KÄHLER GEOMETRY AND HODGE THEORY ANDREASHÖRING The aim of these lecture notes is to give an introduction to analytic geometry, that is the geometry of complex manifolds, with a focus on the Hodge theory of compact Kähler manifolds. It starts with an introduction to complex manifolds and the objects (differential forms, cohomology theories, connections...) naturally attached to them. In Section 3, the analytic results established in the Appendix are used to prove the existence of the Hodge decomposition on compact Kähler manifolds. Finally in Section 4 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. Complex manifolds 2 2. Connections, curvature and Hermitian metrics 36 3. Kähler manifolds and Hodge theory 66 4. Kodaira’s projectivity criterion 93 Appendix A. Hodge theory 104 References 119 Date:4thDecember2009. 1 2 ANDREASHÖRING 1. Complex manifolds In this chapter we will see that complex manifolds are differentiable manifolds whosetransitionfunctionsareholomorphicandwewilladaptthenotionsoftangent bundle and differential forms to this new context. In analogy to the definition of the de Rham cohomology in differential geometry, we will use the calculus of (p,q)-forms and the differential operator ∂ to define the Dolbeault cohomology groups of a complex manifold. The subject of the Hodge Theorem 3.36 is to relate this cohomology theory to de Rham cohomology, but this needs serious technical preparation and will be the subject of the following sections. Throughout the whole text, we assume that the reader is familiar with the basic notions of differential geometry as explained in [Biq08]. We will use the term differentiable as a synonym for smooth or C∞1. Let U ⊂ Cn be an open subset and f : U → C be any complex-valued function. We say that f is differentiable if for some R-linear identification Cn ’ R2n and C ’ R2, the composition f : U ⊂ R2n → C ’ R2 is differentiable. It is straightforward to see that this definition does not depend on the choice of the identifications. 1.A. Holomorphic functions in several variables. In this section we recall very briefly the notions from holomorphic function theory of several variables that wewillneed. Areaderthatisnotsofamiliarwiththissubjectmaywanttoconsult [Voi02,Ch.1]. Foramuchmoreampleintroductiontothefunctiontheoryofseveral complex variables, [Gun90, KK83, LT97] are standard references. 1.1. Definition. LetU ⊂Cn beanopensubset,andletf :U →Cbea(complex- valued) differentiable function. We say that f is holomorphic in the point a∈U if for all j ∈{1,...,n} the function of one variable z 7→f(a ,...,a ,z ,a ,...,a ) j 1 j−1 j j+1 n is holomorphic in a . j 1.2. Exercise. LetU ⊂Cnbeanopensubset,andletf :U →Cbeadifferentiable function. a) Denote by z ,...,z the standard coordinates on U, and by x (resp. y ) their 1 n j j real and imaginary parts. Show that f is holomorphic in a∈U if and only if (cid:18) (cid:19) ∂f 1 ∂f ∂f (a):= +i (a)=0 ∀ j =1,...,n. ∂z 2 ∂x ∂y j j j b) For a∈U, consider the R-linear application given by the differential df :Cn →C. a Show that the function f is holomorphic in a if and only if df is C-linear. (cid:3) a 1Inmanycases,theC∞-conditionisactuallymuchmorethanwhatwewillneed. Forsimplicity