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L6: Almost complex structures To study general symplectic manifolds, rather than Kähler manifolds, it is helpful to extract the “homotopy-theoretic essence” of having a complex structure. An almost complex structure is a bundle automorphism J : T M → T M with J 2 = −I. If M is a complex manifold, with complex co-ordinates zj = xj + iyj , then there is a distinguished J defined by ∂ ∂ J = ; ∂x ∂y µ ¶ J Ã ∂ ∂y ! ∂ =− . ∂x The tangent spaces to M are naturally complex vector spaces, which carry multiplication by i. More generally, for almost complex (M, J), J extends complex-linearly to T M ⊗ C , and splits this space into ±i-eigenspaces T M ⊗ C = ∼ T M , and T 1,0(M ) ⊕ T 0,1(M ). So T 1,0(M ) = R in the complex case can also be viewed as ∂ i, with ∂ = ∂ − i ∂ . If J T 1,0(M ) = C h ∂z ∂zj ∂xj ∂yj j comes from a system of complex co-ordinates, we say it is integrable. Definition: An a.c.s. J on (M, ω) is compatible with a symplectic form ω if ω(Jv, Ju) = ω(u, v); ω(v, Jv) > 0 ∀v 6= 0 The symmetric bilinear form g(u, v) = ω(u, Jv) defines a Riemannian metric in this case. Proposition: (M, ω) admits a compatible J. Proof: First we show this is true for a symplectic vector space V . Choose some metric g on V , and define A by ω(u, v) = g(Au, v). Then A∗ = −A so AA∗ is symmetric positive definite: g(AA∗v, v) = g(A∗v, A∗v) > 0 ∀v 6= 0 −1 is diagoThus AA∗ = B.diag(λ1, . . . , λ2n ).B √ nalisable, with λi > 0, and AA∗ exists and √ √ equals B.diag( λ1, . . . , λ2n).B −1. Set: √ J = ( AA∗)−1A ⇒ JJ ∗ = I and J ∗ = −J Thus J 2 = −I is an a.c.s. and ω(Ju, Jv) = g(AJu, Jv) = g(JAu, Jv) = g(Au, v) √ ω(u, Ju) = g(−JAu, u) = g( AA∗u, u) > 0 so J is compatible. Use this construction on each TxM , since it’s canonical it works globally. The proof actually shows that the space of compatible J is the same as the space of metrics, which is convex, hence contractible. Thus introducing a compatible J involves “essentially no choice”. Definition: a symplectic vector bundle is a vector bundle π : E → B such that each fibre Eb has a linear symplectic form ωb, these form a smooth global section Ω of Λ2E ∗, and locally in B things are trivial: ∼ (U × R 2n, ω ) (π −1(U ), Ω) = 0 Corollary: such an E canonically admits the structure of a complex vector bundle. Hence, it has a first Chern class. This is a “characteristic class”; we assign to each complex vector bundle E → B an element c1(E) ∈ H 2(B; Z) such that c1(f ∗E) = f ∗c1(E) is natural under continuous maps and pullback of vector bundles. Note that choices of a.c.s J on E are all homotopic, so c1(E) does not change as it lives in the integral cohomology which is discrete. If E = T M we write c1(M ) for c1(E). There are many definitions of c1(E): (i) The determinant line bundle Λr E → B is pulled back by a classifying map φE : B → BU (1) = C P∞, i.e. Λr E = φ∗E Ltaut, and we ] is generated by know H ∗(C P∞; Z) = Z[cuniv 1 an element of degree 2; set c1(E) = φ∗E (cuniv ). 