KÄHLER GEOMETRY AND HODGE THEORY OLIVIERBIQUARDANDANDREASHÖRING The aimofthese lecture notes is to giveanintroductionto analytic geometry,that isthegeometryofcomplexmanifolds,withafocusontheHodgetheoryofcompact Kählermanifolds. The text comes in two parts that correspondto the distribution of the lectures between the two authors: - the first part, by Olivier Biquard, is an introduction to Hodge theory, and more generally to the analysis of elliptic operators on compact manifolds. - the second part, by Andreas Höring, starts with an introduction to complex manifolds and the objects (differential forms, cohomology theories, connections...) naturallyattachedtothem. InSection4,theanalyticresultsestablishedinthefirst partareusedtoprovetheexistenceoftheHodgedecompositiononcompactKähler manifolds. Finally in Section 5 we prove the Kodaira vanishing and embedding theorems which establish the link with complex algebraic geometry. Among the numerous books on this subject, we especially recommend the ones by Jean-Pierre Demailly [Dem96], Claire Voisin [Voi02] and Raymond Wells [Wel80]. Indeed our presentation usually follows closely one of these texts. Contents 1. Hodge theory 2 2. Complex manifolds 17 3. Connections, curvature and Hermitian metrics 50 4. Kähler manifolds and Hodge theory 72 5. Kodaira’s projectivity criterion 99 References 110 Date:16thDecember 2008. 1 2 OLIVIERBIQUARDANDANDREASHÖRING 1. Hodge theory ThischapterisanintroductiontoHodgetheory,andmoregenerallytotheanalysis on elliptic operatorson compact manifolds. Hodge theory represents De Rham co- homology classes (that is topological objects) on a compact manifold by harmonic forms (solutions of partial differential equations depending on a Riemannian met- ric on the manifold). It is a powerful tool to understand the topology from the geometric point of view. In this chapter we mostly follow reference [Dem96], which contains a complete concise proof of Hodge theory, as well as applications in Kähler geometry. 1.A. The Hodge operator. Let V be a n-dimensional oriented euclidean vector space (it will be later the tangent space of an oriented Riemannian n-manifold). Therefore there is a canonical volume element vol ∈ ΩnV. The exterior product ΩpV ∧Ωn−pV →ΩnV is a nondegeneratepairing. Therefore,for aformβ ∈ΩpV, one can define ∗β ∈Ωn−pV by its wedge product with p-forms: (1.1) α∧∗β =hα,βivol for all β ∈ΩpV. The operator ∗:Ωp →Ωn−p is called the Hodge ∗ operator. In more concrete terms, if (e ) is a direct orthonormal basis of V, then i i=1...n (eI) is an orthonormal basis of ΩV. One checks easily that I⊂{1,...,n} ∗1=vol, ∗e1 =e2∧e3∧···∧en, ∗vol=1, ∗ei =(−1)i−1e1∧···∧ei···en. More generally, b (1.2) ∗eI =ǫ(I,∁I)e∁I, where ǫ(I,∁I) is the signature of the permutation (1,...,n)→(I,∁I). 