0.1. VECTOR BUNDLES 1. In this chapter, we discuss the Hermitian Geometry on complex manifolds and Hermitian vector bundles.

Transcription

1 01 VECTOR BUNDLES 1 Math 6397 Riemannian Geometry II HERMITIAN MANIFOLDS AND HERMITIAN VECTOR BUNDLES Min Ru In this chapter, we discuss the Hermitian Geometry on complex manifolds and Hermitian vector bundles 01 Vector Bundles Definition 01 Suppose M is a complex manifold, and E is a topological space We call E an holomorphic vector bundle over M, if there exists a continuous mapping π : E M such that (1) E p = π 1 (p); p M, is a linear space with rank r, ie E p = C r (2) There exists an open covering {U α } α I of M, and π 1 (U α ) is holomorphic to topology product U α C r That is φ α : π 1 (U α ) U α C r, α I such that φ α : E P {p} C r C r, p U α is a C linear isomorphism between complex vector space E p and C r (U α, φ α ) can be considered as a neighborhood of local coordinations We call such U α the trivialization neighborhood of E On U α U β φ, for p U α U β, let φ αβ := φ α φ 1 β φ β(e p ) : C r φ 1 β φ α E p C r, then φ αβ (p) GL(r, C) and φ αβ : U α U β GL(r, C) is holomorphic φ αβ is called transitive function of E (3) The φ αβ satisfies the compatible conditions provide by the definition of φ αβ φ αβ (p)φ βγ (p) = φ αγ (p); p U α U β U γ φ αβ (p) = φ βα (p) 1 ; p U α U β then we call E, φ, M or briefly E is an holomorphic bundle with rank r Note that the meaning of φ αβ is that, for any vector on E p, there are different coordinates with respect to the different trivialization neighborhood of E Then the different coordinates will transform according to the transitive function Since the transitive function only depends on p, the transformation of the different coordinates only depends on p and is independent of the vectors For example, for any V E p, p U α U β, let Vα i and V j β, 1 i, j r be the coordinates of V in the neighborhoods φ 1 (U α ) = U α C r and φ 1 (U β ) = U β C r respectively Then V α = φ αβ (p)v β, where V α = (V1 α,, Vr α ) t, V β = (V β 1,, V r β ) t φ αβ (p) GL(r, C) only depends on p It does not depend on the vector V We also note that E is

2 2 also a complex manifold The neighborhood {π 1 (U α ) = U α C r } α I is just its open covering of the local coordinates Moreover the projective mapping π : E M is a holomorphic mapping In general the rank of E is independent of the dimension of the manifold M If rank E = 1, we call this E a holomorphic line bundle In the above definition, the transitive function φ αβ : U α U β GL(r, C) is holomorphic, which means that eevry entry of the matrix representation of φ αβ is holomorphic on U α U β Let M be a complex manifold and {U α } α I be a covering of M, if there are φ αβ, φ αβ : U α U β GL(r, C), such that φ αβ are holomorphic and satisfy (3) Then we can construct a holomorphic vector bundle with transitive functions {φ αβ } To do that, we first introduce an equivalent relation on (U α C r ), as (p, a α ) U α C r, (q, a β ) U β C r α I (p, a α ) (q, a β ) p = q and a α = φ αβ a β Using (3), it is easy to verity is an equivalent relation on (U α C r ) Let α I E = (U α C r )/ and define π : E M by π[p, a α ] = p, where [p, a alpha ] α I is the equivalent class of (p, a α ) Then (E, φ, M) is a holomorphic vector bundle with rank r, U α ; α I are the trivialization neighborhoods of E, and {φ αβ } are the transitive functions of E In general, if all φ αβ belong some category of functions, then we call E a vector bundle in such category For example, if all φ αβ are holomorphic, then E is a holomorphic vector bundle; if all φ αβ are C, then E is a C vector bundle; and if all φ αβ are continuous, then E is a continuous vector bundle Here φ αβ is C or continuous means that every entry of φ αβ are C or continuous respectively In definition 11, E is a holomorphic vector bundle with the base complex manifold M and fibre C r The concept of the vector bundle is not only in the complex categor y, but also in the real category Definition 02 Let π : E M be a holomorphic vector bundle on a complex manifold bundle M Let U be an open set of M A continuous (C, holomorphic) mapping f : U E, such that π f = id U, is called a continuous (C, holomorphic) section over U We use Γ 0 (U, E)(Γ(U, E), Γ(U, E)) to denote the set of continuous (C, holomorphic) sections over U Let U α be a trivialization neighborhood of the holomorphic vector bundle E with rank r We define the canonical sections over U α by e α i (x) = φ 1 (x, (0,, 1, 0,, 0)), x U α, 1 i r Such e α i, 1 i r, are holomorphic sections over U α, and for x U α, e α, 1 i r, generate the fibre E x So we call {e α i } 1 i r the (canonical) local frame of E For every holomorphic vector bundle E, and every trivialization neighborhood U α, a local frame always exists If U α U β φ, then (11) e α k = φ βαjk e β j, j=1

