Harmonic cohomology of symplectic manifolds Stefan Haller - PDF document

Harmonic cohomology of symplectic manifolds Stefan Haller University of Vienna Krynica, April 2003

( M, Λ) Poisson manifold. δ : Ω ∗ ( M ) → Ω ∗− 1 ( M ) , δα := i Λ dα − di Λ α. Then δ 2 = 0, but ∆ := dδ + δd = 0. Call α ∈ Ω ∗ ( M ) harmonic if dα = δα = 0. For ( M 2 n , ω ) symplectic, Libermann introduced ∗ : Ω n − k ( M ) → Ω n + k ( M ) Then ∗ 2 = 1 and δ = ± ∗ d ∗ . Question. [Brylinski] Does every cohomol- ogy class have a harmonic representative? Thm. [Brylinski] Yes, for K¨ ahler ( M, ω ) . Thm. [Mathieu] Yes, iff ( M 2 n , ω ) Lefschetz, i.e. [ ω ] k : H n − k ( M ) → H n + k ( M ) onto ∀ k ≥ 0. Question. Which cohomology classes have harmonic representatives? 1

Put H ∗ hr ( M ) ⊆ H ∗ ( M ) , subspace of harmonic cohomology classes. b k hr := b k hr ( M ) := dim H k hr ( M ) , harmonic Betti numbers. • Compute b k hr or even H ∗ hr ( M )! • How much does H ∗ hr ( M ) depend on ω ? • When do we have f : M 1 → M 2 ⇒ f ∗ : H ∗ hr ( M 2 ) → H ∗ hr ( M 1 )? • What about H ∗ hr ( M 1 × M 2 )? e duality for H ∗ • Which kind of Poincar´ hr ( M )? 2

Def. For m = 0: Z k 0 space of harmonic forms α ∈ Ω k ( M ), dα = δα = 0: 0 δ ← α d → 0 For m > 0: Z k m ⊆ Ω k ( M ) space of α ∈ Ω k ( M ), s.t. dα = 0 and s.t. ∃ α j ∈ Ω k − 2 j ( M ), 1 ≤ j ≤ m , with δα = dα 1 , δα j = dα j +1 , 1 ≤ j ≤ m − 1 and δα m = 0: 0 δ → · · · δ d → δ d → δ d ← α d ← α m ← α 2 ← α 1 → 0 For m < 0: Z k m space of α ∈ Ω k ( M ), s.t. ∃ α j ∈ Ω k +2 j − 1 ( M ), 1 ≤ j ≤ − m , with α = δα 1 , dα j = δα j +1 , 1 ≤ j ≤ − m − 1: α δ → δ d → · · · δ d → δ d ← α 1 ← α 2 ← α − m − 1 ← α − m Finally set Z k m H k m ∩ img d ⊆ H k ( M ) , m ( M ) := Z k those classes in H k ( M ) having representatives in Z k m . 3

Have · · · ⊆ Z ∗ m ⊆ Z ∗ m +1 ⊆ · · · and · · · ⊆ H ∗ m ( M ) ⊆ H ∗ m +1 ( M ) ⊆ · · · filtration of H ∗ ( M ). Set b k m := dim H k m ( M ) . Note that H ∗ hr ( M ) = H ∗ 0 ( M ) and b k hr = b k 0 . Def. H ∗ m ( M ) := H ∗ m ( M ) /H ∗ ˜ m − 1 ( M ) . b k H k ˜ m := dim ˜ m ( M ) 4

Thm. M 2 n symplectic manifold. Then 1. H ∗ m ( M ) does only depend on [ ω ] ∈ H ∗ ( M ) . 2. f : M 2 n 1 → M 2 n 2 , f ∗ [ ω 2 ] = [ ω 1 ] . Then 1 2 f ∗ : H ∗ m − n 2 ( M 2 ) → H ∗ m − n 1 ( M 1 ) . 3. H k < 0 , and H k m ( M ) = 0 for k < m ( M ) = H k ( M ) for k > > 0 . 4. [ ω ] k : ˜ H n + m − k H n + m + k ( M ) → ˜ ( M ) is an m m b n + m − k b n + m + k isomorphism, ˜ = ˜ . m m 5. If H ∗ ( M ) finite dimensional we set ρ i j := [ ω ] j : H i − 2 j ( M ) → H i ( M ) � � rank and get b n + m − k − b n + m − k = ρ n + m + k − b n + m + k m m ρ n + m + k +2 l − ρ n + m + k +2 l � = . k +2 l − 1 k +2 l l ≥ 1 5

