《微分几何：辛几何引论》课程教学资源（英文讲义）Lec05 COMPLEX AND KAHLER MANIFOLDS

LECTURE 5: COMPLEX AND KAHLER MANIFOLDS ¨ Contents 1. Almost complex manifolds 1 2. Complex manifolds 5 3. K¨ahler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds ¶ Almost complex structures. Recall that a complex structure on a (real) vector space V is automorphism J : V → V such that J 2 = −Id. Roughly speaking, a complex structure on V enable us to “multiply √ −1” on V and thus convert V into a complex vector space. Definition 1.1. An almost complex structure J on a (real) manifold M is an assign￾ment of complex structures Jp on the tangent spaces TpM which depend smoothly on p. The pair (M, J) is called an almost complex manifold. In other words, an almost complex structure on M is a (1, 1) tensor field J : TM → TM so that J 2 = −Id. Remark. As in the symplectic case, an almost complex manifold must be 2n dimen￾sional. Moreover, it is not hard to prove that any almost complex manifold must be orientable. On the other hand, there does exists even dimensional orientable manifolds which admit no almost complex structure. There exists subtle topologi￾cal obstructions in the Pontryagin class. For example, there is no almost complex structure on S 4 (Ehresmann and Hopf). Example. As in the symplectic case, any oriented surface Σ admits an almost com￾plex structure: Let ν : Σ → S 2 be the Gauss map which associates to every point x ∈ Σ the outward unit normal vector ν(x). Define Jx : TxΣ → TxΣ by Jxu = ν(x) × u, 1

2 LECTURE 5: COMPLEX AND KAHLER MANIFOLDS ¨ where × is the cross product between vectors in R 3 . It is quite obvious that Jx is an almost complex structure on Σ. Example. We have seen that on S 6 there is no symplectic structure since H2 (S 6 ) = 0. However, there exists an almost complex structure on S 6 . More generally, every ori￾ented hypersurface M ⊂ R 7 admits an almost complex structure. The construction is almost the same as the previous example: first of all, there exists a notion of “cross product” for vectors in R 7 : we identify R 7 as the imaginary Cayley numbers, and define the vector product u × v as the imaginary part of the product of u and v as Cayley numbers. Again we define Jxu = ν(x) × u, where ν : M → S 6 is the Gauss map that maps every point to its unit out normal. Then J is an almost complex structure. Details left as an exercise. Remark. S 2 and S 6 are the only spheres that admit almost complex structures. ¶ Compatible triple. Now let (M, ω) be a symplectic manifold, and J an almost complex structure on M. Then at each tangent space TpM we have the linear symplectic structure ωp and the linear complex structure Jp. Recall from lecture 1 that Jp is tamed by ωp if the quadratic form ωp(v, Jpv) is positive definite, and Jp is compatible with ωp if it is tamed by ωp and is a linear symplectomorphism on (TpM, ωp), or equivalently, gp(v, w) := ωp(v, Jpw) is an inner product on TpM. Definition 1.2. We say an almost complex structure J on M is compatible with a symplectic structure ω on M if at each p, Jp is compatible with ωp. Equivalently, J is compatible with ω if and only if the assignment gp : TpM × TpM → R, gp(u, v) := ωp(u, Jv) defines a Riemannian structure on M. So on M we get three structures: a symplectic structure ω, an almost complex structure J and a Riemannian structure g, and they are related by g(u, v) = ω(u, Jv), ω(u, v) = g(Ju, v), J(u) = ˜g −1 (˜ω(u)), where ˜g and ˜ω are the linear isomorphisms from TM to T ∗M that is induced by g and ω respectively. Such a triple (ω, g, J) is called a compatible triple