3 01 VECTOR BUNDLES 3 where φ αβ = (φ αβij ) 1 i,j r is the matrix representation of the transitive functions of E Let U be an open set of M, s : U E be a section, and {U α } α I be the trivialization neighborhood of E Then the section s U U α may be expressed by (x, s α (x)) U α C r = π 1 (U α ), x U α U s is a continuous (C, holomorphic) section is equivalent to that s α, α I, are continuous (C, holomorphic) on U α U In terms of the canonical local frame, 1 i qr}, we can write {e α i where If U α U β φ, then s U U α = s α = (12) s β,j = ie s α,1 s α,r s α,i e α i, i=1 C r s α,k φ βαjk, k=1 s β = φ βα s α ; on U α U β where where φ αβ = (φ αβij ) 1 i,j r is the matrix representation of the transitive functions of E If s, t Γ 0 (U, E)(Γ(U, E), Γ(U, E)), we define (s+t)(x) = s(x)+t(x); x U Then s + t is still a section over U and s + t Γ 0 (U, E)(Γ(U, E), Γ(U, E)) Moreover it is easy to verify that Γ 0 (U, E)(Γ(U, E), Γ(U, E)) is an abelian group with respect to the above addition operation Hence ( Γ(U, E), r UV ) U UM is a presheaf on M, where r UV is restriction holomorphism We use O(E) to denote the sheaf generated by presheaf ( Γ(U, E), r UV ) U UM and O(E) is called the sheaf of the germs of holomorphic sections of E For every U α the trivialization neighborhood of E, O(E) U α = ro Uα, where r is the rank of vector bundle E Note that a O-sheaf F on a complex manifold X is called locally free if there exist an open covering {U α } α I and an O-isomorphism φ α : p α O U α = F Uα for all α I Trivially, if X is connected, then p α = p β, for α, β I So, in this case, O(E) is a locally free sheaf On the contrary, if F is a locally free O sheaf on a connected complex manifold M, then there exists a holomorphic vector bundle E on M with rank r, such that F = O(E) In fact, if we let E (α) 1,, E r (α) be the canonical sections of ro U α, and E (β) 1,, E r (β) canonical sections of ro U β, on U α U β φ, since (E (α) i generate the ro U α and ro U β respectively, we have (13) E (α) i = j=1 a α βije (β) i ; 1 i r ) 1 i r and (E (β) i be ) 1 i r