Thm. Suppose M 2 n symplectic manifold and m ∈ Z . Then the following are equivalent: 1. H ∗ m ( M ) = H ∗ ( M ) . 2. [ ω ] k : H n + m − k ( M ) → H n + m + k ( M ) is onto for all k ≥ 0 . Particularly H ∗ 0 ( M ) = H ∗ ( M ) iff M Lefschetz [Mathieu]. M 2 n closed symplectic manifold and Thm. m, k ∈ Z . Then the well defined bilinear pairing � H n − k H n + k ˜ − m ( M ) ⊗ ˜ ( M ) → R , ([ α, β ]) := M α ∧ β m is non-degenerate. Moreover if n even � H n H n � ˜ 0 ( M ) ⊗ ˜ 0 ( M ) → R sign( M ) = sign . 6

Thm. M 2 n closed symplectic manifold. Then the well defined bilinear pairing � H k H k ˜ 0 ( M ) ⊗ ˜ 0 ( M ) → R , � � [ α ] , [ β ] � � := M α ∧ ∗ β is non-degenerate. It is symmetric for k even and skew symmetric for k odd. Particularly b k ˜ 0 ( M ) is even for odd k . Prop. Suppose M 1 and M 2 symplectic mani- folds with finite dimensional cohomology. Then p M 1 × M 2 p M 1 p M 2 � ˜ ( t ) = ˜ m 1 ( t ) · ˜ m 2 ( t ) , m m 1 + m 2 = m where p M b k m ( M ) t k . � ˜ ˜ m ( t ) := 7

Consider the Lie algebras: g := sl (2 , R ) = � e, f, h � g ⊇ b := � e, h � g ⊇ b ⊇ h := � h � Standard generators and relations: [ h, e ] = 2 e [ h, f ] = − 2 f [ e, f ] = h Def. V h category of h –modules s.t. V k := { v ∈ V : h · v = kv } V k � V = k ∈ Z only finitely many V k � = 0. V b resp. V g category of b resp. g –modules with underlying h –module in V h . Def. For V ∈ V b and k ∈ Z define V [ k ] ∈ V b by V [ k ] := V as vector space and h · v := hv + kv e · v = ev 8

Lemma. V, W ∈ V g , ϕ : V → W a b –module homomorphism. Then ϕ is a g –module homo- morphism. Prop. [Mathieu] Let V ∈ V b . Then there exists unique filtration of V by b –submodules · · · ⊆ V m ⊆ V m +1 ⊆ · · · s.t. V m = 0 < 0 , m < V m = V m > > 0 , ∀ m ∈ Z . ( V m /V m − 1 )[ − m ] ∈ V g Moreover, as b -modules � ( V m /V m − 1 ) V ≃ m ∈ Z but not canonically. A b –module homomorphism ϕ : V → W is fil- tration preserving: ϕ ( V m ) ⊆ W m 9

Ex. R 2 ∈ V g standard g –representation. Then �� � S k j,l R 2 � V = [ l ] ∈ V b j l with filtration �� � S k j,l R 2 � V m = [ l ] . l ≤ m j [Mathieu] For V ∈ V b and m ∈ Z the Prop. following are equivalent: (i) V m = V e k : V m − k → V m + k is onto ∀ k ≥ 0 (ii) 10

M topological space, ω ∈ H 2 ( M ) Main Ex. and suppose H k ( M ) = 0 for k > > 0. Then V := H ∗ ( M ) ∈ V b via e · α := ω ∪ α for α ∈ H k ( M ) h · α := kα h –eigen spaces V k = H k ( M ). Prop. (Poincar´ e duality) M closed oriented manifold, ω ∈ H 2 ( M ) . Set H ∗ ( M ) m := H ∗ ( M ) m /H ∗ ( M ) m − 1 . ˜ Then Poincar´ e duality factors to non-degenerate pairing H ∗ ( M ) m ⊗ ˜ H ∗ ( M ) n − m → R , ˜ n = dim M . 11

Recall that a symplectic manifold ( M 2 n , ω ) is called Lefschetz if [ ω ] k : H n − k ( M ) → H n + k ( M ) is onto for all k ≥ 0. Equivalently  0 m < n  H ∗ ( M ) m = H ∗ ( M ) m ≥ n  Ex. All K¨ ahler manifolds are Lefschetz. For this talk a symplectic manifold is called weakly Lefschetz if [ ω ] k : H n +1 − k ( M ) → H n +1+ k ( M ) if onto for all k ≥ 0. Equivalently  0 m < n − 1 H ∗ ( M ) m =  H ∗ ( M ) m ≥ n + 1  Ex. Some 6–dimensional nil-manifolds. 12

