii)

    For any principal bundle connection θ on t:s-1(x)S, the horizontal space E¯ of θ determines a multiplicative distribution E on 𝒢S.

Notice that for any choice of θ the multiplicative distribution E is complementary to K:

T𝒢S=KE.

So ET𝒢S𝒢 is a symplectic sub-bundle. Hence, to finish the proof of the lemma, it remains to exhibit a Lagrangian multiplicative distribution LT𝒢S𝒢 such that

EΩ=KL.

For this, we choose a Gx-invariant distribution

L¯Ts-1(x)(s-1(x)×νx(S)),

satisfying

E¯ω¯=K¯L¯

and such that ω¯ vanishes along L¯. Such a distribution can be obtained by a standard averaging argument, using the fact that ω¯ is Gx-invariant and has kernel K¯. ∎

Proof of Proposition A.1.

Let (𝒢,Ω)M be a proper symplectic Lie groupoid with orbit S. By Lemma A.4, we can choose a principal bundle connection θ on t:s-1(x)S, so that its horizontal space determines a multiplicative distribution E on 𝒢S complementary to K:

T𝒢S=KE.

Moreover, we can choose a Lagrangian multiplicative distribution LT𝒢S𝒢 such that

EΩ=KL.

In particular,

T𝒢S𝒢=T𝒢SL.

By Lemma A.2, we can choose a 2-metric η on 𝒢 such that L=(T𝒢S). The exponential map of this 2-metric gives a linearization map:

expη:ν(𝒢S)|V𝒢|U,

where V and U are neighborhoods of 𝒢S in ν(𝒢S) and 𝒢, respectively. Obviously, the differential of this map at a point g𝒢S gives an isomorphism:

dgexpη:Tg𝒢Sνg(𝒢S)Tg𝒢S(Tg𝒢S)=Tg𝒢SLg.

On the other hand, the connection θ determines a closed 2-form Ωlinθ on ν(𝒢S) whose pullback to 𝒢S coincides with the pullback of Ω (see Section 5.8). It follows from the explicit expression of Ωlinθ that in the direct sum decomposition

T𝒢S𝒢=EKL=EKν(GS),

E is symplectic, while K and L=ν(GS) are Lagrangian sub-bundles, for both Ωlinθ and (expη)*Ω.

Now Lemma A.3 gives a vector bundle automorphism Φ:ν(GS)ν(GS) such that

(Φ)*(expη)*Ω|g=Ωlinθ|gfor all g𝒢S.

In the notation above,

Φ=I(expη)*Ω(IΩlinθ)-1.

Since the forms (expη)*Ω and Ωlinθ are multiplicative, it follows that Φ:ν(GS)ν(GS) is a groupoid automorphism.

This completes the proof of Proposition A.1. ∎

References

[1] A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), no. 3, 445–495. Search in Google Scholar

[2] O. Brahic and R. L. Fernandes, Integrability and reduction of Hamiltonian actions on Dirac manifolds, Indag. Math. (N.S.) 25 (2014), no. 5, 901–925. Search in Google Scholar

[3] J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114. Search in Google Scholar

[4] H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J. 123 (2004), no. 3, 549–607. Search in Google Scholar

[5] H. Bursztyn and O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble) 53 (2003), 309–337. Search in Google Scholar

[6] E. Cartan, La topologie des groupes de Lie, Actualités Scient. Industr. 358, Hermann, Paris 1936. Search in Google Scholar

[7] A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, Quantization of singular symplectic quotients, Progr. Math. 198, Birkhäuser, Basel (2001), 61–93.Search in Google Scholar

[8] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631–661. Search in Google Scholar

[9] M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), no. 4, 681–721. Search in Google Scholar

[10] M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620. Search in Google Scholar

[11] M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Differential Geom. 66 (2004), no. 1, 71–137. Search in Google Scholar

[12] M. Crainic and R. L. Fernandes, Rigidity and flexibility in Poisson geometry, Proceedings of the 4th conference on Poisson geometry (Luxembourg 2004), Trav. Math. 16, Université du Luxembourg, Luxembourg (2005), 53–68. Search in Google Scholar

[13] M. Crainic and R. L. Fernandes, Lectures on integrability of Lie brackets, Geom. Topol. Monogr. 17 (2011), 1–107. Search in Google Scholar

