Cohomology theories on locally conformally symplectic manifolds H - PowerPoint PPT Presentation

Cohomology theories on locally conformally symplectic manifolds Hˆ ong Vˆ an Lˆ e Institute of Mathematics of ASCR Zitna 25, 11567 Praha 1, Czech Republic Pacific Rim Geometry Conference, Osaka, December 2011

joint work with Jiˇ ri Vanˇ zura • Motivations • Primitive forms and primitive (co)homology • Primitive cohomology and Lichnerowicz- Novikov cohomology • Examples and historical backgrounds • Open problems

• I. Motivations A differentiable manifold ( M 2 n , ω, θ ) provided with a non-degenerate 2-form ω and a clo- sed 1-form θ is called a locally conformally symplectic (l.c.s.) manifold, if dω = − ω ∧ θ , dθ = 0. The 1-form θ is called the Lee form f and ω = e − f ω 0 , where of ω . Locally θ = d dω 0 = 0. L.c.s. forms were introduced by Lee, and have been extensively studied by Vaisman.

L.c.s. manifolds are phase spaces for a natu- ral generalization of Hamiltonian dynamics, mapping torus of a contactomorphism, sim- ple model for twisted symplectic geometry. They contain the subclass of L.C. K. mani- folds. The Lichnerowicz deformed differential d θ : Ω ∗ ( M 2 n ) → Ω ∗ ( M 2 n ) is defined by d θ ( α ) := dα + θ ∧ α. Note that d 2 θ = 0 and d θ ( ω ) = 0. The resulting Lichnerowicz cohomology groups,

(Novikov cohomology groups) are important conformal invariants of l.c.s. manifolds. Two l.c.s. forms ω and ω ′ on M 2 n are con- formally equivalent, if ω ′ = ± ( e f ) ω for some f ∈ C ∞ ( M 2 n ). In this case θ ′ = θ ∓ d f , hence d θ and d θ ′ are gauge equivalent : f ∧ ) α = e ± f d θ ( e ∓ f α ) . d θ ′ ( α ) = ( d θ ∓ d H ∗ (Ω ∗ ( M 2 n ) , d θ ) = H ∗ (Ω ∗ ( M 2 n ) , d θ ′ ) . Remark : By the Darboux theorem there is no local conformal invariant of l.c.s. ma- nifolds. AIM: construct new cohomological

invariants for l.c.s. manifolds. L : Ω ∗ ( M 2 n ) → Ω ∗ ( M 2 n ) , α �→ ω ∧ α. d θ L = Ld. d k := d kθ . d k L p = L p d k − p . I ω : T x M 2 n → T ∗ x M 2 n , V �→ i V ω . G ω ∈ Γ(Λ 2 TM 2 n ) s.t. i G ω I ω = Id , where x M 2 n → T x M 2 n , V �→ i V ( G ω ( x )) . i G ω : T ∗ ∗ ω : Ω p ( M 2 n ) → Ω 2 n − p ( M 2 n ) , β ∧ ∗ ω α := Λ p G ω ( β, α ) ∧ ω n n ! . ∗ 2 ω = Id. L ∗ : Ω p ( M 2 n ) → Ω p − 2 ( M 2 n ) , α p �→ − ∗ ω L ∗ ω α p .

ω : Ω p ( M 2 n ) → Ω p − 1 ( M 2 n ) , ( d k ) ∗ α p �→ ( − 1) p ∗ ω d n + k − p ∗ ω ( α p ) . π k : Ω ∗ ( M 2 n ) → Ω k ( M 2 n ) be the projection. L ∗ = i ( G ω ), [ L ∗ , L ] = A, [ A, L ] = − 2 L, [ A, L ∗ ] = 2 L ∗ . II Primitive forms and primitive (co)homology x M 2 n , 0 ≤ k ≤ n , is called primitive, α ∈ Λ k T ∗ if L n − k +1 α = 0. α ∈ Λ k T ∗ x M 2 n , n + 1 ≤ k ≤ 2 n , is called primitive, if α = 0. β ∈ Λ k T ∗ x M 2 n is called coeffective, if Lβ = 0.

x ( M 2 n ) : = the set of primitive elements P k in Λ k T ∗ x M 2 n . Lemma An element α ∈ Λ k T ∗ x M 2 n , is primi- tive, if and only if L ∗ α = 0. x M 2 n is coeffective, 2. An element β ∈ Λ k T ∗ if and only if ∗ ω β is primitive. x M 2 n = Lefschetz decomposition Λ n − k T ∗ 3. P n − k ( M 2 n ) ⊕ LP n − k − 2 ( M 2 n ) ⊕ L 2 P n − k − 4 ( M 2 n ) · · · , x x x x M 2 n = L k P n − k Λ n + k T ∗ ( M 2 n ) ⊕ L k +1 P n − k − 2 ( M 2 n ) · · · , x x for n ≥ k ≥ 0. 4. L k : Λ n − k T ∗ x M 2 n is an iso- x M 2 n → Λ n + k T ∗ morphism, for 0 ≤ k ≤ n .

