1 COMPLEX MANIFOLDS, VECTOR BUNDLES AND HODGE THEORY JEAN-LUC BRYLINSKI PHILIP FOTH (cid:176)c Birkhauser Boston 1998. All print and electronic rights and use rightsreserved. Personal, non-commericaluseonly, forindividualswith permission from author or publisher. 2 Contents 1 Holomorphic vector bundles 5 1.1 Vector bundles over smooth manifolds . . . . . . . . . . 5 1.2 Complex manifolds . . . . . . . . . . . . . . . . . . . . . 8 1.3 Holomorphic line bundles . . . . . . . . . . . . . . . . . . 10 1.4 Divisors on Riemann surfaces . . . . . . . . . . . . . . . 15 1.5 Line bundles over complex manifolds . . . . . . . . . . . 17 1.6 Intersection of curves inside a surface . . . . . . . . . . . 24 1.7 Theta function and Picard group . . . . . . . . . . . . . 28 2 Cohomology of vector bundles 31 ˇ 2.1 Cech cohomology for vector bundles . . . . . . . . . . . . 31 2.2 Extensions of vector bundles . . . . . . . . . . . . . . . . 38 2.3 Cohomology of projective space . . . . . . . . . . . . . . 41 2.4 Chern classes of complex vector bundles . . . . . . . . . 47 2.5 Construction of the Chern character . . . . . . . . . . . . 52 2.6 Riemann-Roch-Hirzebruch theorem . . . . . . . . . . . . 53 2.7 Connections, curvature and Chern-Weil . . . . . . . . . . 54 2.8 The case of holomorphic vector bundles . . . . . . . . . . 64 2.9 Riemann-Roch-Hirzebruch theorem for CPn . . . . . . . . 66 2.10 RRH for a hypersurface in projective space . . . . . . . . 69 2.11 Applications of Riemann-Roch-Hirzebruch . . . . . . . . 71 2.12 Dolbeault cohomology . . . . . . . . . . . . . . . . . . . 75 2.13 Grothendieck group . . . . . . . . . . . . . . . . . . . . . 83 2.14 Algebraic bundles over CPn . . . . . . . . . . . . . . . . . 84 3 Hodge theory 89 3.1 Complex and Riemannian structures . . . . . . . . . . . 89 3 4 CONTENTS 3.2 K¨ahler manifolds . . . . . . . . . . . . . . . . . . . . . . 93 3.3 The moduli space of polygons is K¨ahler . . . . . . . . . 104 3.4 Hodge decomposition in dimension 1 . . . . . . . . . . . 107 3.5 Harmonic forms on compact manifolds . . . . . . . . . . 109 3.6 Hodge theory on K¨ahler manifolds . . . . . . . . . . . . . 112 3.7 Hodge Conjecture . . . . . . . . . . . . . . . . . . . . . . 128 3.8 Hodge decomposition and sheaf cohomology . . . . . . . 131 3.9 Formality of cohomology . . . . . . . . . . . . . . . . . . 132 4 Complex algebraic varieties 137 4.1 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.2 Signature . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.3 Examples and Siegel space . . . . . . . . . . . . . . . . . 142 4.4 Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.5 Algebraic cycles . . . . . . . . . . . . . . . . . . . . . . . 161 4.6 Operations on algebraic cycles . . . . . . . . . . . . . . . 171 4.7 Abel-Jacobi theorem . . . . . . . . . . . . . . . . . . . . 175 4.8 K3 surface . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.9 Compact complex surfaces . . . . . . . . . . . . . . . . . 181 4.10 Cohomology of a quadric . . . . . . . . . . . . . . . . . . 186 4.11 Lefschetz theorem . . . . . . . . . . . . . . . . . . . . . . 190 5 Families and moduli spaces 199 5.1 Families of algebraic projective manifolds . . . . . . . . . 199 5.2 The Legendre family of elliptic curves . . . . . . . . . . . 208 5.3 Deformation of complex structures . . . . . . . . . . . . 214 5.4 Vector bundles over an elliptic curve . . . . . . . . . . . 219 5.5 Moduli spaces of vector bundles . . . . . . . . . . . . . . 