4 4 where a α β ij are holomorphic on U α U β Then φ αβ = (a α β ij ) 1 i,j r is a nonsigular r r matrix If we construct a holomorphic vector bundle E by means of the transitive functions φ αβ, then F = O(E) Example 1 The trivial bundlee = M C r, π(p C r ) = p; p M, then (14) g αβ = I r, α, β I where {U α } α I is any given open covering of M Example 2 Tangent bundle This is the most important example of vector bundles, which is also the background of the concept of the vector bundle Suppose M is a differential manifold with real dimensional n, M = U α, where {U α } α I is an open covering of M which consists of the local coordinate neighborhood of M For p M, let T p M be the tangent space of M at point p Then T M = T p M is the tangent bundle of M, where π : T M M is defined by π(t p M) = p Let {U α } α I be a coordinate system of M, ie each U α, α I, is the local coordinate neighborhood Let π 1 (U α ) = T p (M) and φ α : π 1 (U α ) p U α U α R n If (x 1,, x n ) is a local coordinate system of U α, then, for p U α, T p (M) = { n a i x i p a i R, i = 1,, n} For V = n a i x i p i=1 T p M, φ α (V ) = (x 1 (p),, x n (p), a 1,, a n ) If U β is another local coordinates neighborhood of the point p and (y 1,, y n ) is a local coordinate system of U β Then V T p M has another representation V = n b j y j p, and we have (15) a i = n j=1 α I j=1 b j xi y j p; i = 1,, n The following equality is the matrix presentation of (15) (16) a 1 a n = x 1 x y 1 1 y n x n x y n 1 y n Hence the transitive function φ αβ = ( xi y j ) : U α U β GL(n, R) is the Jacobi Matrix of the transformation of local coordinates system It is easy to verify that the transitive functions φ αβ satisfy (12) and C T M is a real vector bundle with rank n p b 1 b n Example 3 Holomorphic Tangent Bundle T 10 (M) i=1 p M

5 01 VECTOR BUNDLES 5 Suppo se that M is a n-dimension complex manifold, {U α } α I is an open covering of M, which consists of the local coordinate neighborhoods of M Let (z 1,, z n ) be local coordinates of U α, Tp 10 (M) be the set of all (10) tangent vectors In terms of the local coordinates of U α, p U α, Tp 10 (M) := { n a i z i p a i C; a i n}, T 10 (M) = Tp 10 (M), and π : T 10 (M) 1 p M M be defined by π(tp 10 (M)) = p Then π 1 (U α ) = p (M) Let φ α : π 1 (U α ) U α C n be defined, for every V = T 10 p U α n a i i=1 z i p Tp 10 (M), φ α (V ) = (p, a 1,, a n ) = (z 1 (p),, z n (p), a 1,, a n ) If (w 1,, w n ) is a local coordinate system of U β, and p U α U β, then each (10) tangent vector V Tp 10 (M) has another representation by means of local coordinates of U β, ie V = n b j w j p Then (17) a 1 a n = j=1 z 1 z w 1 1 w n z n z w n 1 w n and φ αβ = ( zi w ) : U i α U β GL(n, C) φ αβ is the Jacobi Matrix of the transformation of the local coordinates system Since zi w, 1 i, j n, are j holomorphic fu nctions of (w 1,, w n ), φ αβ : U α U β GL(n, C) is holomorphic and φ αβ satisfy (12) Therefore T 10 (M) is a holomorphic vector bundle with rank n Example 4 Holomorphic Cotangent Bundle T (10) (M) Let M be a n-dimension complex manifold and {U α } α I be an open covering of M which consists of the local coordinate neighborhoods For p U α M, in terms of local coordinates z 1,, z n of U α, T p (10) (M) = { n c i dz i c i C; i=1 1 i n}, T (10) (M) = (M), π : T (10) (M) M is given by p M T (10) p p b 1 b n π(t (10) p ) = p, π 1 (U α ) = p U α T (10) p (M), φ α : π 1 (U α ) that, for ω = n i=1 c idz i Tp (10) w 1,, w n is a local coordinate system of U β, then ω T (10) representation, in terms of w 1,, w n, ω = 1 i n, have a matrix representation: U α C n, such (M), φ α(ω) = (p, c 1,, c n ) If p U α U β, and (M) has another (c 1,, c n ) = (d 1,, d n ) n j=1 p d j dw j Then c i = w 1 w z 1 1 z n w n w z n 1 z n n j=1 d j w j z i,