1.1. Exercise. Suppose that in the basis (e ) the quadratic form is given by the i matrixg =(g ), andwrite the inversematrixg−1 =(gij). Provethatfor a 1-form ij α=α ei one has i (1.3) ∗α=(−1)i−1gijα e1∧···∧ei∧···∧en. j 1.2. Exercise. Prove that ∗2 =(−1)p(n−p) on Ωp. b If n is even, then ∗:Ωn/2 →Ωn/2 satisfies ∗2 =(−1)n/2. Therefore, if n/2 is even, the eigenvalues of ∗ on Ωn/2 are ±1, and Ωn/2 decomposes accordingly as (1.4) Ωn/2 =Ω ⊕Ω . + − The elements of Ω are called selfdual forms, and the elements of Ω antiselfdual + − forms. For example, if n=4, then Ω is generated by the forms ± (1.5) e1∧e2±e3∧e4, e1∧e3∓e2∧e4, e1∧e4±e2∧e3. KÄHLER GEOMETRY AND HODGE THEORY 3 1.3. Exercise. If n/2 is even, provethat the decomposition(1.4)is orthogonalfor the quadratic form Ωn/2∧Ωn/2 →Ωn ≃R, and (1.6) α∧α=±|α|2vol if α∈Ω . ± 1.4. Exercise. If u is an orientation-preservingisometry of V, that is u∈SO(V), provethatupreservestheHodgeoperator. Thismeansthefollowing: uinducesan isometryofV∗ =Ω1,andanisometryΩpuofΩpV definedby(Ωpu)(x1∧···∧xp)= u(x1)∧···∧u(xp). Then for any p-form α∈ΩpV one has ∗(Ωpu)α=(Ωn−pu)∗α. This illustrates the fact that an orientation-preserving isometry preserves every object canonically attached to a metric and an orientation. 1.B. Adjoint operator. Suppose (Mn,g) is an oriented Riemannian manifold, and E → M a unitary bundle. Then on sections of E with compact support, one can define the L2 scalar product and the L2 norm: (1.7) (s,t)= hs,ti volg, ksk2 = hs,si volg. E E ZM ZM If E and F are unitary bundles and P :Γ(E) →Γ(F) is a linear operator, then a formal adjoint of P is an operator P∗ :Γ(F)→Γ(E) satisfying (1.8) (Ps,t) =(s,P∗t) E F for all sections s∈C∞(E) and t∈C∞(F). c c 1.5. Example. Consider the differential of functions, d:C∞(M)→C∞(Ω1). Chooselocalcoordinates(xi)inanopensetU ⊂M andsupposethat the function f and the 1-form α = α dxi have compact support in U; write volg = γ(x)dx1 ∧ i ···∧dxn, then by integration by parts: hdf,αivolg = gij∂ fα γdx1···dxn i j ZM Z =− f∂ (gijα γ)dx1···dxn i j Z =− fγ−1∂ (gijα γ)volg. i j Z It follows that (1.9) d∗α=−γ−1∂ (γgijα ). i j More generally, one has the following formula. 1.6. Lemma. Theformaladjointoftheexteriorderivatived:Γ(ΩpM)→Γ(Ωp+1M) is d∗ =(−1)np+1∗d∗. 4 OLIVIERBIQUARDANDANDREASHÖRING Proof. For α∈C∞(Ωp) and β ∈C∞(Ωp+1) one has the equalities: c c hdα,βivolg = du∧∗v ZM ZM = d(u∧∗v)−(−1)pu∧d∗v ZM by Stokes theorem, and using exercice 1.2: =(−1)p+1+p(n−p) u∧∗∗d∗v ZM =(−1)pn+1 hu,∗d∗vivolg. ZM (cid:3) 1.7. Remarks. 1) If n is even then the formula simplifies to d∗ =−∗d∗. 2) The same formula gives an adjoint for the exterior derivative d∇ :Γ(Ωp⊗E)→ Γ(Ωp+1⊗E) associated to a unitary connection ∇ on a bundle E. 3) As a consequence, for a 1-form α with compact support one has (1.10) (d∗α)volg =0 ZM since this equals (α,d(1))=0. 1.8. Exercise. Suppose that (Mn,g) is a manifold with boundary. Note ~n is the normal vector to the boundary. Prove that (1.10) becomes: (1.11) (d∗α)vol=− ∗α=− α vol∂M. ~n ZM Z∂M Z∂M For 1-forms we have the following alternative formula for d∗. 