x M 2 n → Λ n − k T ∗ x M 2 n is injec- 5. L : Λ n − k − 2 T ∗ tive, for k = − 1 , 0 , 1 , · · · , n − 2. p := (Ω ∗ ( M 2 n ) , d p ). K ∗ F 0 K ∗ p := K ∗ p ⊃ F 1 K ∗ p := LK ∗ p − 1 ⊃ · · · ⊃ F k K ∗ p := L k K ∗ p − k ⊃ · · · ⊃ F n +1 K ∗ p = { 0 } . d + k := Π pr d k : Ω q ( M 2 n ) → P q − 1 ( M 2 n ) . d k = d + k + Ld − k ,

d − k : Ω q ( M 2 n ) → Ω q − 1 ( M 2 n ) , 0 ≤ q ≤ n . ( d + k ) 2 ( α q ) = 0 , d − k − 1 d − k ( α q ) = 0 , q ≤ n, k ) α q = 0 , q ≤ n − 1 , k d + k + d + ( d − k − 1 d − ( d k − 1 ) ∗ ω ( d k ) ∗ ω ( α q ) = 0 . Assume that 0 ≤ q ≤ n − 1. k ) := ker d + k ∩ P q ( M 2 n ) H q ( P ∗ ( M 2 n ) , d + . d + k ( P q − 1 ( M 2 n )) ker( d k ) ∗ ω ∩ P q ( M 2 n ) H q ( P ∗ ( M 2 n ) , ( d k ) ∗ ω ) := ω ( P q +1 ( M 2 n )) . ( d k +1 ) ∗

k ) := ker d − k ∩ P q ( M 2 n ) H q ( P ∗ ( M 2 n ) , d − . d − k +1 ( P q +1 ( M 2 n )) Proposition Assume dim( M 2 n , ω, θ ) ≥ 2. 1. If [( k − 1) θ ] � = 0 ∈ H 1 ( M 2 n , R ) then H 1 ( P ∗ ( M 2 n ) , d + k ) = H 1 (Ω ∗ ( M 2 n ) , d k ) . 2. If [( k − 1) θ ] = 0 ∈ H 1 ( M 2 n , R ) then H 1 ( P ∗ ( M 2 n ) , d + k ) = H 1 (Ω ∗ ( M 2 n ) , d θ ) if [ ω ] � = 0 ∈ H 2 (Ω ∗ ( M 2 n ) , d θ ) H 1 ( P ∗ ( M 2 n ) , d + k ) = H 1 (Ω ∗ ( M 2 n ) , d θ ) ⊕ R

if [ ω ] = 0 ∈ H 2 (Ω ∗ ( M 2 n ) , d θ ) . Proposition Assume that 0 ≤ k ≤ n . If α ∈ P k ( M 2 n ), then for all l l ( α k ) = ( d l ) ∗ ω ( α k ) d − n − k + 1 . Hence H k ( P ∗ ( M 2 n ) , d − l ) = H k ( P ∗ ( M 2 n ) , ( d l ) ∗ ω ). Proposition Let ( M 2 n , ω, θ ) be a compact l.c.s manifold. Then H k ( P ∗ ( M 2 n ) , d + l ) = H k ( P ∗ ( M 2 n ) , ( d − l + k − n ) ∗ ω ) for all l and 0 ≤ k ≤ n − 1.

III The relations between primitive coho- mology and Lichnerowicz-Novikov coho- mology The spectral sequence { E p,q k,r , d k,r : E p,q k,r → E p + r,q − r +1 } , r ≥ 0, is associated to the fil- k,r tration ( F ∗ K ∗ k , d k ). k, 0 ∼ E p,q = P q − p ( M 2 n ) if n ≥ q ≥ p E p,q k, 0 = 0 otherwise . E p,q k, 1 = H q − p ( P ∗ ( M 2 n ) , d + k − p ) if 0 ≤ p ≤ q ≤ n − 1 ,