ofnotations,wewillneverthelessmakethisassumptionthroughoutthewholetext. KÄHLER GEOMETRY AND HODGE THEORY 3 1.3. Definition. Let a ∈ Cn be a point. The polydisc around a with multiradius R∈(R+)n is the set D(a,R):={z ∈Cn | |z −a |<R for all j ∈{1,...,n}}. j j j If R=(1,...,1) and a=0, we abbreviate D(0,R) by Dn and call Dn the unit disc in Cn. 1.4. Theorem. [Voi02, Thm.1.17]LetU ⊂Cn beanopensubset, andletf :U → Cbeadifferentiablefunction. Thefunctionf isholomorphicineverypointz ∈U 0 if and only if it satisfies one of the following conditions: (1) For every point a ∈ U there exists a polydisc D ⊂ U such that the power series X f(a+z)= α zI, I I converges for every a+z ∈D. (2) If D =D(a,r) is a polydisc contained in U, then for every z ∈D (cid:18) 1 (cid:19)nZ dζ dζ f(z)= f(ζ) 1 ∧...∧ n 2πi ζ −z ζ −z |ζj−aj|=rj 1 1 n n 1.5. Exercise. (Maximum principle) Let f : U ⊂ Cn → C be a holomorphic function. If |f| admits a local maximum in a point z ∈U, there exists a polydisc 0 D around z such that f| is constant. (cid:3) 0 D The notion of holomorphic function immediately generalises to the case of a map with values in Cm. 1.6. Definition. Let U ⊂ Cn be an open subset, and let f : U → Cm be a differentiable map. We say that f is holomorphic in the point z ∈U if f ,...,f 0 1 m are holomorphic in z for every j =1,...,m. 0 A holomorphic map f : U → Cn is locally biholomorphic in the point z ∈ U if 0 there exists a neighbourhood V ⊂U of z such that f| is bijective onto f(V) and 0 V f|−1 is holomorphic. It is biholomorphic if it is bijective on its image and locally V biholomorphic in every point. 1.7. Definition. Let U ⊂ Cn be an open subset, and let f : U → Cm be a holomorphic map. The Jacobian matrix of f at a point a∈U is the matrix (cid:18) (cid:19) ∂f J (a)= k(a) . f ∂z j 1(cid:54)k(cid:54)m,1(cid:54)j(cid:54)n Asfordifferentiablemaps,aholomorphicmapwhoseJacobianmatrixhasconstant rank admits locally a canonical representation: 1.8. Theorem. (Ranktheorem, [KK83, Thm.8.7])LetU ⊂Cn beanopensubset, letf :U →Cmbeaholomorphicmap,andletz ∈U beapointsuchthatJ (z)has 0 f constant rank k in a neighbourhood of z . Then there exist open neighbourhoods 0 4 ANDREASHÖRING z ∈ V ⊂ U and f(z ) ∈ W ⊂ Cm and biholomorphic mappings φ : Dn → V and 0 0 ψ :W →Dm such that φ(0)=z , ψ(f(z ))=0 and 0 0 ψ◦f ◦φ:Dn →Dm is given by (z ,...,z )7→(z ,...,z ,0,...,0). 1 n 1 k 1.9. Exercise. Let f : U ⊂ Cn → Cn be a holomorphic map. Show that f is locally biholomorphic in the point z ∈U if and only if 0 detJ (z )6=0. f 0 (cid:3) 1.10. Exercise. (Cauchy-Riemann equations) Let U ⊂Cn be an open subset, and let f =(f ,...,f ):U →Cm be a differentiable function such that 1 m f (z)=f (z ,...,z ). j j 1 n Set x =Re(z ),y =Im(z ) and u =Re(f ),v =Im(f ) k k k k j j j j forallj ∈{1,...,m}andk ∈{1,...,n}. Showthatf isholomorphicifandonlyif ∂u ∂v ∂u ∂v j = j, j =− j ∂x ∂y ∂y ∂x k k k k for all j ∈{1,...,m} and k ∈{1,...,n}. (cid:3) 1.11. Exercise. LetV ⊂Cn beasimplyconnectedopensubset. Letφ:V →Rbe adifferentiablepluriharmonicfunction,i.e. afunctionsuchthatforeverya,b∈Cn the restriction of φ to the line V ∩{a+bζ | ζ ∈C} is harmonic. Then there exists a holomorphic function f :V →C such that Re(f)=φ. (cid:3) 1.B. Complex manifolds. 