1.9. Lemma. LetE beavectorbundlewithunitaryconnection∇,thentheformal adjoint of ∇:Γ(M,E)→Γ(M,Ω1⊗E) is n ∇∗α=−Trg(∇u)=− (∇ α)(e ). ei i 1 X Proof. Takealocalorthonormalbasis(e )ofTM,andconsideranE-valued1-form i α = α ei. We have ∗α = (−1)i−1α e1∧···∧ei∧···∧en. One can suppose that i i just at the point p one has ∇e (p) =0, therefore dei(p) =0 and, still at the point i p, b n d∇∗α= (∇ α )e1∧···∧en. i i 1 X Finally ∇∗α(p)=− n(∇ α )(p). (cid:3) 1 i i P KÄHLER GEOMETRY AND HODGE THEORY 5 1.10. Remark. Actually the same formula is also valid for p-forms. Indeed, d∇ : Γ(M,Ωp) → Γ(M,Ωp+1) can be deduced from the covariant derivative ∇ : Γ(M,Ωp)→Γ(M,Ω1⊗Ωp) by the formula1 d∇ =(p+1)a◦∇, where a is the antisymmetrization of a (p+1)-tensor. Also observe that if α ∈ Ωp ⊂⊗pΩ1, its norm as a p-form differs from its norm as a p-tensor by |α|2 =p!|α|2 . Ωp ⊗pΩ1 Putting together this two facts, one can calculate that d∗ is the restriction of ∇∗ to antisymmetric tensors in Ω1⊗Ωp. We get the formula n (1.12) d∗α=− e y∇ α. i i 1 X Of course the formula remains valid for E-valued p-forms, if E has a unitary con- nection ∇. 1.11. Exercise. Consider the symmetric part of the covariantderivative, δ∗ :Γ(Ω1)→Γ(S2Ω1). Prove that its formal adjoint is the divergence δ, defined for a symmetric 2-tensor h by n (δh) =− (∇ h)(e ,X). X ei i 1 X 1.C. Hodge-de Rham Laplacian. 1.12. Definition. Let(Mn,g)beanorientedRiemannianmanifold. TheHodge-De Rham Laplacian on p-forms is defined by ∆α=(dd∗+d∗d)α. Clearly, ∆ is a formally selfadjoint operator. The definition is also valid for E- valued p-forms, using the exterior derivative d∇, where E has a metric connection ∇. 1.13. Example. Onfunctions∆=d∗d;using(1.9),weobtaintheformulainlocal coordinates: 1 (1.13) ∆f =− ∂ gij det(g )∂ f . i ij j det(g ) ij q (cid:0) (cid:1) In particular, for the flat metrpic g = n(dxi)2 of Rn, one has 1 n P ∆f =− ∂2f. i 1 X In polar coordinates on R2, one has g =dr2+r2dθ2 and therefore 1 1 ∆f =− ∂ (r∂ f)− ∂2f. r r r r2 θ 1Thisformulaistrueassoonas∇isatorsionfreeconnection onM. 6 OLIVIERBIQUARDANDANDREASHÖRING More generally on Rn with polar coordinates g =dr2+r2g , one has Sn−1 1 1 ∆f =− ∂ (rn−1∂ f)+ ∆ f. rn−1 r r r2 Sn−1 Similarly, on the real hyperbolic space Hn with geodesic coordinates, g = dr2 + sinh2(r)g and the formula reads Sn−1 1 1 ∆f =− ∂ (sinh(r)n−1∂ f)+ ∆ f. sinh(r)n−1 r r r2 Sn−1 1.14. Exercise. On p-forms in Rn prove that ∆(α dxI)=(∆α )dxI. I I 1.15. Exercise. Prove that ∗ commutes with ∆. 1.16. Exercise. If (Mn,g) has a boundary, prove that for two functions f and g one has ∂f (∆f)gvol= hdf,dgivol− gvol∂M. ∂~n ZM ZM Z∂M Deduce ∂g ∂f (∆f)gvol= f∆gvol+ f − g vol∂M. ∂~n ∂~n ZM ZM Z∂M (cid:0) (cid:1) 1.17. Exercise. Prove that the radial function defined on Rn by (V being the n volume of the sphere Sn) 1 if n>2 G(r)= (n−2)Vn−1rn−2 ( 1 logr if n=2 2π satisfies ∆G=δ (Dirac function at 0). Deduce the explicit solutionof ∆f =g for 0 g ∈C∞(Rn) given by the integral formula c f(x)= G(|x−y|)g(y)|dy|n. ZRn The function G is called Green’s function. Similarly, find the Green’s function for the real hyperbolic space. 