LECTURE 5: COMPLEX AND KAHLER MANIFOLDS 3 ¨ ¶ Almost complex = almost symplectic. According to proposition 3.4 and its corollary in lecture 1 we get immediately Proposition 1.3. For any symplectic manifold (M, ω), there exists an almost com￾plex structure J which is compatible with ω. Moreover, the space of such almost complex structures is contractible. Remark. Obviously the proposition holds for any non-degenerate 2-form ω on M which does not have to be closed. Such a pair (M, ω) is called an almost symplectic manifold. Conversely, one can prove (exercise) Proposition 1.4. Given any almost complex structure on M, there exists an almost symplectic structure ω which is compatible with J. Moreover, the space of such almost symplectic structures is contractible. So the set of almost symplectic manifolds coincides with the set of almost com￾plex manifolds. Example. For the almost complex structures on surfaces (or hypersurfaces in R 7 ) that we described above, ωx(v, w) = hν(x), v × wi defines a compatible almost symplectic structure (which is symplectic for surfaces but not symplectic for S 6 ). The following question is still open: Donaldson’s question: Let M be a compact 4-manifold and J an almost complex structure on M which is tamed by some symplectic structure ω. Is there a symplectic form on M that is compatible with J? An important progress was made by Taubes who answered the problem affirma￾tively for generically almost complex structures with b + = 1. ¶ Almost complex submanifolds. Almost complex structure provides a method to construct symplectic submani￾folds. Definition 1.5. A submanifold X of an almost complex manifold (M, J) is an almost complex submanifold if J(T X) ⊂ T X. Proposition 1.6. Let (M, ω) be a symplectic manifold and J a compatible almost complex structure on M. Then any almost complex submanifold of (M, J) is a symplectic submanifold of (M, ω). Proof. Let ι : X → M be the inclusion. Then ι ∗ω is a closed 2-form on X. It is non-degenerate since ωx(u, v) = gx(Jxu, v)

4 LECTURE 5: COMPLEX AND KAHLER MANIFOLDS ¨ and gx|TxX is nondegenerate. ¶ The splitting of tangent vectors. Let (M, J) be an almost complex manifold. Denote by TCM = TM ⊗ C the complexified tangent bundle. We extend J linearly to TCM by J(v ⊗ z) = Jv ⊗ z, v ∈ TM, z ∈ C. Then again J 2 = −Id, but now on a complex vector space TpM ⊗ C instead of on a real vector space. So for each p ∈ M the map Jp has eigenvalues ±i, and we have an eigenspace decomposition TM ⊗ C = T1,0 ⊕ T0,1, where T1,0 = {v ∈ TM ⊗ C | Jv = iv} is the +i-eigenspace of J and T0,1 = {v ∈ TM ⊗ C | Jv = −iv} is the −i-eigenspace of J. We will call vectors in T1,0 the J-holomorphic tangent vectors and vectors in T0,1 the J-anti-holomorphic tangent vectors. Lemma 1.7. J-holomorphic tangent vectors are of the form v ⊗1−Jv ⊗i for some v ∈ TM, while J-anti-holomorphic tangent vectors are of the form v ⊗ 1 + Jv ⊗ i for some v ∈ TM. Proof. Obviously for any v ∈ TM, J(v ⊗ 1 − Jv ⊗ i) = Jv ⊗ 1 + v ⊗ i = i(v ⊗ 1 − Jv ⊗ i) while J(v ⊗ 1 + Jv ⊗ i) = Jv ⊗ 1 − v ⊗ i = −i(v ⊗ 1 + Jv ⊗ i). The conclusion follows from dimension counting. As a consequence, we see Corollary 1.8. If we write v = v1,0 + v0,1 according to the splitting above, then v1,0 = 1 2 (v − iJv), v0,1 = 1 2 (v + iJv).