1.12. Definition. A complex manifold of dimension n is a connected Hausdorff topological space X such that there exists a countable covering (U ) by open i i∈I sets and homeomorphisms φ : U → V onto open sets V ⊂ Cn such that for all i i i i (i,j)∈I×I, the transition functions φ ◦φ−1| :φ (U ∩U )→φ (U ∩U ) j i φi(Ui∩Uj) i i j j i j arebiholomorphic. Wecallthecollection(U ,φ ) acomplexatlasofthemanifold. i i i∈I A complex manifold is compact if the underlying topological space is compact. As in the case of differential geometry [Biq08, Ch.1.2], we say that two atlas are equivalentiftheirunionisstillanatlas. Thisdefinesanequivalencerelationonthe set of complex atlas on X. 1.13. Definition. A complex structure on X is the data of an equivalence class of a complex atlas on X. KÄHLER GEOMETRY AND HODGE THEORY 5 1.14. Remark. Note that in contrast to differentiable manifolds, it is in general notpossibletochoosethewholeaffinespaceisacoordinatechart: justtakeX =D the unit disc, then by Liouville’s theorem there is no non-constant holomorphic map C → D. We say that a complex manifold X is (Brody-)hyperbolic if it does not admit non-constant holomorphic maps f : C → X. Deciding whether certain manifolds are hyperbolic is a very active (and difficult!) research subject. 1.15. Examples. 1. Let U ⊂Cn be an open set. Then U is a complex manifold, an atlas is given by one chart. 2. More generally let X be a complex manifold of dimension n, and let U ⊂ X be a connected open set. Then U has an induced structure of complex manifold of dimension n. 3. Let Λ ⊂ Cn be a lattice of rank 2n. Then the quotient group X := Cn/Λ endowed with the quotient topology has a unique holomorphic structure induced by the standard holomorphic structure on Cn (cf. Exercise 1.16). We call X a complex torus2. 4. Let V be a complex vector space of dimension n+1 and let P(V) be the set of complex lines in V passing through the origin, i.e. the set of complex subvector spaces of dimension one. If v ∈ V \0 is a point, then Cv, the complex vector space generated by v is an element of P(V) which we denote by [v]. Furthermore if v0 = λv for some λ ∈ C∗, then [v0] = [v]. Vice versa if l ∈ P(V), there exists a v ∈ V \0 such that l = [v] and v is unique up to multiplication by an element λ∈C∗. Therefore we have a surjective map π :V \0→P(V),v 7→[v] and we endow P(V) with the quotient topology defined by π and the standard topology on V. Let V ’ Cn+1 be a C-linear isomorphism, then we can write v = (v ,...,v ) and 0 n we call [v :...:v ] 0 n homogeneous coordinates of [v]∈P(V). As in the case of the real projective space, we can then define a structure of complex manifold on P(V) as follows: for every i∈{0,...,n}, set U :={[v]∈P(V) | v 6=0} i i and v v v φ :U →Cn, [v] 7→ ( 0,...,ci,..., n). i i v v v i i i With this definition we have φ (U ∩U )={(z ,...,z ) | z 6=0}, i i j 1 n j 2InthesequelwheneverwewriteX:=Cn/Λ,weimplicitlyassumethatthelatticehasmaximal rank. EquivalentlyX isalwayssupposedtobecompact. 