Poisson manifolds of compact type David Mart nez Torres, IST Lisbon - PowerPoint PPT Presentation

Introduction and outline of results Basic properties of PMCT Integral affine structures and PMCT The non-regular case Construction of PMCT Poisson manifolds of compact type David Mart´ ınez Torres, IST Lisbon Joint work with M. Crainic and R. L. Fernandes David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Integral affine structures and PMCT The non-regular case Construction of PMCT ⊲ A Poisson manifold of compact type (PMCT) ( M , π ) is the Lie algebroid of a compact (Hausdorff) symplectic groupoid ( G , Ω) ⇒ ( M , π ). ⊲ A Poisson manifold of strong compact type (PMCT) is the Lie algebroid of a compact (Hausdorff) source 1-connected symplectic groupoid (Σ( M ) , Ω) ⇒ ( M , π ). ⊲ For an appropriate class of Poisson manifolds, is there a Poisson topology ? David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Integral affine structures and PMCT The non-regular case Construction of PMCT ⊲ Focus our attention on strong compact type Poisson manifolds: We do not know of other integrations of ( T ∗ M , [ · , · ] π ); if there are they might not be symplectic. We have an explicit (but complicated) model for Σ( M ) ⇒ M which will allow characterizing PMCT. We can “exponentiate” constructions from ( T ∗ M , [ · , · ] π ) to Σ( M ), but not to other integrations G ⇒ M , so we can draw more consequences from CT condition. David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Integral affine structures and PMCT The non-regular case Construction of PMCT Redefine Poisson manifold of compact type ∼ = integrable with compact source 1-connected integration David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Integral affine structures and PMCT The non-regular case Construction of PMCT ⊲ Goals of the talk Describe properties of PMCT: 1 Poisson cohomology behaves well, Poisson actions are Hamiltonian. Regular PMCT: Description of nearby regular Poisson structures, openness of integrable regular Poisson structures. David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Integral affine structures and PMCT The non-regular case Construction of PMCT ⊲ Goals of the talk Describe properties of PMCT: 1 Poisson cohomology behaves well, Poisson actions are Hamiltonian. Regular PMCT: Description of nearby regular Poisson structures, openness of integrable regular Poisson structures. There are no non-regular PMCT. 2 David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Integral affine structures and PMCT The non-regular case Construction of PMCT ⊲ Goals of the talk Describe properties of PMCT: 1 Poisson cohomology behaves well, Poisson actions are Hamiltonian. Regular PMCT: Description of nearby regular Poisson structures, openness of integrable regular Poisson structures. There are no non-regular PMCT. 2 If ( G , Ω) is a compact symplectic groupoid with 1-connected s -fibers, then G is regular. David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Integral affine structures and PMCT The non-regular case Construction of PMCT ⊲ Goals of the talk Describe properties of PMCT: 1 Poisson cohomology behaves well, Poisson actions are Hamiltonian. Regular PMCT: Description of nearby regular Poisson structures, openness of integrable regular Poisson structures. There are no non-regular PMCT. 2 If ( G , Ω) is a compact symplectic groupoid with 1-connected s -fibers, then G is regular. Present a construction of non-trivial (i.e. not symplectic) PMCT 3 related to quasi-Hamiltonian Abelian spaces. ⊲ Integral affine structures are the key tool for global results. David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Characteristic foliation and isotropy groups Integral affine structures and PMCT Cohomology The non-regular case Semi-local normal form Construction of PMCT ⊲ Generalize ( M , π ) PMCT in two directions: PMCT � � Proper Properties of Σ( M ): 1 ( M , π ) PMCT = ( M , π ) s -proper + M compact. Example (PMCT) ( S , ω ), S compact symplectic manifold with finite π 1 . Example ( s -proper PM) ( S × g , ω × π lin ), S compact symplectic manifold with finite π 1 , g semisimple of compact type. Example (Proper PM) ( S × g , ω × π lin ), S symplectic manifold with finite π 1 , g semisimple of compact type. Poisson � Dirac: DMCT ,... Examples above with ω closed. 