Symplectic Geometry, Lecture 8

SYMPLECTIC GEOMETRY, LECTURE 8 1. Almost-complex Structures Recall compatible triples (ω, g, J), wherein two of the three determine the third (g(u, v) = ω(u, Jv), ω(u, v) = g(Ju, v), J(u) = g̃ −1 (ω̃(u)) where g̃, ω̃ are the induced isomorphisms T M → T ∗ M ). Proposition 1. For (M, ω) a symplectic manifold with Riemannian metric g, ∃ a canonical almost complex structure J compatible with ω. Idea. Do polar decomposition on every tangent space. � Corollary 1. Any symplectic manifold has compatible almost-complex structures, and the space of such struc­ tures is path connected. Proof. For the first part, using a partition of unity gives a Riemannian metric, so the rest follows from the proposition. For the second part, given J0 , J1 , let gi = ω(·, Ji ·) for i = 0, 1 and set gt = (1 − t)g0 + tg1 . Each of these (for t ∈ [0, 1]) is a metric, and gives an ω-compatible J˜t by polar decomposition, with J˜0 = J0 and � J˜1 = J1 . The mechanism of the proof also gives Proposition 2. The set J (Tx M, ωx ) of ωx -compatible complex structures on Tx M is contractible, i.e. ∃ht : J (Tx M, ωx ) → J (Tx M, ωx ) for t ∈ [0, 1], h0 = id, h1 = J → J0 , ht (J0 ) = J0 ∀t. Corollary 2. The space of compatible almost-complex structures on (M, ω) is contractible. It is the space of sections of a bundle whose fibers are contractible by the previous proposition. More generally, let E → M be a vector bundle. Definition 1. A metric on E is a family of positive-definite scalar products �·, ·�x : Ex × Ex → R. E is symplectic (resp. complex) if there is a family of nondegenerate skew-symmetric forms ωx : Ex × Ex → R (resp. complex structures Jx : Ex → Ex , Jx2 = −1). Then metrics always exist, and every sympletic vector bundle is a complex vector bundle and vice versa. Proposition 3. For (M, J) an almost-complex manifold, ω0 , ω1 two symplectic forms compatible with J, ωt = (1 − t)ω0 + tω1 is symplectic and J-compatible ∀t ∈ [0, 1] (i.e. the space of J-compatible ω is convex). Note that • The space of such ω might be empty, as there are almost complex manifolds (like S 6 ) which have no symplectic structures. • Not every manifold has an almost-complex structure (e.g. S 4 , by the Ehresman-Hopf theorem). Problem. ∃ an almost-complex structure ⇔ ∃ a nondegenerate 2-form. • The proposition works if we put tame instead of compatible, i.e. require ω(u, Ju) > 0 ∀u = � 0 but not symmetry. Proof. ωt is closed and ωt (u, Ju) = (1 − t)ω0 (u, Ju) + tω1 (u, Ju) > 0 ∀u �= 0, so ωt is nondegenerate and thus � symplectic. Moreover, gt (u, v) = ωt (u, Jv) = (1 − t)g0 (u, v) + tg1 (u, v) is a metric. Definition 2. X ⊂ (M, J) is an almost-complex submanifold if J(T X) = T X, i.e. ∀x ∈ X, v ∈ Tx X, Jv ∈ Tx X. 1 2 Proposition 4. If X is an almost-complex submanifold in compatible (M, ω, J), then X is symplectic (i.e. ω|X is nondegenerate). Proof. ∀u ∈ Tx X, u �= 0, Ju ∈ Tx X and ω(u, Ju) > 0, so ∀u ∈ Tx X � {0}, ω(u, ·)|Tx X ∈ Tx∗ X is nonzero, giving � us an isomorphism T X → T ∗ X as desired. Let (R2n , Ω0 , J0 , g0 ) be the standard symplectic structure, complex