《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:COMPLEXAND KAHLERMANIFOLDSCONTENTS11.3Almost complex manifolds52.Complex manifolds93.Kahler manifolds114.Dolbeault cohomology1.ALMOSTCOMPLEXMANIFOLDS Almost complex structures.Recall that a complex structure on a (real)vector space V is automorphismJ: VV such thatJ2 = -Id.Roughly speaking, a complex structure on V enable us to “multiply -" on Vand thus convertVintoacomplexvector space.Definition 1.1. An almost compler structure J on a (real) manifold M is an assign-ment of complex structuresJponthetangentspacesT,M whichdepend smoothlyon p. The pair (M, J) is called an almost compler manifold.In otherwords, an almost complex structure on M is a (1,1)tensor field J :TM → TM so that J? = -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 mustbe orientable.On the other hand,there does exists even dimensional orientablemanifolds which admit no almost complex structure. There exists subtle topologi-cal obstructions in the Pontryagin class. For example, there is no almost complexstructure on s4 (Ehresmann and Hopf).Erample.As in the symplectic case,any oriented surface admits an almost com-plex structure: LetV:E→S2be the Gauss map which associates to every point r e the outward unit normalvectorv(r).DefineJ:T→TbyJru =v(r) x u,1

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

2LECTURE5:COMPLEXANDKAHLERMANIFOLDSwhere x is the cross product between vectors in R3. It is quite obvious that J isan almost complex structure on Z.Erample.Wehave seen that ons6thereis no symplectic structure sinceH?(s6)=0.However, there exists an almost complex structure on s6.More generally, every ori-ented hypersurface M c R7 admits an almost complex structure. The constructionis almost the same as the previous example: first of all, there exists a notion of"cross product"for vectors in R7:weidentify R as the imaginary Cayley numbers,and define the vector product u x as the imaginary part of the product of u andw as Cayley numbers.Again we defineJru =v(r) x u,where v :M → s6 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? and 6 are the only spheres that admit almost complex structures.Compatibletriple.Now let (M,w) be a symplectic manifold, and J an almost complex structureon M.Then at each tangent spaceT,M we have the linear symplectic structure wpand the linear complex structureJp.Recall fromlecture 1 that J,is tamed by wp ifthe quadratic form wp(u, Jpu)is positive definite,and J is compatible with wp if itis tamed by wp and is a linear symplectomorphism on (T,M,wp),or equivalently,gp(v,w) :=wp(, Jpw)is an inner product on T,M.Definition 1.2. We say an almost complex structure J on M is compatible with asymplectic structurew on M if at eachp,Jpis compatiblewithwp.Equivalently, J is compatible with w if and only if the assignment9p: TpM × TpM →R, gp(u, u) :=wp(u, Ju)defines a Riemannian structure on M.So on M we get three structures: a symplecticstructure w, an almost complex structure J and a Riemannian structure g, and theyare related byg(u,v) = w(u, Ju),w(u,v) = g(Ju, v),J(u) = g-1(w(u),where g and are the linear isomorphisms from TM toT*M that is induced by gand w respectively. Such a triple (w, g, J) is called a compatible triple