M symplectic. M → P → B Hamiltonian fibra- tion, i.e. structure group is reduced to Hamil- tonian group. Question. [Lalonde, McDuff] Does every Hamiltonian fibration c–split, i.e. do we always have: H ∗ ( P ) = H ∗ ( B ) ⊗ H ∗ ( M ) Thm. [Blanchard] Yes for Lefschetz M . Thm. Suppose ( M, ω ) weakly Lefschetz. Then every Hamiltonian fibration M → P → B c– splits. For the proof we use deep Thm. [Lalonde, McDuff] ( M, ω ) closed sym- plectic manifold, M → P → B Hamiltonian fibration, where B CW–complex, dim B ≤ 3 . Then the fibration c–splits. 13

Proof. Will show that the spectral sequence collapses at the E 2 -term. McDuff and Lalonde’s theorem ⇒ E 2 = E 3 = E 4 . Consider E 4 = H ∗ ( M ) ⊗ H ∗ ( B ) ∈ V b . Hamiltonian fibration ⇒ ∂ 4 [ ω ] = 0 and thus ∂ 4 ( e · α ) = e · ∂ 4 α. Moreover ∂ 4 : ( E 4 ) k → ( E 4 ) k − 3 So ∂ 4 ( E 4 m ) ⊆ E 4 m − 3 . Weakly Lefschetz ⇒  0 m < n − 1  E 4 m = E 4 m ≥ n + 1  So ∂ 4 = 0 and thus E 4 = E 5 . Similarly ∂ k = 0 for k ≥ 4. � 14

For symplectic ( M, ω ) two filtrations on H ∗ ( M ): 1. H ∗ m ( M ) via harmonic forms. 2. H ∗ ( M ) m the filtration stemming from the b -module structure on H ∗ ( M ), defined via [ ω ] ∈ H 2 ( M ). Thm. We have H ∗ n + m ( M ) = H ∗ ( M ) m . Proof. Check, that H ∗ m ( M ) are b –submodules. Check, that H ∗ m ( M ) = 0 for m < < 0. Check, that H ∗ m ( M ) = H ∗ ( M ) for m > > 0. One then can explicitly extend the b –module structure on ( H ∗ m ( M ) /H ∗ m − 1 ( M ))[ n − m ] to a g –module structure. Now done, since a filtration with these proper- ties is unique. � 15

Prop. (K¨ unneth Theorem) M i closed sym- plectic manifolds. Then p M 1 × M 2 p M 1 p M 2 � ˜ ( t ) = ˜ m 1 ( t ) · ˜ m 2 ( t ) . m m 1 + m 2 = m Proof. Ordinary K¨ unneth Theorem ⇒ H ∗ ( M 1 × M 2 ) = H ∗ ( M 1 ) ⊗ H ∗ ( M 2 ) Little inspection ⇒ this is isomorphism of b – modules. Thus � H ∗ ( M 1 × M 2 ) m = � � ˜ H ∗ ( M 1 ) ⊗ H ∗ ( M 2 ) m H ∗ ( M 1 ) m 1 ⊗ ˜ H ∗ ( M 2 ) m 2 � ˜ = m 1 + m 2 = m Since the two filtrations agree we get H ∗ H ∗ H ∗ ˜ � ˜ m 1 ( M 1 ) ⊗ ˜ m ( M 1 × M 2 ) = m 2 ( M 2 ) m 1 + m 2 = m � 16

e duality) M 2 n closed sym- Thm. (Poincar´ plectic. Then H n + k H n − k − m ( M ) → R ˜ ( M ) ⊗ ˜ m is well defined and non-degenerate. Proof. Ordinary Poincar´ e duality ⇒ � ∗ H n − k ( M ) = � H n + k ( M ) Little inspection ⇒ this is isomorphism of b – modules. Thus � ∗ H n + k ( M ) H n − k ( M ) m = � m and � � ∗ � ∗ H n − k ( M ) m = � � H n + k ( M ) − m ˜ H n + k ( M ) ˜ m = Since the two filtrations agree � ∗ H n + k H n − k � ˜ ˜ ( M ) = − m ( M ) m This is the statement. � 17