221 5.6 Unitary bundles and representations of π . . . . . . . . 224 1 5.7 Symplectic structure on moduli spaces . . . . . . . . . . 232 5.8 Verlinde formula . . . . . . . . . . . . . . . . . . . . . . 237 5.9 Non-abelian Hodge theory . . . . . . . . . . . . . . . . . 243 5.10 Hyper-K¨ahler manifolds . . . . . . . . . . . . . . . . . . 252 5.11 Monodromy groups . . . . . . . . . . . . . . . . . . . . . 255 Chapter 1 Holomorphic vector bundles Weaseling out of things - this is what separates us from the animals (except for weasel). Homer Simpson 1.1 Real and complex vector bundles over smooth manifolds Vector bundles arise in geometry in several contexts. One may remem- ber from the study of smooth manifolds that the notion of tangent bundle inevitably appeared as a powerful tool of differential geometry. If the dimension of a manifold M is k then the dimension of the total space of the tangent bundle to M is twice as big. The first and simple example arises when we take M = Rk. Here the tangent bundle is just the direct product of two copies of Rk. So, TRk = {(x,y); x,y ∈ Rk}. A vector bundle always comes with the projection map p to the manifold. In turn, the manifold is imbedded into the bundle as its zero-section σ : 0 TM σ ↑↓ p 0 M In fact, for every point x ∈ M, the fiber p−1(x) is the tangent space T M. x 5 6 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES To define a vector bundle properly, we also need the local triviality condition. The map p is a submersion and represents a locally triv- ial fibration meaning the following. Any point x ∈ M has an open neighbourhood U, such that we have a trivialization: p−1(U) (cid:39) U ×Rk (cid:38) (cid:46) U Next we introduce the important notion of a section of the tangent bundle. A section of TM is a smooth mapping v : M → TM such that p·v = Id . A section of TM is exactly a smooth vector field on the M manifold M. We denote by Γ(TM) the space of all smooth sections of TM. Apparently, it has the structure of a vector space. Besides, if we take a smooth function f ∈ C∞(M) and a section v ∈ Γ(TM), then f.v is a section of TM too, so Γ(TM) also is a module over C∞(M). In addition, Γ(TM) has a Lie algebra structure under the bracket of vector fields. Further, one meets the first example of a dual vector bundle as one consideres the cotangent bundle T∗M, which is dual to TM. The fiber of T∗M over a point x ∈ M is the cotangent space T∗M = (T M)∗. x x The sections of T∗M are the smooth 1-forms on M. An interesting fact is that the manifold T∗M has a canonical structure of a symplectic manifold. It means that there exists a two-form ω on M such that dω = 0 and ω ∧...∧ω is a volume form. To get ω explicitly, we take (cid:124) (cid:123)(cid:122) (cid:125) k the Liouville 1-form α on T∗M defined as follows. For x ∈ M, ξ ∈ T∗M x and v, a tangent vector to T∗M at (x,ξ) we let (cid:104)α ,v(cid:105) = (cid:104)ξ,dp(v)(cid:105), (x,ξ) where dp(v) ∈ T M. In local coordinates (x ,...,x ,ξ ,...,ξ ) on T∗U x 1 k 1 k with U - small open subest of M diffeomorphic to Rk we have α = (cid:80) k ξ dx . The symplectic form is now taken as ω = dα. (In local i=1 i i (cid:80) coordinates, ω = k dξ dx . It is easy to see that dω = 0 and ωk = i=1 i i (−1)kk!dx ∧···∧dx ∧dξ ∧···dξ .) 