1 (ii) Take a generic section s : B → Λr E which is transverse to the zero-section, which implies its zero-set Z(s) is a smooth submanifold of B of real codimension 2. Set c1(E) = PD[Z(s)]. (iii) Pick a connexion dA on E, given by a matrix of one-forms (θij ) w.r.t. a basis sj ∈ Γ(E) of local sections of E: dA : Γ(E) → Γ(E ⊗ T ∗B); sj 7→ X i si ⊗ θij The curvature FA = dA ◦ dA, locally dθ + θ ∧ θ, is a matrix-valued 2-form in Ω2(End(E)), so T r(FA) ∈ Ω2(B) is a 2-form. The Bianchi identity says this is closed, so defines a cohomology class, and c1(E) = [T r(FA)/2iπ]. One can check this class is independent of choice of connexion dA. We noted c1(E) ∈ H 2(M ; Z) is an integral class, so does not change as we vary JE continuously. Here are some other properties: (i) c1(E ∗) = −c1(E); (ii) c1(E ⊕E 0) = c1(E)⊕c1(E 0); given any short exact sequence 0 → E → E → E 0 → 0, the same formula holds; (iii) c1(E ⊗ F ) = rk(E).c1(F ) + rk(F ).c1(E); in particular for a line bundle c1(L⊗k ) = kc1(L). (iv) If M is compact, there is an isomorphism {C −vector bundles/M } c1 −→ H 2(M ; Z) Isomorphism If M is complex, this can fail holomorphically. Examples: (i) c1(Σg ) = 2 − 2g. This is Gauss-Bonnet: on a surface, both c1 and the Euler characteristic count (signed) zeroes of a generic vector field. ∼ (ii) c1(Ltaut → C Pn) = −[H] ∈ H 2(C Pn) = ZhHi, where [H] = PD[C Pn−1] is the hyperplane class; c1(C Pn) = n + 1, by considering poles of a holoc n-form (dz0∧· · ·∧dzn)/(z0 . . . zn). Adjunction formula: if C ⊂ (X 4, J) is an almost complex curve in an almost complex surface, i.e. if J preserves T C ⊂ T X, then 2g(C) − 2 = −c1(X) · [C] + [C]2 So the genus of an almost complex curve is determined by its homology class. Proof: We have a SES of complex v.bundles 0 → T C → T X|C → νC/X → 0 Now deduce c1(T X|C ) = c1(T C)⊕c1(νC/X ) and evaluate the terms. A small (smooth, generic) displacement of C in X gives a section of νC/X with exactly [C]2 zeroes, to sign. ¥ A four-manifold X has a (non-degenerate, symmetric) intersection form on H 2(X; R ). Let b+ and b− denote the number of positive and negative eigenvalues, so the signature σ(X) is the difference b+ −b−, whilst b+ +b− = b2(X). Now give X an a.c.s J, defining c1 = c1(T X, J). Signature theorem (Hirzebruch): c2 1 = 2e(X) + 3σ(X) is a topological invariant. Despite this, there are lots of possible c1’s. Say M 4 is even if every class a ∈ H 2(M ) has even square. Algebraic topology shows (i) if M is even and H 2(M ; Z) has no 2-torsion, any h s.t. h2 = 2e + 3σ and h ≡ 0 ∈ H 2(M ; Z2) is c1 of some almost complex structure, and (ii) for any a.c.s. c2 1 ≡ σ (8). Corollary: if X 4 admits an a.c.s. then the selfconnect sum X]X does not, X]X]X does, etc. Proof: The condition c2 1 ≡ σ(8) & the signature theorem show 1 − b1 + b+ must be even. But b1(X]X) = 2b1(X), b+(X]X) = 2b+(X) etc, so for X]X we have 1 − b1 + b+ is odd. S 4 was not symplectic since H 2(S 4) = 0. Now we see C P2]C P2 is not symplectic since it has no a.c.s. The proof that C P3]C P3]C P3 is not symplectic takes most of gauge theory and a few hundred pages. In dimension 2n > 4, there is no known example of a triple (X, [ω], J) with [ω] ∈ H 2(X) s.t. [ω]n 6= 0 and J an a.c.s. and X known to admit no symplectic structure.