1.D. Statement of Hodgetheory. Let(Mn,g)beaclosedRiemannianoriented manifold. Consider the De Rham complex 0→Γ(Ω0)→d Γ(Ω1)→d ···→d Γ(Ωn)→0. Remind that the De Rham cohomology in degree p is defined by Hp = {α ∈ C∞(M,Ωp),dα=0}/dC∞(M,Ωp−1). Other situation: (E,∇) is a flat bundle, we have the associated complex 0→Γ(Ω0⊗E)d→∇ Γ(Ω1⊗E)d→∇ ···d→∇ Γ(Ωn⊗E)→0 and we can define the De Rham cohomology with values in E in the same way. In both cases, we have the Hodge-De Rham Laplacian ∆=dd∗+d∗d. 1.18. Definition. A harmonic form is a C∞ form such that ∆α=0. KÄHLER GEOMETRY AND HODGE THEORY 7 1.19. Lemma. If α ∈ C∞(M,Ωp), then α is harmonic if and only if dα = 0 and c d∗α = 0. In particular, on a compact connected manifold, any harmonic function is constant. Proof. It is clear that if dα = 0 and d∗α = 0, then ∆α = d∗dα + dd∗α = 0. Conversely,if ∆α=0, because (∆α,α)=(d∗dα,α)+(dd∗α,α)=kdαk2+kd∗αk2, we deduce that dα=0 and d∗α=0. (cid:3) 1.20. Remark. The lemma remains valid on complete manifolds, for L2 forms α such that dα and d∗α are also L2. This is proved by taking cut-off functions χ , such that χ−1(1) are compact domains which exhaust M, and |dχ | remains j j j bounded by a fixed constant C. Then h∆α,χ αivol= hdα,d(χ α)i+hd∗α,d∗(χ α)i vol j j j ZM ZM (cid:0) (cid:1) = χ (|dα|2+|d∗α|2)+hdα,dχ ∧αi−hd∗α,∇χ yαi vol j j j ZM (cid:0) (cid:1) Using |dχ |6C and taking j to infinity, one obtains (∆α,α)=kdαk2+kd∗αk2. j Note Hp the space of harmonic p-forms on M. The main theorem of this section is: 1.21. Theorem. Let (Mn,g) be a compact closed oriented Riemannian manifold. Then: (1) Hp is finite dimensional; (2) one has a decomposition C∞(M,Ωp) = Hp ⊕ ∆(C∞(M,Ωp)), which is orthogonal for the L2 scalar product. This is the main theorem of Hodge theory, and we will prove it later, as a conse- quence of theorem 1.42. Just remark now that it is obvious that ker∆ ⊥ im∆, because ∆ is formally selfadjoint. Also, general theory of unbounded operators gives almost immediately that L2(M,Ωp) = Hp ⊕im∆. What is non trivial is: finite dimensionality of Hp, closedness of im∆, and the fact that smooth forms in the L2 image of ∆ are images of smooth forms. Now we will derive some immediate consequences. 1.22. Corollary. Same hypothesis. One has the orthogonaldecomposition C∞(M,Ωp)=Hp⊕d C∞(M,Ωp−1) ⊕d∗ C∞(M,Ωp+1) , where (cid:0) (cid:1) (cid:0) (cid:1) (1.14) kerd=Hp⊕d C∞(M,Ωp−1) , (1.15) kerd∗ =Hp⊕d∗(cid:0) C∞(M,Ωp+1(cid:1)) . (cid:0) (cid:1) 8 OLIVIERBIQUARDANDANDREASHÖRING Note that since harmonic forms are closed, there is a natural map Hp →Hp. The equality (1.14) implies immediately: 1.23. Corollary. Same hypothesis. The map Hp →Hp is an isomorphism. Using exercice 1.15, we obtain: 1.24. Corollary.