6 6 or (18) c 1 c n = w 1 w z 1 1 z n w n w z n 1 z n t d 1 d n So gαβ = φ αφ 1 β = ( wi z ) t : U j α U β Gl(r, C) is holomorphic and satisfy (12) Hence T (10) (M) is a holomorphic vector bundle with rank n Comparing (17) and (18), the relation of the matrix representation of the transitive functions is gαβ = (gt αβ ) = (g 1 αβ )t Example 5 Let (A, π, M) and (B, π, M) be two vector bundles over a manifold M We can construct new vector bundles from A and B For example (1) The dual bundle A of A, where π : A M is given by π(a p) = p, and A p = (A p ) is the dual space of A p (2) A B, where (A B) p = A p B p, and π : A B M is given by π((a B) p ) = p (3) A B, where (A B) p = A p B p, and π : A B M is given by π((a B) p ) = p Let {U α } α I be an open covering of M, which consists of the common trivialization neighborhoods of A and B, and let {φ αβ } and {ψ αβ } be the transitive functions of A and B respectively ( Then ) the transitive functions of A, A B, and A B are (g 1 gαβ 0 αβ )t,, and g 0 Ψ αβ ψ αβ respectively, where αβ g αβ ψ αβ is decided by t he order of the arrangement of the locally frame of A B Example 6 Λ q T (01) (M), the q extra-product of T (01) (M) For p M, (Λ q T (01) (M)) p = { a i1,,ı q (p)d z i1 d z iq a i1,,i q (p) C and a i1 i q is anti sysmetric for i 1,, i q } Λ q T (01) (M) is a C vector bundle on M with rank ( n q) Let π : E M be a holomorphic vector bundle on complex manifold M with rank r, U be an open set of M, a (smooth) E-valued (0, q) form ω is ω = k j=1 ω j s j, where s j Γ(U, E), Γ(U, E) denotes the group of the smooth sections of E over U, and ω j are smooth (0, q) forms on U We use ε 0,q (U, E) to denote the set of all (smooth) E-valued (0, q) forms on U, and denote by ε 0,q (E) the sheaf of germs of E-valued (0, q) forms, generated by the presheaf (ε 0,q (U, E), r UV ) U UM Let U be an open set, then Γ(U, ε 0,q (E)) = ε 0,q (U, E) If (U, z 1,, z n ) is a local coordinate, then ω has a local expression as follows ω = s i1 i q d z i1 d z iq 1 i 1< <i q m

7 01 VECTOR BUNDLES 7 where s i1,ı q Γ(U, E) Let U α be a trivialization neighborhood of E, then ω has a local expression on U α ω U Uα = s (α) i 1,,i q d z i1 d z iq 1 p 1< <p q m where s (α) i 1,,i q Γ(U U α, E), 1 i 1 < < i q m, and or ω U Uα = s (α) i 1,,i q = k=1 1 p 1< <p q m k=1 s α,k i 1,,i q e α k, For a (smooth) E-valued (0, q) form ω on U, we write we define ω = ω = s α,k i 1,,i q (d z i1 d z iq ) e α k ωk α e α k, k=1 ( ω k α ) e α k k=1 We show that is well-define In fact, if U α U β φ, ω has another expression and from (11), ω = ω β j = ω β j eβ j, j=1 ωk α φ βαjk k=1 Since φ βαjk are holomorphic, φ βαjk = 0 Thus, ω β j = This, together with (11) implies, ( ω k α )φ βαjk k=1 ( ω k α ) e α k = k=1 ( ω β j ) eβ j So is well-define Hence we have the following exact sequence of sheaves (113) 0 O(E) i ε (0,0) (E) j=1 ε (0,1) (E) ε (0,2) (E)

