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):
Let S be an orbit of a proper symplectic groupoid . Then one can choose a 2-metric η on , a principal connection θ on , and a groupoid automorphism , such that the composition
is a groupoid isomorphism and satisfies
We will not go into details on 2-metrics on a groupoid . For our purpose, it suffices to know that it amounts to a metric on the objects M, a metric on the arrows and a metric on the composable arrows which are compatible with all structure maps. The details can be found in . We will focus our attention on the metric on the arrows , which we write simply as η. As we mentioned above, the main property of a 2-metric is that the exponential map is a groupoid morphism.
Recall that a multiplicative distribution in a groupoid is, by definition, a subgroupoid of the tangent groupoid . We start by remarking that for a proper groupoid, if one is given a multiplicative distribution in complementary to , then we can choose a 2-metric which is adapted to the decomposition .
If S is an orbit of a proper Lie groupoid and in is a multiplicative distribution complementary to , then we can choose a 2-metric in such that
The existence of a 2-metric on a proper groupoid is proved in  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 to a quasi action . We choose a splitting as follows: first, we restrict our attention to the subgroupoid and choose an Ehresmann connection H for the source map. Then we observe that 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 :
Second, we need to choose a metric η on for which the source map becomes a Riemannian submersion. We choose η so that along S we have additionally
Again, this is possible because and .
Then, we proceed as in . On the fiber product
we consider the restriction of the product metric . The groupoid acts (on the right) on in a proper and free fashion and the quotient is . If we average the product metric on , using the lifted tangent action, we obtain a metric that descends to the quotient . The resulting metrics on , and give a 2-metric. The important remark is that because and are multiplicative distributions, averaging does not alter the orthogonal decomposition:
Let be a proper symplectic Lie groupoid with orbit S. Note that is a coisotropic submanifold and that the restriction of the symplectic form to has kernel
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:
Let be a symplectic vector bundle and assume that there are sub-bundles such that
is a coisotropic sub-bundle;
, where ;
, where is a Lagrangian sub-bundle.
Then there is a canonical isomorphism of symplectic vector bundles
where denotes the standard symplectic form on .
Note that is symplectic. Hence, one has the direct sum decomposition
The bundle isomorphism A is obtained by combining this direct sum decomposition with the isomorphism , . In other words, it is the unique isomorphism that makes the following diagram commute:
We now turn to multiplicative versions of these results:
Let be a proper symplectic Lie groupoid with orbit S. Then one can choose multiplicative distributions such that
, where and E is a symplectic sub-bundle;
, where L is a Lagrangian sub-bundle.
We can assume that the groupoid has already be linearized. Hence, we look at a multiplicative symplectic form Ω in the groupoid .
Fixing a point , our groupoid can also be identified with
In other words, is a quotient by a proper and free -action by groupoid automorphisms of the pair groupoid:
One now checks easily that a multiplicative form Ω on corresponds to a -basic multiplicative closed 2-form
where is a -basic form on .
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 -invariant distribution in gives rise to a multiplicative -invariant distribution in , by setting
The quotient is a multiplicative distribution on . Two instances of this are as follows:
The restriction of the symplectic form Ω to has kernel the multiplicative distribution:
This multiplicative distribution corresponds to the -invariant distribution in
defined by the tangent spaces to the -orbits: , where .
For any principal bundle connection θ on , the horizontal space of θ determines a multiplicative distribution E on .
Notice that for any choice of θ the multiplicative distribution E is complementary to K:
So is a symplectic sub-bundle. Hence, to finish the proof of the lemma, it remains to exhibit a Lagrangian multiplicative distribution such that
For this, we choose a -invariant distribution
and such that vanishes along . Such a distribution can be obtained by a standard averaging argument, using the fact that is -invariant and has kernel . ∎
Proof of Proposition A.1.
Let be a proper symplectic Lie groupoid with orbit S. By Lemma A.4, we can choose a principal bundle connection θ on , so that its horizontal space determines a multiplicative distribution E on complementary to K:
Moreover, we can choose a Lagrangian multiplicative distribution such that
By Lemma A.2, we can choose a 2-metric η on such that . The exponential map of this 2-metric gives a linearization map:
where V and U are neighborhoods of in and , respectively. Obviously, the differential of this map at a point gives an isomorphism:
On the other hand, the connection θ determines a closed 2-form on whose pullback to coincides with the pullback of Ω (see Section 5.8). It follows from the explicit expression of that in the direct sum decomposition
E is symplectic, while K and are Lagrangian sub-bundles, for both and .
Now Lemma A.3 gives a vector bundle automorphism such that
In the notation above,
Since the forms and are multiplicative, it follows that is a groupoid automorphism.
This completes the proof of Proposition A.1. ∎
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