[Poincaréduality]Samehypothesis. TheHodge∗operatorinduces an isomorphism ∗ : Hp → Hn−p. In particular the corresponding Betti numbers are equal, b =b . p n−p 1.25. Remark. As an immediate consequence, if M is connected then Hn = R since H0 = R. Since ∗1 = volg and volg > 0, an identification with R is just M given by integration of n-forms on M. R 1.26. Remark. In Kähler geometry there is a decomposition of harmonic forms using the (p,q) type of forms, Hk⊗C=⊕kHp,k−p, and corollary1.24 canthen be 0 refined as an isomorphism ∗:Hp,q →Hm−q,m−p, where n=2m. 1.27. Remark. Suppose that n is a multiple of 4. Then by exercises 1.3 and 1.15, one has an orthogonal decomposition (1.16) Hn/2 =H ⊕H . + − Underthewedgeproduct,thedecompositionisorthogonal,H ispositiveandH + − is negative, therefore the signature of the manifold is (p,q) with p = dimH and + q =dimH . − 1.28. Exercise. Suppose again that n is a multiple of 4. Note d : Γ(Ωn/2−1) → ± Γ(Ω )theprojectionofdonselfdualorantiselfdualforms. Provethaton(n/2−1)- ± forms, one has d∗d =d∗d . Deduce that the cohomology of the complex + + − − (1.17) 0→Γ(Ω0)→d Γ(Ω1)→d ···→d Γ(Ωn/2−1)→d+ Γ(Ω )→0 + is H0, H1, ..., Hn/2−1, H . + 1.29. Exercise. Using exercise 1.14, calculate the harmonic forms and the coho- mology of a flat torus Rn/Zn. 1.30. Exercise. Let (M,g) be a compact oriented Riemannian manifold. 1)Ifγ isanorientation-preservingisometryof(M,g)andαaharmonicform,prove that γ∗α is harmonic. 2) (requires some knowledge of Lie groups) Prove that if a connected Lie group G acts on M, then the action of G on H•(M,R) given by α→γ∗α is trivial2. 3) Deduce that harmonic forms are invariant under Isom(M,g)o, the connected component of the identity in the isometry group of M. Apply this observation to 2If ξ belongs to theLiealgebraofGandXξ isthe associated vector fieldon M givenbythe infinitesimalaction of G(that is defined byXξ(x)= ddtetξx|t=0), then one has ddt(etξ)∗α|t=0 = LXξα=iXξdα+diXξα. Deducethatifαisclosed,thentheinfinitesimalactionofGonH•(M,R) istrivial. KÄHLER GEOMETRY AND HODGE THEORY 9 giveaproofthatthecohomologyofthen-spherevanishesindegreesk =1,...,n−1 (prove that there is no SO(n+1)-invariant k-form on Sn using the fact that the representationof SO(n) on ΩkRn is irreducible and therefore has no fixed nonzero vector). 1.E. Bochnertechnique. Let(E,∇)beabundleequippedwithaunitaryconnec- tion over an oriented Riemannian manifold (Mn,g). Then ∇:Γ(E)→Γ(Ω1⊗E) and we can define the rough Laplacian ∇∗∇ acting on sections of E. Using a local orthonormalbasis (e ) of TM, from lemma 1.9 it follows that i n (1.18) ∇∗∇s= −∇ ∇ s+∇ s. ei ei ∇eiei 1 X If we calculate just at a point p and we choose a basis (e ) which is parallel at p, i then the second term vanishes. In particular, using the Levi-Civita connection, we get a Laplacian ∇∗∇ acting on p-forms. It is not equal to the Hodge-De Rham Laplacian, as follows from: 1.31. Lemma.[Bochnerformula]Let(Mn,g)beanorientedRiemannianmanifold. Then for any 1-form α on M one has ∆α=∇∗∇α+Ric(α). 