8 8 ε (0,3) (E) ε (0,n) (E) 0 where O(E) is the sheaf of germ of the holomorphic section of E Since = 0 and by the Dalbeaultd Theorem, (113) is an exact sequence of sheaves We define that (115) H (0,q) (M, E) = H q (M, ε 0, (E)) and = {ω Γ(M, ε0,q (E) ω = 0} {η η Γ(M, ε 0,q 1 (E))} ; H 0,0 (M, E) Γ(M, O(E)) 1 q n The fundamental problem (Hodge theory) is to study H q (M, O(E)), 0 q n, and to prove dim C H 0,q (M, E) < +, when M is a compact complex manifold The practice method is to connect such cohomology group with the harmonic forms and to discuss the resolutions of the harmonic operator To discuss the resolutions of the harmonic operator, we need to extend Γ(M, ε 0,q (E)) to a Hilbert space, on which the Harmonic operator operates Then similar to what we have done in the L 2 estimate, we apply the L 2 methods in Hilbert space 02 Hermitian Manifolds and Hermitian Vector Bundles At first we need to endow w ith a metric on the holomorphic vector bundle E, as well as a metric on the base manifold M Let M be a complex manifold with dimensional n, T (10) (M) be its (1,0)- tangent vector bundle of M Sometimes we call T 1,0 (M) the holomorphic tangent bundle of M Definition 03 A Hermitian metric, denoted by ds 2, is a set of innerproduct {, p } p M such that (1) For p M,, p is a Hermitian inner product on T p (1,0) (M), ie η, ζ Tp 1,0 (M), c 1, c 2 C, ξ, ξ > 0, as ξ 0; c 1 ξ + c 2 ξ, ζ = c 1 ξ, ζ + c 2 η, ζ and ξ, η = η, ξ (2) If ξ, η are C section of T 1,0 (M) over an open set U, then ξ, ζ is the C function on U If z 1,, z n is a local coordinate system of M, then z, 1 i n, are i holomorphic sections on this local coordinates neighborhood U, and (116) g i,j = z i, z j ; 1 i, j n

9 02 HERMITIAN MANIFOLDS AND HERMITIAN VECTOR BUNDLES 9 is the C function on U with g i, j = g j,ī We can write this Hermitian metric as ds 2 = n i,j=1 g i, jdz i d z j Since ξ, ξ > 0 for ξ 0, the matrix g = (g i, j) 1 i,j n > 0, ie, g = (g i, j) 1 i,j n is a positive definite Hermitian matrix A complex manifold with a given Hermitian metric is said a Hermitian manifold If M is a n-dimensional complex manifold, then M is a 2n-dimensional real manifold If z 1,, z n is a local complex coordinate system, writ e z i = x i + 1y i, 1 i n, then (x 1, x 2,, x n, y 1, y n ) is a real local system in the same coordinate neighborhood For p M, T p (M) is the tangent space of M at p Since M is real 2n dimensional manifold, T p (M) = R 2n, and x 1 p x 2 p x n p y 1 p y is a frame of T n p (M) Because M is a complex manifold, there is a naturally almost complex structurej : T p M p T p M; p M, where J is a linear transformation of T p M such that (118) { J( x ) = i y ; i J( y ) = i x ; 1 i n, i It is easy to verify the (118) is independent of the complex coordinates z 1,, z n, although the definition of (118) is by means of local coordinates z 1,, z n If we extend J : C T (M) C T p (M), and set z i = 1 2 ( x i 1 y i ) and z i = 1 2 ( x i + 1 ); 1 i n, yi then (116) J z i = 1 2 (J( x i ) 1( y i )) = 1 2 ( y i + 1 x i ) and = 1 2 ( x i 1 y i ) = 1 z i, (117) J z i = 1 2 (J( x i ) + 1( y i )) = 1 2 ( y i 1 x i ) = 1 2 ( x i + 1 y i ) = 1 z i Since J 2 = id, z 1 z are eigenvectors of J with eigenvalue 1, and n z,, 1 z are eigenvectors of J with eigenvalue 1 Moreover all the n vectors of Tp 10 (M) are eigenvectors of J with eigenvalue 1 For a n-dimensional Hermitian manifold M, which is naturally a real 2ndimension smooth manifold, if ds 2 = n i,j=1 g īj dz i d z j, is a Hermitian metric on M, the Hermitian metric matrix G = (g ij ) is a n n positive Hermitian matrix