6 ANDREASHÖRING so the transition functions φ ◦φ−1| :φ (U ∩U )→φ (U ∩U ) given by j i φi(Ui∩Uj) i i j j i j z z z 1 z z (z ,...,z )7→[z :...:z :1:z :...:z ]7→( 1,...,cj,..., i−1, , i,..., n) 1 n 1 i−1 i n z z z z z z j j j j j j are well-defined and biholomorphic. One checks easily that the complex structure on P(V) does not depend on the choice of the isomorphism V ’Cn+1. Very often we will denote by Pn the projective space P(Cn+1). 1.16. Exercise. Let X be a complex manifold, and let Γ be a subgroup of the group of automorphisms of X. We say that Γ acts properly discontinuous on X if for any two compact subsets K ,K ⊂X, we have 1 2 γ(K )∩K 6=∅ 1 2 for at most finitely many γ ∈Γ. The group acts without fixed points if γ(x)6=x ∀ γ ∈Γ. Suppose that Γ acts properly discontinuous and without fixed points on X, and denote by X/Γ the set of equivalence classes under this action. Show that X admits a unique complex structure such that the natural map π : X → X/Γ is holomorphic and locally biholomorphic. (cid:3) 1.17. Exercise. Show that as a differentiable manifold, we have Pn ’S2n+1/S1, where S2n+1 ⊂ Cn+1 ’ R2n+2 denotes the unit sphere and S1 ⊂ C acts on Cn+1 by scalar multiplication S1×Cn+1 →Cn+1,(λ,x)7→λ·x. In particular the topological space Pn(C) is compact. (cid:3) 1.18. Exercise. (Grassmannian) Let V be a complex vector space of dimension n. For an integer 0<r <n, we define the Grassmannian as the set G (V):={S ⊂Vsubspace of dimension r}. r Fix a Hermitian product on V and denote by U ⊂ GL(V,C) the unitary group V with respect to this metric. Show that we have a surjective map U →G (V). V r We endow G (V) with the quotient topology induced by the surjection U → r V G (V). Show that G (V) is a compact topological space. r r We define an atlas on G (V) as follows: for any T ⊂ V a subspace of dimension r i n−r, set U :={S ⊂Vof dimension r | S∩T =0}. i i Choose an arbitrary S ∈U , then we can define i i φ :U →Hom(S ,T )’Cr(n−r) i i i i KÄHLER GEOMETRY AND HODGE THEORY 7 by associating to S ∈U the unique linear map f ∈Hom(S ,T ) such that i i i S ⊂V =S ⊕T i i is the graph of f. Show that the maps φ = φ (S ,T ) define a complex atlas on i i i i G (V). (cid:3) r 1.19. Exercise. (Hopf varieties) Let λ be a real number such that 0<λ<1. We define a group action Z×(Cn\0)→(Cn\0),(m,z)7→λmz, and denote by H the quotient (Cn\0)/Z. Show that H admits the structure of a complex manifold and is diffeomorphic to S2n−1 ×S1. Hint: note that Cn \0 is diffeomorphic to S2n−1×R+. (cid:3) 1.20. Definition. Let X (resp. Y) be a complex manifold of dimension n (resp. m) and denote by (U ,φ : U → V ) (resp. (M ,ψ : M → N ) ) the i i i i i∈I j j j j j∈J corresponding atlas. A holomorphic map from X to Y is a continuous map f : X →Y such that for every (i,j)∈I×J, the map φ ◦f ◦φ−1 :V ⊂Cn →N ⊂Cm j i i j is holomorphic. AholomorphicfunctiononacomplexmanifoldX isaholomorphicmapf :X →C. 1.21. Example. The Hopf varieties (cf. Exercise 1.19) admit a holomorphic map f :H →Pn−1 defined as follows: by definition H =(Cn\0)/Z and Pn−1 =(Cn\0)/C∗ and it is straightforward to see that the projection π : Cn \ 0 → Pn−1 factors through the projection π :Cn\0→H. 1.22. Exercise. Show that the fibres of f are elliptic curves. What can we say about their j-invariant ? (cid:3) 1.23. Exercise. Let V be a complex vector space of