1 k 1 k LetgiverigorousdefinitionofarealvectorbundleE overamanifold M. First, we require the existence of a smooth map p: E ↓ p M 1.1. VECTOR BUNDLES OVER SMOOTH MANIFOLDS 7 Next, we define a manifold E× E ⊂ E×E consisting of pairs (v ,v ) M 1 2 in the same fiber: E × E = {(v ,v ) ∈ E ×E, such that pv = pv }. M 1 2 1 2 We must have the smooth addition map: + E × E → E M (cid:38) (cid:46) M and the smooth dilation map R×E → E. We impose the requirement that each fiber has a vector space structure. Besides, we need the local triviality condition as in the case with the tangent bundle: p−1(U) (cid:39) U ×Rk (cid:38) (cid:46) U It follows that E × E is a closed submanifold of E ×E. We can put M the field of complex numbers C instead of R to obtain the definition of a smooth complex vector bundle over M. We notice that the number k in the definition is usually referred to as the rank of the bundle E. Asectionofavectorbundlep : E → M isasmoothmaps : M → E such that p◦s = Id . M Havingabundle,wecandefinethedualvectorbundle,itssymmetric and exterior powers. Also, we can have the direct sum of two bundles as well as the tensor product of them. Though it is more or less clear in the former case, we claim that there exists unique manifold structure on E⊗F, such that if v,w are smooth sections of E and F respectively, then v⊗w is a smooth section of E⊗F. In terms of trivializations we have E (cid:39) U ×Rp and F (cid:39) U ×Rq imply (E ⊗F) (cid:39) U ×Rpq. |U |U |U One can form from two bundles E and F on M a ”hom bundle”. By definition, Hom(E,F) = E∗⊗F. There exists a vector bundle map Hom(E,F)⊗E → F. It is formed via Hom(E,F)⊗E (cid:39) E∗ ⊗F ⊗E (cid:39) (E ⊗E∗)⊗F → R⊗F (cid:39) F. So,onenaturallyhasthemappingΓ(Hom(E,F)) → Hom(Γ(E),Γ(F)). 8 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES 1.2 Complex manifolds The basic difference with the real case is that the transition functions are biholomorphic. For any open set U ⊂ M we have an algebra H(U) of holomorphic functions over U. The complex structure on M is com- pletely defined by a linear map J : TM → TM of its tangent bundle, such that J2 = −Id . So, TM becomes a complex vector bundle. TM Furthemore, J must satisfy some integrability condition. If M is a complex manifold we introduce the notion of holomorphic vector bundle E. E ↓ p M First, E is a complex vector bundle and the total space is a complex manifold. We require that the map p, the addition and dilation maps are holomorphic. Example. We consider in detail the complex projective space CPn, because it has a lot of structures and interesting holomorphic vector bundles. The complex manifold CPn is defined as the set of lines in Cn+1 through the origin and is covered by n + 1 open sets U ,U ,...,U . 0 1 n Each U is biholomorphic to Cn and is defined as the set of lines in Cn+1 i spanned by a vector (z ,z ,...,z ) with z (cid:54)= 0. The map ψ : U → Cn 0 1 n i i i may be viewed as the one sending the line passing through the point (z ,z ,...,z ) ∈ Cn+1 to the point (z0,..., zi−1, zi+1,..., zn) ∈ Cn. The 0 1 n zi zi zi zi inverse map ψ−1 : Cn → U sends the point (u ,...,u ) to the line in i i 1 n CPn+1 passing through the point (u ,...,u ,1,u ,...,u ). 1 i i+1 n It is useful to introduce the homogeneous coordinate notation. A point on CPn is denoted by [z : z : ··· : z ], where at least one 0 1 n coordinateisnotzero, andrepresentsthelinepassingthroughthepoint (z ,...,z ) in Cn+1. In our new notation the point [z : z : ··· : z ] is 0 n 0 1 n the same as the point [λz : λz : ··· : λz ] for any λ (cid:54)= 0. 0 1 n The topology on CPn as well as the complex manifold structure is determined by those on U . Each U is an open set and we carry over i i the topology from Cn to U . The complex manifold structure is given i by the atlas ψ ,...,ψ . Let us show it explicitly in low dimensions. 