P n − p ( M 2 n ) E p,n k, 1 = , if 0 ≤ p ≤ n, d + k − p ( P n − p − 1 ( M 2 n )) E p,q k, 1 = 0 otherwise . d l + p, 1 : E p,q l + p, 1 → E p +1 ,q l + p, 1 is defined for 0 ≤ p ≤ q ≤ n by H q − p ( P ∗ ( M 2 n ) , d + l ) → H q − p − 1 ( P ∗ ( M 2 n ) , d + l − 1 ) , α ] �→ [ d − [˜ l ˜ α ] . Corollary Assume that 1 ≤ p ≤ q ≤ n − 1. Then E p,q l, 2 = E p − 1 ,q − 1 . l, 2

Theorem The spectral sequences E p,q k,r on ( M 2 n , ω, θ ) and on ( M 2 n , ω ′ , θ ′ ) are isomor- phic, if ω and ω ′ are conformal equivalent. Furthermore, the E k, 1 -terms of the spectral sequences on ( M, ω, θ ) and ( M, ω ′ , θ ′ ) are iso- morphic, if ω ′ = ω + d θ ρ for some ρ ∈ Ω 1 ( M 2 n ). Theorem Assume that ω = d 1 τ . ( M 2 n ) ⊕ H q − p − 1 1. E p,q l + p, 1 = H q − p ( M 2 n ) for l − 1 l 0 ≤ p ≤ q ≤ n − 1. 2. E p,q l, 2 = 0, if 1 ≤ p ≤ q ≤ n − 1. 3. If 0 ≤ q ≤ n , then E 0 ,q l, 2 = H q l ( M 2 n ).

l + p, 2 = H n + p 4. If 0 ≤ p ≤ n then E p,n l + p ( M 2 n ). 5. The spectral sequence { E p,q l,r , d l,r } stabili- zes at the term E l, 2 . l := ker d − l ∩ Ω k ( M 2 n ) C k d l (Ω k − 1 ( M 2 n )) . Lemma For 0 ≤ p ≤ q ≤ n − 1 the following sequences is exact 0 → (Ω q − ( p +1) ( M 2 n ) , d l − 1 ) L → (Ω q +1 − p ( M 2 n ) , d l ) → Π L p → ( E p,q +1 l + p, 0 , d l + p, 0 ) → 0 .

L p ¯ δ p,q · · · → H q − p → H q − ( p +1) → E p,q ( M 2 n ) ( M 2 n ) → l + p, 1 l − 1 l ¯ L p ¯ → H q +1 − p L → E p,q +1 ( M 2 n ) l + p, 1 → · · · l δ p,n − 1 ( M 2 n ) [ L ] · · · → E p,n − 1 H n − ( p +2) → C n − p → → . l + p, 1 l − 1 l If moreover ω = d 1 τ the following sequences are exact L p ¯ l + p, 1 → H q − ( p +1) 0 → H q − p → E p,q ( M 2 n ) ( M 2 n ) → 0 , l − 1 l → E p − 1 ,q l + p, 2 → H q − p ( M 2 n ) δ → H q − p ( M 2 n ) → l l l + p, 2 → H q − ( p +1) ( M 2 n ) δ → E p,q → . l − 1

For 0 ≤ p ≤ n − 1 we have [ L p ] δ p,n 0 → C n − p → T n − ( p +1) → E p,n → 0 , l + p, 1 l − 1 l T n − ( p +1) := ker[ L p +1 ] : C n − ( p +1) → H n + p +1 . l − 1 l − 1 l + p Theorem Assume that ω T = d T ρ and T ≥ 2. Then the spectral sequence ( E p,q l,r , d l,r ) stabi- lizes at terms E ∗ , ∗ l,T +1 . The main idea is to find a short exact se- quence, whose middle term is E ∗ , ∗ l,T , and mo- reover, this short exact sequence is induced

from the trivial action of the operator L T on (a part of) complexes entering in the derived exact couples. Theorem Assume that ( M 2 n , ω, θ ) is a com- pact connected globally conformally symplec- tic manifold. Then the spectral sequence ( E p,q k,r , d k,r ) stabilizes at the E ∗ , ∗ k, 2 -term. The main idea: For symplectic manifolds ( M 2 n , ω ) the term E p,p k , 0 ≤ p ≤ 1 and k ≥ 1, is generated by ω p , which acts on E 0 ,r injec- k tively, if p + r ≤ n .

IV Examples and historical backgrounds • For θ = 0 there is known construction of coeffective cohomology groups (Bouche, Fernandez, De Leon) which are dual to the primitive cohomology groups (Tseng-Yau) via the the symplectic star operators. • There is a compact 6-dimensional nilmani- fold M 6 equipped with a family of symplectic forms ω t , t ∈ [0 , 1] , with varying cohomo- logy classes [ ω t ] ∈ H 2 ( M 6 , R ). Fernandez at all. showed that the coeffective cohomology