dimension n, and fix an integer 0 < r < n. Show that there exists a natural biholomorphism between the Grassmannians (cf. Exercise 1.18) G (V)→G (V∗), r n−r where V∗ is the dual space of V. (cid:3) 1.24. Exercise. Showthataholomorphicfunctiononacompactcomplexmanifold is constant. (cid:3) 1.25. Remark. We define the category of complex manifolds as the topological spacesthatlocallylooklikeopensetsinsomeCn andtheholomorphicfunctionsas theholomorphicmapstoC. Whilethisapproachisveryclosetothecorresponding definitions in differential geometry, an equivalent approach that is closer to the spiritofmodernalgebraicgeometryistodefineacomplexmanifoldasaringedspace (X,O )whereX isatopologicalspaceandO isthestructuresheaf(cf. Definition X X 8 ANDREASHÖRING 1.57), i.e. the sheaf of rings whose sections we define to be the holomorphic ones. For more details on this point of view, cf. [Wel80, Ch.1]. 1.26. Definition. A holomorphic map f : X → Y is a submersion (resp. immer- sion) if for every x ∈ X, there exists a coordinate neighbourhood of x such that the Jacobian of f has the maximal rank dimY (resp. dimX). A holomorphic map f :X →Y is an embedding if it is an immersion and f is a homeomorphism from X onto f(X). 1.27. Remark. It is an easy exercise to check that the rank of the Jacobian does not depend on the choice of the coordinate charts. 1.28. Definition. Let X be a complex manifold of dimension n, and let Y ⊂ X bea(closed)subset. ThenY isa(closed)submanifoldofX ofcodimensionk iffor each point x ∈ Y, there exist an open neighbourhood U ⊂ X and a holomorphic submersion f :U →Dk such that U ∩Y =f−1(0). 1.29. Remark. Sincemostofthesubmanifoldsthatwewillencounterarecompact we will very often omit to mention that they are closed. 1.30. Example. LetX andY becomplexmanifoldsofdimensionnandmrespec- tively. Let f : X → Y be a holomorphic map, and y ∈ Y such that the Jacobian J has rank m for every x ∈ f−1(y). Then the fibre f−1(y) is a submanifold of f dimension n−m. 1.31. Exercise. a) Show that a submanifold of a complex manifold is a complex manifold. b) Show that the image of an embedding f :X →Y is a submanifold of Y. (cid:3) 1.32. Exercise. Let X be a compact complex submanifold of Cn. Show that X has dimension zero (Hint: cf. Exercise 1.24). (cid:3) 1.33. Exercise. (1-dimensional complex tori) Let Λ⊂C be a lattice, and let X :=C/Λ be the associated complex torus. a) Show that X is diffeomorphic to S1×S1. b) Let ϕ : C/Λ → C/Λ0 be a biholomorphic map such that ϕ(0) = 0. Show that there exists a unique α∈C∗ such that αΛ=Λ0 and such that the diagram C z7→αz C π π0 C/Λ ϕ C/Λ0 commutes. Hint: recall (or prove) that the group of biholomorphic automorphisms of C is Aut(C)={z 7→αz+β | α∈C∗,β ∈C}. c) Show that X is biholomorphic to a torus of the form X(τ):=C/(Z+Zτ) where τ ∈C such that Im(τ)>0. KÄHLER GEOMETRY AND HODGE THEORY 9 d) Let H := {τ ∈ C | Im(τ) > 0} be the Poincaré upper half plane. We define a group action (cid:18) (cid:19) a b aτ +b SL(2,Z)×H→H,( ,τ) 7→ . c d cτ +d Show that the biholomorphic equivalence classes of complex tori of dimension 1 have a natural bijection with H/SL(2,Z). Remark: the set H/SL(2,Z) has a natural complex structure. „The J-invariant “ defines a biholomorphism H/SL(2,Z)→C (cf. e.g. [Eke06]). (cid:3) 1.34. Definition. A projective manifold is a submanifold X ⊂PN such that there exist homogeneous polynomials f ,...,f ∈ C[X ,...,X ] of degree d ,...,d 1 k 0 N 1 k such that X ={x∈PN | f (x)=...