0 n 1.2. COMPLEX MANIFOLDS 9 In the case n = 1 we have CP1 = U ∩ U , U ψ(cid:39)0 C, U ψ(cid:39)1 C, 0 1 0 1 U := U ∩U ⊂ CP1: 0,1 0 1 U →ψ0 C∗ 0,1 ψ (cid:38) (cid:46) φ 1 C∗ The map φ has to be compatible with the complex manifold structure. Now we have ψ [z : z ] = z /z ∈ C∗ and ψ [z : z ] = z /z ∈ 0 0 1 1 0 1 0 1 0 1 C∗. This means that φ is a holomorphic map from C∗ to C∗: φ(u) = u−1. The manifold CP1 is called also the ”Riemann sphere”, partly because it is homeomorphic to the usual sphere S2 and Riemann was among the first who treated it as a complex manifold. We see that it is possible to represent CP1 as the union of U and the point ”at 0 infinity”. Alsothereexistsawell-knownstereographicprojectionwhich holomorphically identifies U with C. 0 This discussion gives us the idea how to prove that CPn = (Cn+1 \ {0})/C∗ with the quotient topology is compact in general. The space Cn+1 \ {0} contains the sphere S2n+1 = {(z ,...,z ) ∈ Cn+1; |z |2 + 0 n o ... + |z |2 = 1}. Besides, we have the direct product decompositions n Cn+1 \ {0} = R∗S2n+1 and C∗ = R∗T, where T = {z ∈ C,|z| = 1} - + + the unit circle. Now we represent CPn as the quotient S2n+1/T, which is clearly compact. The next case to consider is n = 2. Here we have CP2 = U ∪U ∪U . 0 1 2 The subset U identifies with C2 via ψ [z : z : z ] = (z /z ,z /z ). 0 0 0 1 2 1 0 2 0 Analogously, U (cid:39) C2 is provided by ψ [z : z : z ] = (z0, z2). We 1 1 0 1 2 z1 z1 have: U →ψ0 C2 \(u = 0) (u,v) ∈ C2 0,1 ψ (cid:38) (cid:46) φ 1 0,1 C2 \(α = 0) (α,β) ∈ C2 We notice that 1 v φ (u,v) = ( , ) 0,1 u u is a holomorphic map. Its inverse 1 β φ−1(α,β) = ( , ) 0,1 α α 10 CHAPTER 1. HOLOMORPHIC VECTOR BUNDLES is a holomorphic map too. We conclude that φ is biholomorphic. 0,1 Another way to deduce biholomorphicity is to consider the differential map dφ (u,v) : C2 → C2. We have 0,1 (cid:195) (cid:33) (cid:181) (cid:182) ∂α ∂β − 1 − v dφ = ∂u ∂v = u2 u2 . 0,1 ∂α ∂β 0 1 ∂v ∂u u The determinant of dφ (u,v) is equal to − 1 and is never zero. Since 0,1 u3 φ is holomorphic and bijective, then it is automatically a biholomor- 0,1 phic map. We have a well-known Fact. Any holomorphic function on a compact connected manifold M is constant. Proof. Let f : M → C be a holomorphic function. Since M is compact, |f| attains its maximum value at some point. The maximum principle implies that f is constant in some neighbourhood of this point. Now theprincipleofanalyticalcontinuationtellsusthatf islocallyconstant on M. (cid:176) Exercise. Another consequence of the maximum principle is that a compact connected complex Lie group is always abelian. 1.3 Holomorphic line bundles Now we are ready to consider the first example of a non-trivial line bundle E on CPn, which is called the tautological line bundle. The bun- dle E is defined as a subbundle of the trivial bundle Cn+1×CPn. More precisely, E = {(v,l) ∈ Cn+1 ×CPn;v ∈ l}. We can show that this line bundle satisfies the local triviality condition by finding a non-vanishing section over each open set of some covering. Let us define the section σ i overU ⊂ CPn byσ [z : ... : z ] =((z /z ,...,z /z ,1,z /z ,...,z /z ), i i 0 n 0 i i−1 i i+1 i n i line through (z ,...,z )). 0 n It turns out that the line bundle E has no non-zero global sections. LEMMA 1.3.1 Γ (E) = {0}. hol Proof. We have the inclusion Γ (E) ⊂ Γ (Cn+1 ×CPn). The latter hol hol space is the same as the space of (n+1)-tuples (f ,...,f ) of functions 1 n+1