structure, and metric on R2n . • Sp(2n, R) is the group of linear symplectomorphisms of (R2n , Ω0 ), i.e. {A ∈ GL(2n, R)|Ω0 (Au, Av) = Ω(u, v) ∀u, v}. • GL(n, C) is the group of C-linear automorphisms of (R2n , J0 ), i.e. {A|AJ0 = J0 A}. • O(2n) is the group of isometries of (R2n , g0 ), i.e. {A|At A = 1}. • U (n) = GL(n, C) ∩ O(2n). Proposition 5. Sp(2n) ∩ O(2n) = Sp(2n) ∩ GL(n, C) = O(2n) ∩ GL(n, C) = U (n). Proof. The intersection of any two of these sets is the set of automorphisms preserving two of the three in a compatible triple, and thus must preserve all of them. � • For (V, Ω, J) a symplectic vector space with compatible almost-complex structure, ∃ an isomorphism ∼ (V, Ω, J) → (R2n , Ω0 , J0 ). • The space Ω(V ) of all symplectic structures on V is ∼ = GL(V )/Sp(V, Ω0 ) ∼ = GL(2n, R)/Sp(2n), as GL(V ) ∗ acts transitively on Ω(V ) by φ �→ φ Ω0 with stabilizer Sp(V, Ω). • The space J (V ) of almost-complex structures on V is ∼ = GL(V )/GL(V, J) ∼ = GL(2n, R)/GL(n, C). ∼ • The space J (V, Ω) of Ω-compatible J’s on V is = Sp(V, Ω)/Sp(V, Ω) ∩ GL(V, J) ∼ = Sp(2n, R)/U (n). • The constractibility of J (V, Ω) is now the fact that Sp(2n, R) retracts onto its subgroup U (n). 2. Vector Bundles and Connections For E → M a real or complex vector bundle, we have an exact sequence (1) dπ 0 → Ex → Tp E → Tx M → 0 for each p ∈ E, x = π(p). Here, Ex ⊂ Tp E gives the set of vertical directions: we would like a splitting Tp E = Ex ⊕ (Tp E)horiz , i.e. a way to transport from one fiber to another. The data required to do this is a connection. Definition 3. A connection � on E is an R or C-linear mapping C ∞ (M, E) → C ∞ (M, T ∗ M ⊗ E) = Ω1 (M, E) s.t. �(f σ) = df · σ + f �σ. For v ∈ Tx M , we let �v denote the mapping σ �→ �σ(v). � r r ∼ Choose a local�trivialization � of E, i.e. a frame of sections ei s.t. R (or C )×U = E|U , (ξ1 , . . . ,1ξr ) �→ ξi ei . Then �σ = �( ξi ei ) = (dξi )ei + ξi �ei , i.e. locally � = d + A, where A = (aij ) ∈ Ω (M, End(E)) is a matrix-valued 1-form (the connection 1-form) with aij the component of �ej along ei . Globally, given �, �� , �(f s) − �� (f s) = f (�s − �� s), so � − �� is C ∞ (M, E)-linear and the space of connections is an affine space modeled on Ω1 (M, End(E)). 2.1. Horizontal Distribution. Let σ : M → E be a section, dx σ : Tx M → Tσ(x) E the induced map. Then �σ(x) ∈ Tx∗ M ⊗ Ex depends only on dσ(x). Thus, we can also think of � as a projection π � : Tσ(x) E → Ex , with �v σ = π � (dσ(v)). Then H� = Ker π � is the horizontal subspace at p(x). Definition 4. For �·, ·� a Euclidean or Hermitian metric on E, � is compatible with the metric if d�σ, σ � � = ��σ, σ � � + �σ, �σ � �. As above, locally one can find an orthonormal frame of sections (ei ), �ei , ej � = δi,j . Writing � = d + A in this trivialization, the compatibility becomes (2) ��ξ, η� + �ξ, �η� = �dξ, η� + �Aξ, η� + �ξ, dη� + �ξ, Aη� Since d�ξ, η� = �dξ, η� + �ξ, dη�, this means that the connection 1-form A must be skew-symmetric (or antiHermitian). Also note that � on E induces a �∗ on E ∗ by d(φ(σ)) = ��∗ φ, σ� + �φ, �σ�, and similarly for E ⊗ F , etc.