=f (x)=0}. 1 k Let f ,...,f ∈ C[X ,...,X ] be homogeneous polynomials. We will establish a 1 k 0 N sufficient (but not necessary!) condition for the closed set X ={x∈PN | f (x)=...=f (x)=0} 1 k to be a submanifold of PN. Let π :CN+1\0→PN be the projection map, we call π−1(X) the affine cone over X. It is straightforward to see that π−1(X)={x∈(CN+1\0) | f (x)=...=f (x)=0}, 1 k where we consider the f as polynomials on CN+1. Suppose now that for every j x∈π−1(X), the Jacobian matrix (cid:18) (cid:19) ∂f J = j ∂z l 1(cid:54)j(cid:54)k,0(cid:54)l(cid:54)N has rank k. Then π−1(X) is a submanifold of (CN+1\0) of dimension N +1−k. A straightforward computation shows that X is a submanifold of PN of dimension N −k. 1.35. Definition. A projective submanifold X ⊂ Pn of dimension m defined by n−mhomogeneouspolynomialsofdegreed ,...,d suchthattheJacobianhas 1 n−m rank n−m in every point is called a complete intersection. 1.36. Exercise. Let f ,...,f ∈ C[X ,...,X ] be homogeneous polynomials of 1 k 0 N degree d ,...,d and set 1 k X :={x∈PN | f (x)=...=f (x)=0}. 1 k Show that X is a submanifold of codimension m if and only if the Jacobian matrix (cid:18) (cid:19) ∂f J = j ∂z l 1(cid:54)j(cid:54)k,0(cid:54)l(cid:54)N has rank m for every point in the affine cone π−1(X). (cid:3) 10 ANDREASHÖRING 1.37. Exercise. Let f ∈C[X ,...,X ] be a homogeneous polynomial and set 0 N X :={x∈PN | f(x)=0}. Then the hypersurface X is smooth, i.e. a submanifold, if and only if ∂f ∂f {x∈(CN+1\0) | (x)=...= (x)=0} ∂X ∂X 0 N is empty. (cid:3) 1.38. Exercise. Show that C :={[X :Y :Z :T]∈P3 | XT −YZ =Y2−XZ =Z2−YT =0} is a submanifold of dimension one of P3. Can you find two homogenuous polyno- mials f ,f such that, as a set, 1 2 C ={[X :Y :Z :T]∈P3 | f ([X :Y :Z :T])=f ([X :Y :Z :T])=0} ? 1 2 What is the rank of the Jacobian matrix? (cid:3) 1.39. Exercise. LetV beacomplexvectorspaceofdimensionn,andfixaninteger 0<k <n. We define a map k ^ ψ :G (V)→P( V) k as follows: let U ⊂V be a subspace of dimension k and let u ,...,u be a basis of 1 k U. The multivector u ∧...∧u 1 k gives a point in P(VkV). a) Show that ψ is well-defined, i.e. does not depend on the choice of the basis. Show that ψ defines an embedding, the Plücker embedding. b) Show that G (V) is a projective manifold. k Hint: show that imψ can be identified with the set of multivectors w ∈VkV that are decomposable, i.e. there exists vectors v ,...,v ∈V such that 1 k w =v ∧...∧v . 1 k For every w ∈VkV consider the linear map k+1 ^ φ :V → V,v 7→v∧w. w and prove that w is decomposable if and only if rkφ (cid:54)n−k. w c)SetV :=C4,andlete ,...,e bethecanonicalbasis. Every2-vectorw ∈V2C4 1 4 has a unique decomposition w =X e ∧e +X e ∧e +X e ∧e +X e ∧e +X e ∧e +X e ∧e . 0 1 2 1 1 3 2 1 4 3 2 3 4 2 4 5 3 4 Showthatforthehomogeneouscoordinates[X :...:X ]onP(V2C4),thePlücker 0 5 embedding of G(2,C4) in P(V2C4)’P5 has the equation X X −X X +X X =0. 0 5 1 4 2 3 (cid:3)