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 z j = x j + iy j, then there is a distinguished J defined by J ( x ) = y ; J ( 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 1,0 (M) T 0,1 (M). So T 1,0 (M) = R T M, and in the complex case can also be viewed as T 1,0 (M) = C z, with j z = j x i j y. If J 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 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 0 Thus AA = B.diag(λ 1,..., λ 2n ).B 1 is diagonalisable, with λ i > 0, and AA exists and equals B.diag( λ 1,..., λ 2n ).B 1. Set: J = ( AA ) 1 A 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 T x M, 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 E b has a linear symplectic form ω b, these form a smooth global section Ω of Λ 2 E, and locally in B things are trivial: (π 1 (U), Ω) = (U R 2n, 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 c 1 (E) H 2 (B; Z) such that c 1 (f E) = f c 1 (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 c 1 (E) does not change as it lives in the integral cohomology which is discrete. If E = T M we write c 1 (M) for c 1 (E).
There are many definitions of c 1 (E): (i) The determinant line bundle Λ r E B is pulled back by a classifying map φ E : B BU(1) = C P, i.e. Λ r E = φ E L taut, and we know H (C P ; Z) = Z[c univ 1 ] is generated by an element of degree 2; set c 1 (E) = φ E (cuniv 1 ). (ii) Take a generic section s : B Λ r 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 c 1 (E) = PD[Z(s)]. (iii) Pick a connexion d A on E, given by a matrix of one-forms (θ ij ) w.r.t. a basis s j Γ(E) of local sections of E: d A : Γ(E) Γ(E T B); s j i s i θ ij The curvature F A = d A d A, locally dθ + θ θ, is a matrix-valued 2-form in Ω 2 (End(E)), so T r(f A ) Ω 2 (B) is a 2-form. The Bianchi identity says this is closed, so defines a cohomology class, and c 1 (E) = [T r(f A )/2iπ]. One can check this class is independent of choice of connexion d A.
We noted c 1 (E) H 2 (M; Z) is an integral class, so does not change as we vary J E continuously. Here are some other properties: (i) c 1 (E ) = c 1 (E); (ii) c 1 (E E ) = c 1 (E) c 1 (E ); given any short exact sequence 0 E E E 0, the same formula holds; (iii) c 1 (E F ) = rk(e).c 1 (F ) + rk(f ).c 1 (E); in particular for a line bundle c 1 (L k ) = kc 1 (L). (iv) If M is compact, there is an isomorphism {C vector bundles/m} c 1 H 2 (M; Z) Isomorphism If M is complex, this can fail holomorphically. Examples: (i) c 1 (Σ g ) = 2 2g. This is Gauss-Bonnet: on a surface, both c 1 and the Euler characteristic count (signed) zeroes of a generic vector field. (ii) c 1 (L taut C P n ) = [H] H 2 (C P n ) = Z H, where [H] = PD[C P n 1 ] is the hyperplane class; c 1 (C P n ) = n + 1, by considering poles of a holo c n-form (dz 0 dz n )/(z 0... z n ).
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 = c 1 (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 ν C/X 0 Now deduce c 1 (T X C ) = c 1 (T C) c 1 (ν C/X ) and evaluate the terms. A small (smooth, generic) displacement of C in X gives a section of ν C/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 = b 2 (X). Now give X an a.c.s J, defining c 1 = c 1 (T X, J). Signature theorem (Hirzebruch): c 2 1 = 2e(X) + 3σ(X) is a topological invariant.
Despite this, there are lots of possible c 1 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. h 2 = 2e + 3σ and h 0 H 2 (M; Z 2 ) is c 1 of some almost complex structure, and (ii) for any a.c.s. c 2 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 c 2 1 σ(8) & the signature theorem show 1 b 1 + b + must be even. But b 1 (X X) = 2b 1 (X), b + (X X) = 2b + (X) etc, so for X X we have 1 b 1 + b + is odd. S 4 was not symplectic since H 2 (S 4 ) = 0. Now we see C P 2 C P 2 is not symplectic since it has no a.c.s. The proof that C P 3 C P 3 C P 3 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 0 and J an a.c.s. and X known to admit no symplectic structure.