LECTURE 5: COMPLEX AND KAHLER MANIFOLDS 5 ¨ ¶ The splitting of differential forms. Similarly one can split the complexified cotangent space T ∗M ⊗ C as T ∗M ⊗ C = T 1,0 ⊕ T 0,1 , where T 1,0 = (T1,0) ∗ = {η ∈ T ∗M ⊗ C | η(Jw) = iη(w), ∀w ∈ TM ⊗ C} = {ξ ⊗ 1 − (ξ ◦ J) ⊗ i | ξ ∈ T ∗M} is the dual space of T1,0, and T 0,1 = (T0,1) ∗ = {η ∈ T ∗M ⊗ C | η(Jw) = −iη(w), ∀w ∈ TM ⊗ C} = {ξ ⊗ 1 + (ξ ◦ J) ⊗ i | ξ ∈ T ∗M} is the dual space of T0,1. More over, any covector η has a splitting η = η 1,0 + η 0,1 , where η 1,0 = 1 2 (η − iη ◦ J), η0,1 = 1 2 (η + iη ◦ J). The splitting of covectors gives us a splitting of k-forms Ω k (M, C) = ⊕l+m=kΩ l,m(M, C), where Ωl,m(M, C) = Γ∞(ΛlT 1,0 ∧ Λ mT 0,1 ) is the space of (l, m)-forms on M. For β ∈ Ω l,m(M, C) ⊂ Ω k (M, C), we have dβ ∈ Ω k+1(M, C). So we have a splitting dβ = (dβ) k+1,0 + (dβ) k,1 + · · · + (dβ) 1,k + (dβ) 0,k+1 . Definition 1.9. For β ∈ Ω l,m(M, C), ∂β = (dβ) l+1,m, ¯∂β = (dβ) l,m+1 . Note that for functions we always have df = ∂f + ¯∂f, while for more general differential forms we don’t have d = ∂ + ¯∂. 2. Complex manifolds ¶ Complex manifolds. Recall that a smooth manifold is a topological space that locally looks like R n , with diffeomorphic transition maps. Definition 2.1. A complex manifold of complex dimension n is a manifold that locally homeomorphic to open subsets in C n , with biholomorphic transition maps

6 LECTURE 5: COMPLEX AND KAHLER MANIFOLDS ¨ Obviously any complex manifold is a real manifold, but the converse is not true. As in the symplectic case, a complex manifold must be of even dimensional if view as a real manifold, and must be orientable. In fact, we have Proposition 2.2. Any complex manifold has a canonical almost complex structure. Proof. Let M be a complex manifold and (U, V, ϕ) be a complex chart for M, where U is an open set in M, and V an open set in C n . We denote ϕ = (z1, · · · , zn), with zi = xi + √ −1yi . Then (x1, · · · , xn, y1, · · · , yn) is a coordinate system on U when we view M as a real manifold. So TpM = R-span of  ∂ ∂xi , ∂ ∂yi     i = 1, · · · , n . We define J on U by the recipe J( ∂ ∂xi ) = ∂ ∂yi , J( ∂ ∂yi ) = − ∂ ∂xi for i = 1, · · · , n, and extends to TpM by linearity. Obviously J 2 = −Id. It remains to prove that J is globally well-defined, i.e. it is independent of the choice of complex coordinate charts. Suppose (U 0 , V 0 , ϕ0 ) is another coordinate chart, with ϕ 0 = (w1, · · · , wn) and wi = ui + √ −1vi . Then on the overlap U ∩ U 0 the transition map ψ : ϕ(U ∩ U 0 ) → ϕ 0 (U ∩ U 0 ), z 7→ w = ψ(z) is biholomorphic. If we write the map as ui = ui(x, y), vi = vi(x, y) in real coordinates, then the real tangent vectors are related by ∂ ∂xk = X j ∂uj ∂xk ∂ ∂uj + ∂vj ∂xk ∂ ∂vj ∂ ∂yk = X j ∂uj ∂yk ∂ ∂uj + ∂vj ∂yk ∂ ∂vj , while the Cauchy-Riemann equation gives ∂uj ∂xk = ∂vj ∂yk , ∂uj ∂yk = − ∂vj ∂xk . It follows that J 0 ( ∂ ∂xk ) = J 0 ( X j ∂uj ∂xk ∂ ∂uj + ∂vj ∂xk ∂ ∂vj ) = X j ∂vj ∂yk ∂ ∂vj + ∂uj ∂yk ∂ ∂uj = ∂ ∂yk . Since J 0 = −Id, we must also have J 0 ( ∂ ∂yi ) = − ∂ ∂xi . It follows J 0 = J.