[14] M. Crainic, R. L. Fernandes and D. Martínez Torres, Regular Poisson manifolds of compact types (PMCT 2), preprint (2016), . Search in Google Scholar

[15] M. Crainic, R. L. Fernandes and D. Martínez Torres, Singular Poisson manifolds of compact types (PMCT 3), in progress. Search in Google Scholar

[16] M. Crainic and I. Struchiner, On the linearization theorem for proper Lie groupoids, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 5, 723–746. Search in Google Scholar

[17] M. del Hoyo and R. L. Fernandes, Riemannian metrics on Lie groupoids, J. reine angew. Math. (2015), 10.1515/crelle-2015-0018. 10.1515/crelle-2015-0018Search in Google Scholar

[18] J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), no. 2, 259–268. Search in Google Scholar

[19] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer, Berlin 2000. Search in Google Scholar

[20] S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 417–436. Search in Google Scholar

[21] M. Fernández, A. Gray and J. W. Morgan, Compact symplectic manifolds with free circle actions, and Massey products, Michigan Math. J. 38 (1991), no. 2, 271–283. Search in Google Scholar

[22] R. L. Fernandes, J.-P. Ortega and T. S. Ratiu, The momentum map in Poisson geometry, Amer. J. Math. 131 (2009), no. 5, 1261–1310. Search in Google Scholar

[23] P. Frejlich and I. Marcut, The normal form theorem around Poisson transversals, preprint (2013), . Search in Google Scholar

[24] V. L. Ginzburg and A. Weinstein, Lie–Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc. 5 (1992), no. 2, 445–453. Search in Google Scholar

[25] M. Gualtieri and S. Li, Symplectic groupoids of log symplectic manifolds, Int. Math. Res. Not. IMRN 2014 (2014), no. 11, 3022–3074. Search in Google Scholar

[26] V. Guillemin, E. Lerman and S. Sternberg, Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge 1996. Search in Google Scholar

[27] V. Guillemin, E. Miranda and A. R. Pires, Symplectic and Poisson geometry on b-manifolds, Adv. Math. 264 (2014), 864–896. Search in Google Scholar

[28] D. Kotschick, Free circle actions with contractible orbits on symplectic manifolds, Math. Z. 252 (2006), no. 1, 19–25. Search in Google Scholar

[29] J.-H. Lu and A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526. Search in Google Scholar

[30] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, Cambridge 1987. Search in Google Scholar

[31] I. Mǎrcut, Deformations of the Lie–Poisson sphere of a compact semisimple Lie algebra, Compos. Math. 150 (2014), no. 4, 568–578. Search in Google Scholar

[32] D. Martínez Torres, A Poisson manifold of strong compact type, Indag. Math. (N.S.) 25 (2014), no. 5, 1154–1159. Search in Google Scholar

[33] D. McDuff, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), no. 2, 149–160. Search in Google Scholar

[34] D. McDuff and D. Salamon, Introduction to symplectic topology, 2nd ed., Oxford University Press, New York 1998. Search in Google Scholar

[35] K. Mikami and A. Weinstein, Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci. 24 (1988), no. 1, 121–140. Search in Google Scholar

[36] Y. Mitsumatsu, Leafwise symplectic structures on Lawson’s foliation, preprint (2011), . Search in Google Scholar

[37] I. Moerdijk and J. Mrcun, Introduction to foliations and Lie groupoids, Cambridge University Press, Cambridge 2003.Search in Google Scholar

[38] P. Ševera and A. Weinstein, Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl. 144 (2001), 145–154. Search in Google Scholar

[39] A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 1, 101–104. Search in Google Scholar

[40] A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), no. 3–4, 379–394. Search in Google Scholar

[41] A. Weinstein, Linearization of regular proper groupoids, J. Inst. Math. Jussieu 1 (2002), no. 3, 493–511. Search in Google Scholar

[42] N.-T. Zung, Proper groupoids and momentum maps: Linearization, affinity, and convexity, Ann. Sci. École Norm. Supér. (4) 39 (2006), no. 5, 841–869. Search in Google Scholar

Received: 2016-03-28
Revised: 2017-01-20
Published Online: 2017-04-06
Published in Print: 2019-11-01

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