Almost complex manifolds An almost-complex manifold is a smooth real manifold 2 M equipped with a smooth endomorphism field J : T M T M satisfying J = Ix for → x − all x M. The linear algebra introduced above may be applied pointwise to the tangent bundle∈ of M. The complexified tangent bundle is TCM := T M C, where C is regarded as a trivial 1,0 0,1 ⊗ vector bundle over M. Since TC,xM = Tx M Tx M for any point x M as above, the tensor field J splits the complexified tangent bundle⊕ into bundles of eigen∈spaces 1,0 0,1 TCM = T M T M, ⊕ and the smooth complex vector bundles (TM,J) and (T 1,0M, i) are C-linearly isomorphic. We notice that a real smooth manifold carrying an almost complex structure may not be a complex manifold. We’ll discuss later that a real smooth manifold carrying an integral almost complex structure is a complex manifold. [Example] (Almost complex structure on spheres) It is natural to ask which spheres S2n admit almost-complex structures or complex structure (or other ones such as K¨ahler structure). 2 CP1 p m R R S = is the Riemann sphere, a complex manifold. It is known HDR(S , ) = • for p =0or p = m; 0 otherwsie. On the other hand, if X is a compact K¨ahler manifold, it is known that the Betti number b (X) 1. Thus S2 is the only sphere that admits 2 ≥ a K¨ahler structure. S2 has only one almost-complex structure up to equivalence. S4 does not admit any almost complex structure. • S6 admits many non-equivalent almost complex structure, but it is an open problem • whether S6 admits an integrable complex structure18. As above, S6 cannot admit K¨ahler structure. For Sn with n 8, deep results from algebraic topology imply that there is no almost • ≥ complex structure there. n [ Example] (C is an almost-complex manifold) Let zα = xα + √ 1yα be the usual − coordinates on Cn, identified with coordinates (x, y) on R2n. 18For the definition of integrable complex structure, see the end of this section. 44 The real tangent bundle T Cn has frame ∂ , ∂ . The almost complex structure ∂xα ∂yα • J : T Cn T Cn is given by: { } → ∂ ∂ ∂ ∂ J( )= , J( )= . ∂xα ∂yα ∂yα −∂xα n n TCC , the complexification of T C , has frame • ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ = √ 1 , = + √ 1 ∂zα 2∂xα − − ∂yα ∂zα 2∂xα − ∂yα n n The extended almost complex structure J : TCC TCC is given by → ∂ 1 ∂ ∂ ∂ J( )= + √ 1 = √ 1 , ∂z 2 ∂y − ∂x − ∂z α α α α ∂ 1 ∂ ∂ ∂ J( )= √ 1 = √ 1 . ∂z 2 ∂y − − ∂x − − ∂z α α α α T 1,0Cn has frame ∂ , and the almost complex structure is given by J( ∂ ) = • { ∂zα } ∂zα √ 1 ∂ . − ∂zα T 0,1Cn has frame ∂ , and the almost complex structure is given by J( ∂ ) = • { ∂zα } ∂zα √ 1 ∂ . − − ∂zα Similarly a complex manifold carries a natural almost complex structure. A map f :(M,J) (M, J) between almost-complex manifolds is pseudoholomorphic if → (f∗)J = J(f∗). f e The followinge is a straightforward consequence of the chain rule and the Cauchy-Riemann equations: Let ∆n Cn be a polydisk. A map f : ∆n Cm is pseudoholomorphic if and ⊂ → only if f is holomorphic. Holomorphic tangent bundles Let X be a complex manifold which is equipped −1 with holomorphic charts (Uα, τα) of X such that the transition maps ταβ = τα τβ are { n } ◦ holomorphic. From τα : Uα C , at each point x Uα X we have the Jacobian → ∈ ⊂ n dτα : TxX C . → 45 −1 The Jacobain of the transition