10 10 Write G = A + 1B Since G = t Ḡ, A t = A and B = B t If we write ds 2 in terms of matrix representation, then ds 2 = dz t Gd z Using dz k = dx k + 1dy k, 1 k n, then x 1,, x n, y 1,, y n are real local coordinates of M, (118) ds 2 = (dx + (dx + 1 dy) t (A + 1B)(dx 1dy) ( ) ( ) = (dx t, dy t A B dx ), B A dy where ds 2 is a Riemainnian metric on M, we call this metric is the ( underlying ) A B Riemannian metric on the Hermitian manifold The matrix is B A trivially symmetric, for v = n (a i x + b i i y ) 0 i ( (119) v 2 = v, v = (a t, b t A B ) B A 1 = a t Aa + b t Bb b t Ba + a t Bb = (a + 1b) t G(a 1b) > 0 ) ( a b Therefore (119) is exactly a Riemannian metric on M, where (a + 1b) t = (a 1 + 1b 1,, a n + 1b n ) and (a 1b) in the corresponding column vector This underlying Riemannian metric possess a J-invariant property, ie for V, W T p (M) and p M (1020) V, W = JV, JW ) If V = n then and (a i i=1 x i + b i y i ), W = n JV = JW = (c i i=1 x i + d i y i ) n ( b i x i + ai y i ) i=1 n ( d i x i + ci y i ) i=1 Therefore ( JV, JW = ( b t, a t A B ) B A ( = a t, b t 0 I I 0 ) ( d c ) ( A B B A ) ) ( 0 I I 0 ) ( c d )

11 02 HERMITIAN MANIFOLDS AND HERMITIAN VECTOR BUNDLES11 ( = a t, b t A B B A = V, W ) ( c d On the contrary, if M is a n-dimensional complex manifold and with a J- invariant Riemannian metric, then this J-invariant metric is a underlying Rienannian metric of an Hermitian manifold Suppose (z 1,, z n ) is a local coordinate system and z i = x i + 1y i, 1 i n, and the expression of the Riemannian metric ( ) ( ) ds 2 = (dx t, dy t A B dx ), C D dy then the J-invariant property of the Riemannian metric implies A = A t, A = D, B = B t and B = C Hence ds 2 = dz t (A + 1B)dz is an Hermitian metric, and the J-invariant Riemannian metric is the underlying Riemannian metric of this Hermitian metric Let E be a holomorphic vector bundle on a n-dimensional complex manifold M with rank r, we can endow with the Hermitian metric on E, similar to what we did on T 10 (M) Definition 04 A Hermitian metric on the vector bundle E is a set of innerproducts, p on E p, p M which satisfying following conditions (1) {, } p is a Hermitian inner-products on E p (2) {, } p is smooth on M, ie for any open set U of M, and s 1, s 2 Γ(U, E), then s 1, s 2 p is a smooth function on U Let {U α } α I be a trivialization neighborhoods of E, e α,1,, e α,r be the holomorphic frame on π 1 (U α ), and {φ αβ } be the transitive functions with respect to {U α } α I Setting h (α) λ µ = {e α,λ, e α,µ }; ξ, η E p, ξ = r ξ(α) λ e α,λ, η = r η µ (α) e α,µ, then 1 1 ) 1 λ, µ r (121) {ξ, η} p = ξ λ (α) ηµ (α) h(α) λ µ (p) where h (α) = (h (α) λ µ ) 1 λ,µ r is a r r positive Hermitian matrix So where ξ (α) = ξ 1 (α) ξ r (α) {ξ, η} p = ξ t (α) h(α) η (α), is the column vector, so is η (α) Let U β be another trivialization neighborhood of E and p U α U β Then ξ = ξ(α) λ e α,λ = ξ(β) λ e β,λ λ=1 λ=1