LECTURE 5: COMPLEX AND KAHLER MANIFOLDS 7 ¨ Conversely, not every almost complex manifold admits a complex structure. One example is CP2#CP2#CP2 . We have seen that S 2 and S 6 are the only spheres that admits almost complex structure. We will see below that S 2 admits a complex structure. One of the major open question in complex geometry is Open problem: Is there a complex structure on S 6 ? ¶ Differential forms on complex manifolds. Now suppose M is a complex manifold and J its canonical almost complex structure. Then in local coordinates TpM ⊗ C = C-span of  ∂ ∂xi , ∂ ∂yi     i = 1, · · · , n . and the two eigenspaces of J are T1,0 = C-span of  1 2  ∂ ∂xi − i ∂ ∂yi      i = 1, · · · , n , T0,1 = C-span of  1 2  ∂ ∂xi + i ∂ ∂yi      i = 1, · · · , n , We define ∂ ∂zj = 1 2  ∂ ∂xj − i ∂ ∂yj  , ∂ ∂z¯j = 1 2  ∂ ∂xj + i ∂ ∂yj  , then T1,0 = C-span of  ∂ ∂zj     j = 1, · · · , n , T0,1 = C-span of  ∂ ∂z¯j     j = 1, · · · , n , Similarly if we put dzj = dxj + idyj , dz¯j = dxj − idyj , then T 1,0 = C-span of {dzj | j = 1, · · · , n} , T0,1 = C-span of {dz¯j | j = 1, · · · , n} , Note that under these notions, the fact df = ∂f + ∂ ¯f is given explicitly as df = X j  ∂f ∂xj dxj + ∂f ∂yj dyj  = X j  ∂f ∂zj dzj + ∂f ∂z¯j dz¯j  . As a consequence, any (l, m)-form β ∈ Ω l,m(M, C) can be expressed locally as β = X |J|=l,|K|=m bJ,KdzJ ∧ dz¯K for some smooth functions bJ,K ∈ C ∞(U, C), where we use the notion dzJ to represent dzj1 ∧ · · · ∧ dzjl for a multi-index J = (j1, · · · , jl), and likewise for dz¯K. This nice local expression implies Theorem 2.3. On complex manifolds d = ∂ + ¯∂ for any (l, m)-forms

8 LECTURE 5: COMPLEX AND KAHLER MANIFOLDS ¨ Proof. The local expression above for β ∈ Ω l,m(M, C) gives dβ = X |J|=l,|K|=m dbJ,K ∧ dzJ ∧ dz¯K. The conclusion follows from the facts dbJ,K = ∂bJ,K + ¯∂bJ,K and ∂bJ,K = X ∂bJ,K ∂zj dzj , ¯∂bJ,K = X ∂bJ,K ∂z¯j dz¯j . ¶ Integrability. The canonical almost complex structure on a complex manifold that we con￾structed above is just the map “multiplication by √ −1” in coordinate charts. Definition 2.4. An almost complex structure J on M is called integrable if it is a complex structure, i.e. there exists local complex coordinates on M so that (M, J) = (C n , √ −1). So a complex structure is an integrable almost complex structure. It is hard to use the definition above to detect whether an almost complex structure is complex or not. However, we do have a very useful criteria. Definition 2.5. The Nijenhuis tensor NJ is NJ (u, v) = [Ju, Jv] − J[Ju, v] − J[u, Jv] − [u, v]. We will leave the proof of the next two exercises as an exercise. Proposition 2.6. NJ is a tensor, i.e. Nj (fu, gv) = fgNj (u, v) for vector fields u, v and smooth functions f, g. Proposition 2.7. One has NJ (u, v) = −8Re([u1,0, v1,0])0,1). As a consequence, we get Corollary 2.8. NJ = 0 ⇐⇒ [T1,0, T1,0] ⊂ T1,0. On a complex manifold [ ∂ ∂zj , ∂ ∂zk ] = 0. It follows Corollary 2.9. NJ = 0 for the canonical almost complex structure J on complex manifolds. The following theorem is a hard theorem which gives an easy-to-use characteri￾zation of integrability of almost complex structures: Theorem 2.10 (Newlander-Nirenberg). Let J be an almost complex structure on M. Then J is integrable ⇐⇒ NJ = 0 ⇐⇒ d = ∂ + ¯∂.