2 David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Characteristic foliation and isotropy groups Integral affine structures and PMCT Cohomology The non-regular case Semi-local normal form Construction of PMCT ⊲ Generalize ( M , π ) PMCT in two directions: PMCT � s -proper � Proper Properties of Σ( M ): 1 ( M , π ) PMCT = ( M , π ) s -proper + M compact. Example (PMCT) ( S , ω ), S compact symplectic manifold with finite π 1 . Example ( s -proper PM) ( S × g , ω × π lin ), S compact symplectic manifold with finite π 1 , g semisimple of compact type. Example (Proper PM) ( S × g , ω × π lin ), S symplectic manifold with finite π 1 , g semisimple of compact type. Poisson � Dirac: DMCT ,... Examples above with ω closed. 2 David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Characteristic foliation and isotropy groups Integral affine structures and PMCT Cohomology The non-regular case Semi-local normal form Construction of PMCT ⊲ Generalize ( M , π ) PMCT in two directions: PMCT � s -proper � Proper Properties of Σ( M ): 1 ( M , π ) PMCT = ( M , π ) s -proper + M compact. Example (PMCT) ( S , ω ), S compact symplectic manifold with finite π 1 . Example ( s -proper PM) ( S × g , ω × π lin ), S compact symplectic manifold with finite π 1 , g semisimple of compact type. Example (Proper PM) ( S × g , ω × π lin ), S symplectic manifold with finite π 1 , g semisimple of compact type. Poisson � Dirac: DMCT ,... Examples above with ω closed. 2 David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Characteristic foliation and isotropy groups Integral affine structures and PMCT Cohomology The non-regular case Semi-local normal form Construction of PMCT ⊲ ( M , π ) s -proper, F characteristic foliation: Isotropy groups are compact. Leaves of F are compact with finite π 1 . M / F is Hausdorff. Use t (i) G x � s − 1 ( x ) − → F x , (ii) s − 1 ( x ) compact and 1-connected . ⊲ If ( M , π ) regular, then M / F admits orbifold structure (this will be true as well in the non-regular case). David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Characteristic foliation and isotropy groups Integral affine structures and PMCT Cohomology The non-regular case Semi-local normal form Construction of PMCT ⊲ ( M n , π ) s -proper: H ∗ π ( M ) = Γ( E ∗ ) Σ( M ) , E → M vector bundle, E x = H ∗ ( s − 1 ( x )). Poisson cohomology complex ≡ right invariant fiberwise (w.r.t. s ) differential forms. H 1 π ( M ) = 0 ⇒ ( M , π ) unimodular, Poisson actions are Hamiltonian. M orientable, Poincar´ e duality pairing: � � ( M ) → E k Σ( M ) × E n − k Σ( M ) s − 1( x ) M / F H k π ( M ) × H n − k C ∞ ( M / F ) → → R π M compact (PMCT), µ Hamiltonian invariant volume form, � ( i P ∧ Q µ ) µ, n ∈ N ∗ , ( P , Q ) �→ n M non-degenerate pairing. π ( M ) ∼ M orientable, H k = H n − k ( M ) ⇒ H n − 1 ( M ) = 0. π π M PMCT, [ π ] � = 0 ∈ H 2 π ( M ) [Crainic-Fernandes]. David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Characteristic foliation and isotropy groups Integral affine structures and PMCT Cohomology The non-regular case Semi-local normal form Construction of PMCT ⊲ ( M n , π ) s -proper: H ∗ π ( M ) = Γ( E ∗ ) Σ( M ) , E → M vector bundle, E x = H ∗ ( s − 1 ( x )). Poisson cohomology complex ≡ right invariant fiberwise (w.r.t. s ) differential forms. H 1 π ( M ) = 0 ⇒ ( M , π ) unimodular, Poisson actions are Hamiltonian. M orientable, Poincar´ e duality pairing: � � ( M ) → E k Σ( M ) × E n − k Σ( M ) s − 1( x ) M / F H k π ( M ) × H n − k C ∞ ( M / F ) → → R π M compact (PMCT), µ Hamiltonian invariant volume form, � ( i P ∧ Q µ ) µ, n ∈ N ∗ , ( P , Q ) �→ n M non-degenerate pairing. π ( M ) ∼ M orientable, H k = H n − k ( M ) ⇒ H n − 1 ( M ) = 0. π π M PMCT, [ π ] � = 0 ∈ H 2 π ( M ) [Crainic-Fernandes]. David Mart´ ınez Torres, IST Lisbon PMCT

Introduction and outline of results Basic properties of PMCT Characteristic foliation and isotropy groups Integral affine structures and PMCT Cohomology The non-regular case Semi-local normal form Construction of PMCT ⊲ Linearization around F leaf of ( M , π ), s -proper [Zung]: David Mart´ ınez Torres, IST Lisbon PMCT