12 12 and ξ (α) = φ αβ ξ (β) (122) {ξ, η} p = ξ t (α) h(α) η (α) = ξ t (β) φt αβh (α) φ αβ η (β) = ξ t (β) h(β) η (β) so that (123) h (β) = φ t αβh (α) φ αβ ; onu α U β A holomorphic vector bundle endowed with a Hermitian metric is called Hermitian vector bundle Definition 05 The linear operator D : Γ(M, E) Γ(M, ε 1 (E)) is a connection on E if D satisfies (124) D(fs) = df S + fds, for s Γ(M, E) and f Γ(M, ε 0 ) Let U α be a trivialization neighborhood of E, and e α,1,, e α,r be a frame on π 1 (U α ) According definition 15 For s Γ(M, E), write s = If we write then Ds = De α,i = l i=1 i=1 j=1 θ (α) ij e α,j s (α) i e α,i on U α Then ds (α) i e α,i + e(α) = i,j=1 e α,1 e α,r s (α) i θ (α) ij e α,j Ds = ds(α) t e (α) + s(α) t θ (α) e (α), where s t = (s 1,, s r ), θ(α) = (θ (α) ij ) is the r r matrix, whose entiers are all 1-forms with respect to the frame e(α), θ(α) is called the connection matrix with respect to the frame e(α) If U β is another trivialization neighborhood of E, and U α U β 0, then on U α U β, (125) Ds = ds α i e α,i + i=1 i=1 s (α) i De α,i

13 02 HERMITIAN MANIFOLDS AND HERMITIAN VECTOR BUNDLES13 = ds(α) t e(α) + s(α) t De(α) = ds(α) t e(α) + s(α) t θ(α)e(α) = ds(β) t e(β) + s(β) t θ(β)e(β) Since s(β) = φ βα s(α) and e(α) = φ t αβ 1 e(β) So we have ds(β) t e(β) + s(β) t θ(β)e(β) ds(β) t = ds(α) t φ t βα + s(α) t dφ t βα = (ds(α) t φ t βα + s(α) t dφ t βα)φ t βα 1 e(α) + s(α) t φ t βαθ(β)φ t βα 1 e(α) = ds(α) t e(α) + s(α) t (dφ t βα, φ t βα 1 ) + φ t βα θ(β)φ t βα 1 e(α) (126) θ(α) = dφ t βα φ t βα 1 + φ t βα θ(β)φ t βα 1 The formula (126) is the relation between two connection matrix θ(α) and θ(β) with respect to the holomorphic frames e(α) and e(β) In fact, it is easy to verify that the formula is also valid for any two frames, e = e 1 e r and d = which are two C frames on π 1 (U) Let θ(e) and θ(d) be the connection matrices with respect to frames e and d, where U is an open set of M, such that e = Ad, d 1 d r, where A is a r r non-sigular matrix and all entries of A are C on U Then (127) θ(e) = da A 1 + Aθ(d)A 1 There are many connections on E, we can make the requirements that dictate a canonical choice of the connection on a Hermitian vector bundle: (1) we write D = D +D, where D : ε 1 (E) ε 10 (E) and D : ε 1 (E) ε 01 (E) are the projective mapping We say a connection D is compatible with the complex structure if D = (2) D is said to be compatible with the Hermitian metric if (128) d{ξ, η} = {Dξ, η} + {ξ, Dη} where ξ, η Γ(M, E) Lemma 06 Let E be a Hermitian matrix vector bundle Then there is a unique connection D on E compatible with both the metric and the complex structure This connection D is called a Hermitian connection on E