maps τij := τi τ is the matrix ◦ j k ∂τij (τj(z)). ∂zl k,l The holomorphic tangent bundle of a complex manifold X is the holomorphic vector bundle X on X which is given by the transition matrices T k ∂τij τij(z) := (τj(z)). ∂zl k,l Just like line bundles, transition matrices satisfying transition conditions (i.e., τjj = Id,τij · τji = Id and τij τjk τki = Id) is equivalent to existence of vector bundles. · · n We denote by TX the underlying real tangent space. As in the C case, if (z1, ..., zn) are holomorphic coordinates on an open subset U X and zj := xj +iyj, then (x1,y1, ..., xn,yn) ⊂ ∂ ∂ ∂ ∂ define real coordinates of X on U and TX admits a basis ( , , ..., , ). The U ∂x1 ∂y1 ∂xn ∂yn | ∂ ∂ ∂ ∂ almost complex structure J : TX U TX U is given by J = , J = . The | → | ∂xj ∂yj ∂yj − ∂xj complexified tangent space C TX has the eigenspace of the eigenvalues i and i, denoted 1,0 0,1 ⊗ C 1,0 0,1 −1,0 by TX and TX so that we have the spliting direct sum TX = TX TX . T X has a basis ∂ := 1 ∂ i ∂ , and T 0,1X has a basis ∂ := 1 ⊗∂ + i ∂ . ⊕ ∂zk 2 ∂xk − ∂yk ∂zk 2 ∂xk ∂yk We claim: The subspace T 1,0X is canonically isomorphic to the holomorphic tangent space X (as a complex vector bundle). In fact, let X = iUi be an open covering by T Cn −1 ∪∗ 1,0 1,0 holomorphic charts φi : Ui Vi := φi(Ui) . Then (φi ) (T X Ui ) T Vi and the latter is canonically trivialized.→ With respect⊂ to these canonical trivializations| ≃ the induced − isomorphisms T 1,0 V T 1,0 V are given by the Jacobian J(φ φ 1) φ (x). Thus both φj (x) j φi(x) i i j j 1,0 ≃ ◦ ◦ bundles T X and X are associated to the same cocycle J(φij) φj . Then they are T { ◦ } isomorphic. Hence we can also call T 1,0X the holomorphic tangent bundle of X. 0,1 Similarly the subspace T X is canonically isomorphic to the complex conjugate X (with complex structure J via the C-linear embeddings). Hence we can call T 0,1X theT anti-holomorphic tangent− bundle of X. The dual spaces V ∗, (V ∗)1,0 and (V ∗)0,1 of a real vector space V Let V be a finite dimensional real vector space endowed with an almost complex structure J. Then the dual space ∗ V = HomR(V, R) 46 has a natural almost complex structure given by J : V ∗ V ∗, f J(f) where J(f)(v) := ∗ → → ∗ f(J(v)). The induced decomposition on (V )C = HomR(V, C)=(VC) is given by ∗ ∗1,0 ∗0,1 (V )C = V V ⊕ where ∗1,0 1,0 ∗ V = f HomR(V, C) f(I(v)) = if(v) =(V ) , { ∈ | } ∗0,1 0,1 ∗ V = f HomR(V, C) f(I(v)) = if(v) =(V ) . { ∈ | − } The natural decomposition of its exterior algebra of V is •V = d k V. ∧ ⊕k=0 ∧ Also the natural decomposition of its exterior algebra of VC is • d k VC = VC. ∧ ⊕k=0 ∧ ∗ ∗ • Moreover, VC = V R C, and V is the real subspace of VC that is left invariant under complex∧ conjugation.∧ ⊗ ∧ ∧ We define p,q p 1,0 q 0,1 V := V C V , ∧ ∧ ⊗ ∧ where the exterior products of V 1,0 and V 0,1 are taken as exterior products of complex vector spaces. An element α V p,qV is of bidegree (p, q). ∈ Proposition 5.1 If V admits an almost complex structure, then p,q p+q 1. V is in a canonical way a subspace of VC. ∧ ∧ k p,q 2. VC = p q k V . ∧ ⊕ + = ∧ ∗ p,q q,p 3. Complex conjugation on VC defines an anti- C linear isomorphism V V , i.e., p,qV = q,pV . ∧ ∧ ≃ ∧ ∧ ∧ 4. The exterior product is of bidegree (0, 0), i.e., (α, β) α β maps p,q r,sV to the subspace p+r,q+sV . → ∧ ∧ × ∧ ∧ 47.