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

3LECTURE5:COMPLEXANDKAHLERMANIFOLDS Almost complex = almost symplectic.According to proposition 3.4 and its corollary in lecture 1 we get immediatelyProposition 1.3. For any symplectic manifold (M,w), there erists an almost com-pler structure J which is compatible with w. Moreover, the space of such almostcompler structures is contractibleRemark. Obviously the proposition holds for any non-degenerate 2-form w on Mwhich does not have to be closed. Such a pair (M,w) is called an almost symplecticmanifold.Conversely, one can prove (exercise)Proposition 1.4. Given any almost compler structure on M, there erists an almostsymplectic structure w which is compatible with J.Moreover, the space of suchalmost symplectic structures is contractible.So the set of almost symplectic manifolds coincides with the set of almost com-plex manifolds.Eample. For the almost complex structures on surfaces (or hypersurfaces in R7)that we described above,wr(v,w) = (v(r), x w)defines a compatible almost symplectic structure (which is symplectic for surfacesbut not symplectic for s6).The following question is still open:Donaldson's question: Let M be a compact 4-manifold and J an almost complexstructure on M which is tamed by some symplectic structure w.Is there a symplecticformonMthatiscompatiblewithJ?An important progress was madeby Taubes who answered theproblem affirma-tively forgenerically almost complex structureswith 6+=1.Almostcomplexsubmanifolds.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 analmost comple submanifold if J(TX) c TX.Proposition 1.6. Let (M,w) be a symplectic manifold and J a compatible almostcompler structure on M. Then any almost compler submanifold of (M, J) is asymplectic submanifold of (M,w).Proof. Let t : X -→ M be the inclusion. Then t*w is a closed 2-form on X. It isnon-degenerate sincewr(u,v) = gr(Jru,v)

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)

4LECTURE5:COMPLEXANDKAHLERMANIFOLDS口and gaTrx is nondegenerate.Thesplittingoftangentvectors.Let (M,J)be an almost complexmanifold.DenotebyTcM =TM C thecomplexified tangent bundle.We extend J linearly toTcM byJ(uz)= JUQz, UETM,zEC.Then again J? = -Id, but now on a complex vector space T,M C instead of on areal vector space. So for each p E M the map J, has eigenvalues ±i, and we havean eigenspace decompositionTMC=Ti,o@To.1,whereTi,o = [uE TM@ CI Ju = iu]is the +i-eigenspace of J andToi=[uETM@CI Ju=-iu]is the -i-eigenspace of J. We will call vectors in Ti.o the J-holomorphic tangentvectors and vectors in To,i the J-anti-holomorphic tangent vectors.Lemma 1.7. J-holomorphic tangent vectors are of the form v@1-Ju@i for someU E TM, while J-anti-holomorphic tangent vectors are of the form 1+ Jvifor some w ETM.Proof. Obviously for any u E TM,J(v@l-Jui)=Jv@l+vi=i(vl-Ju@i)whileJ(u?l+Ju@i=Ju?l-vi=-i(ul+Jui)口TheconclusionfollowsfromdimensioncountingAs a consequence, we seeCorollary 1.8. If we write u = V1,o + vo,1 according to the splitting above, then11(u+iJu).iJu)，2U1,0=Vo,1 =22

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).

LECTURE5:COMPLEXANDKAHLERMANIFOLDS5 The splitting of differential forms.Similarly one can split the complexified cotangent space T*M C asT*MC= T1,0@ T0.1,whereTl,0 = (Ti.0)*= (n ET*M @C I n(Jw) = in(w), Vw ETM @C)={@1-(EoJ)@iIEET*M)is thedual spaceof Ti.o,andT0,1 = (To,1)*= (n E T*M C I n(Jw) =-in(w), Vw E TM @ C)={El+(EoJ)@i|EET*M)is the dual space of To,1. More over, any covector n has a splittingn= n1.0 + no.1,wheren10=n-ioJ)，no1(ninJ)2mThe splitting of covectors gives us a splitting of k-forms2*(M,C) = i+m=2lm(M, C),where 2lm(M, C) = F(A'T1,0 ^ AmT0,1) is the space of (l, m)-forms on M.For β E 2lm(M,C) C 2k(M,C), we have dβ E 2k+1(M,C). So we have asplittingdβ = (dp)*+1,0 + (dβ)k1 +... + (dp)1k + (dp)0,k+1.Definition 1.9. For β lm(M,C),Oβ = (dβp)+1,m, β = (dβ),m+1.Note that for functions we always havedf=of+ofwhileformoregeneral differential forms wedon'thaved=+.2.COMPLEXMANIFOLDS Complex manifolds.Recall that a smooth manifold is a topological space that locallylooks like Rn,with diffeomorphic transition maps.Definition 2.1. A compler manifold of complex dimension n is a manifold thatlocally homeomorphic to open subsets in Cn, with biholomorphic transition maps

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