LECTURE 5: COMPLEX AND KAHLER MANIFOLDS 9 ¨ An an example, we immediately get Theorem 2.11. Any almost complex structure on a surface is integrable. Proof. A direct computation gives NJ (v, v) = 0 and NJ (v, Jv) = 0. Details left as an exercise. 3. Kahler manifolds ¨ ¶ K¨ahler manifolds. Definition 3.1. A K¨ahler manifold is a triple (M, ω, J), where ω is a symplectic form on M, and J an integrable complex structure on M which is compatible with ω. In this case we will call ω a K¨ahler form. Example. (C n , Ω0, J0) is a K¨ahler manifold. Example. Any oriented surface Σ carries a K¨ahler structure: one just choose ω to be the area form and choose J to be an almost complex structure that is compatible with ω. Example. Complex tori M = C n/Z n : Since both the symplectic structure and the complex structure on C n are invariant under translations along real directions, the standard symplectic and complex structures on C n give us a K¨ahler structure on M. Remark. By definition a K¨ahler manifold is both a symplectic manifold and a com￾plex manifold. In 1976 Thurston constructed an example that is both symplectic and complex, but admits no K¨ahler structure. People also constructed symplectic manifolds which do not admit any com￾plex structure (Fernandez-Gotay-Gray 1988), and complex manifolds that admits no symplectic structure (Hopf surface S 1 × S 3 ' C 2 − {0}/{(z1, z2) ∼ (2z1, 2z2)}). ¶ The K¨ahler form. Now let ω be a K¨ahler form on M. Then ω is a real-valued non-degenerate closed 2-form on M. Let’s see what does these conditions give us: • Since M is complex, locally ω = Xajkdzj ∧ dzk + Xbjkdzj ∧ dz¯k + Xcjkdz¯j ∧ dz¯k. • Since J is a symplectomorphism, J ∗ω = ω. On the other hand it is easy to check J ∗dzj = idzj and J ∗dz¯j = −idz¯j . So J ∗ω = X−ajkdzj ∧ dzk + Xbjkdzj ∧ dz¯k − Xcjkdz¯j ∧ dz¯k. It follows ω = Xbjkdzj ∧ dz¯k,

10 LECTURE 5: COMPLEX AND KAHLER MANIFOLDS ¨ i.e. ω ∈ Ω 1,1 (M, C) ∩ Ω 2 (M). We will write bjk = i 2 hjk, so that ω = i 2 Xhjkdzj ∧ dz¯k, hjk ∈ C ∞(U). • Since ω is real-valued, ¯ω = ω. But ω¯ = − i 2 Xhjkdz¯j ∧ dzk = i 2 Xhkjdzj ∧ dz¯k, so at each point p ∈ M the matrix (hjk(p)) is Hermitian. • Moreover, one can check ω n = n!(i 2 ) n det(hjk)dz1 ∧ dz¯1 ∧ · · · ∧ dzn ∧ dz¯n, so the non-degeneracy condition of ω is equivalent to the fact that the matrix (hjk) is non-singular. • The tamed condition ω(v, Jv) > 0 for each v 6= 0 implies that at each p, the matrix (hjk) is positive definite. • Finally, since 0 = dω = ∂ω + ¯∂ω, and ∂ω ∈ Ω 2,1 (M, C) and ¯∂ω ∈ Ω 1,2 (M, C), we get ∂ω = 0, ¯∂ω = 0. In conclusion, we get Theorem 3.2. K¨ahler forms are ∂- and ¯∂-closed (1, 1) forms which are given locally by ω = i 2 Xhjkdzj ∧ dz¯k, hjk ∈ C ∞(U), where at each p, the matrix (hjk) is a positive-definite Hermitian matrix. In general one cannot hope that two symplectic forms in the same cohomolo￾gy class are symplectomorphic, unless they are connected by a path of symplectic structures. As an application of the previous theorem, we have Corollary 3.3. Let M be compact and ω1, ω2 be K¨ahler forms on M with [ω1] = [ω2] ∈ H2 (M), then (M, ω1) and (M, ω2) are symplectomorphic. Proof. On a local chart ωi = i 2 Xh i jkdzj ∧ dz¯k, hi jk ∈ C ∞(U). We let ωt = i 2 X((1 − t)h 1 jk + th0 jk)dzj ∧ dz¯k, hi jk ∈ C ∞(U). Then ((1 − t)h 1 jk + th0 jk) is a positive definite Hermitian matrix, so ωt ’s are all symplectic. Now apply Moser’s trick.