1.32. Remark. There is a similar formula (Weitzenböck formula) on p-forms: the difference ∆α−∇∗∇α is a zero-th order term involving the curvature of M. Proof of the lemma. We have dα =(∇ α) −(∇ α) , therefore X,Y X Y Y X n n d∗dα =− (∇ dα) = −(∇ ∇ α) +(∇ ∇ α) , X ei ei,X ei ei X ei X ei 1 1 X X where in the last equality we calculate only at a point p, and we have chosen the vector fields (e ) and X parallel at p. i Similarly, d∗α=− n(∇ α) , therefore 1 ei ei n n P dd∗α =− ∇ ((∇ α) )=− (∇ ∇ α) . X X ei ei X ei ei 1 1 X X Therefore, still at the point p, comparing with (1.18), n (1.19) (∆α) =(∇∗∇α) + (R α) =(∇∗∇α) +Ric(α) . X X ei,X ei X X 1 X (cid:3) 1.33. Remark. There is a similar formula if the exterior derivative is coupled with a bundle E equipped with a connection ∇. The formula for the Laplacian ∆=(d∇)∗d∇+d∇(d∇)∗ becomes (1.20) ∆α=∇∗∇α+Ric(α)+R∇(α), 10 OLIVIERBIQUARDANDANDREASHÖRING where the additional last term involves the curvature of ∇, n (1.21) R∇(α) = R∇ α(e ). X ei,X i 1 X The proof is exactly the same as above, a difference arises just in the last equality of (1.19), when one analyses the curvature term: the curvature acting on α is that of Ω1⊗E, so equals R⊗1+1⊗R∇, from which: n n (R α) =Ric(α) + R∇ α(e ). ei,X ei X ei,X i 1 1 X X Now let us see an application of the Bochner formula. Suppose M is compact. By Hodge theory, an element of H1(M) is represented by a harmonic 1-form α. By the Bochner formula, we deduce ∇∗∇α+Ric(α) = 0. Taking the scalar product with α, one obtains (1.22) k∇αk2+(Ric(α),α)=0. If Ric>0, this equality implies ∇α=0 and Ric(α)=0. If Ric>0, then α=0; if Ric > 0 we get only that α is parallel, therefore the cohomology is represented by parallelforms. SupposethatM isconnected,thenaparallelformisdeterminedby its values at one point p, so we get an injection H1 ֒→Ω1. p Therefore dimH1 6 n, with equality if and only if M has a basis of parallel 1- forms. This impliesthatM isflat, andbyBieberbach’stheoremthatM is atorus. Therefore we deduce: 1.34. Corollary. If(Mn,g)isacompactconnectedorientedRiemannianmanifold, then: • if Ric>0, then b (M)=0; 1 • if Ric > 0, then b (M) 6 n, with equality if and only if (M,g) is a flat 1 torus. ThiscorollaryisatypicalexampleofapplicationofHodgetheorytoprovevanishing theorems for the cohomology: one uses Hodge theory to represent cohomology classes by harmonic forms, and then a Weitzenböck formula to prove that the harmonic forms must vanish or be special under some curvature assumption. For examples in Kähler geometry see [Dem96]. 1.F. Differential operators. A linear operator P :Γ(M,E)→Γ(M,F) between sectionsoftwo bundles E and F is a differential operator of order d if, in anylocal trivialisation of E and F over a coordinate chart (xi), one has Pu(x)= aα(x)∂ u(x), α |αX|6d where α = (α ,...,α ) is a multiindex with each α ∈ {1...n}, |α| = k, ∂ = 1 k i α ∂ ...∂ , and aα(x) is a matrix representing an element of Hom(E ,F ). α1 αn x x