14 14 Proof Let e = (e 1,, e r ) be a holomorphic frame on E, and h α β = {e α, e β } If such a D exists, the connection matrix θ with respect to e must have type (1, 0), since D e = 0 Since dh α β = {e α, e β } = {De α, e β } + {e α, De β } = { θ αr e r, e β } + {e αk θ βr e r } = r θ αr h r β + h αβ θ βr we have h α β = r θ αr h r β, ie h = θh So θ = h h 1 is the unique solution, where h = (h α β) is a r r Hermitian metric matrix Let e = e 1,, e r be a unitary frame for E, ie {e α, e β } = δ α,β ; 1 α, β r then 0 = d{e i, e j } = {De α, e β } + {e α, De β } = {θ αr e r, e β } + {e α, θ βr e r } = θ αβ + θ βα So the connection matrix θ with respect to every unitary frame is skew-hermitian We now extend the connection operator to D : Γ(M, ε p (E)) Γ(M, ε p+1 (E)); 1 p 2n, using Lebnitz s rule D(ψ σ) = dψ σ + ( 1) p ψ Dσ where ψ Γ(M, ε p ) and σ Γ(M, E) In particular, we discuss D 2 : Γ(M, E) Γ(M, ε 2 (E)) Let f Γ(M, ε 0 ) and σ Γ(M, E), (129) D 2 (fσ) = D(df σ + fdσ) = df Dσ + dfdσ + fd 2 σ = fd 2 σ (129) indicated an important property that D 2 is linear over Γ(M, ε 0 ) If e = e 1,, e r is a local frame of E, and where e = e 1 e r D 2 e = Θ (e) e is represented by a column vector, Θ (e) is called the curvature matrix of Hermitian vector bundlee with respect to the frame e Θ (e)

15 02 HERMITIAN MANIFOLDS AND HERMITIAN VECTOR BUNDLES15 is r r matrix, whose entiers are 2-forms If e = e 1,, e r is another frame and e i = a j i e j 1 i r Then A = (a j i ) is a r r non-sigular matrix Moreover, by (129), Therefore, we have D 2 e = Θ(e )e = AΘ(e)A 1 e (130) Θ (e ) = AΘ (e) A 1 On the other hand so D 2 e = D(De) = D(θ (e) e) = (dθ (e) θ (e) θ (e) )e, (131) Θ (e) = dθ (e) θ (e) θ (e) By (131) we have dθ (e) = dθ (e) θ (e) + θ (e) dθ (e) = (Θe + θ (e) θ (e) ) θ(e) + θ (e) Θ (e) + θ (e) θ (e) so that = Θ (e) θ (e) + θ (e) Θ (e) (132) dθ e θ (e) Θ (e) + (Θ (e) θ (e) ) = 0, which is the Bianchi equality Let e be a holomorphic frame for the Hermitian vector bundle E θ (e) = h h 1 and Then (133) Θ e = dθ (e) θ (e) θ (e) = θ (e) h h 1 hh 1 h h 1 Since I = hh 1, we have hh 1 + h h 1 = 0, so = hh 1 h h 1 = hh 1 h h 1 = h h 1 Hence Θ e h 1 = θ (e) Let e be a unitary frame, then θ(e) + θ (e) t = 0 Θ (e) +Θ (e) t = dθ(e) +dθ (e) t θ(e) θ (e) θ (e) θ (e) t = θ(e) θ(e)+θ(e) t θ(e) t = 0, so the curvature matrix Θ (e) is skew-hermitian with respect to any unitary frame

16 16 We now discuss two special holomorphic vector bundles on a complex manifold M (1) The holomorphic tangent bundle T 10 (M), which is a holomorphic vector bundle with rank r = dim C M Assume that M is endowed with a Hermitian metric,, w hose local expression is ds 2 = g ij dz i d z j, where g ij = z, i z = g j jī We use ω to denote its Hermitian connection matrix and Ω to denote its curvature matrix under the natural frame ( z,, 1 z ) Then we have n and ω = g g 1 Ω = dω ω ω = (g g 1 ) = (ω), where g = (g ij ) is the Hermitian metric matrix on M In general, let e be a frame for T 10 (M), the Hermitian connection matrix ω (e) and curvature matrix Ω (e) under the frame e, then Ω (e) = dω (e) ω (e) ω (e) (2) The holomorphic line bundle L, ie, r = 1 In this case, the Hermitian matrix is locally a positive function h Let {U α } α I be trivialization neighborhoods of L Then the Hermitian metric {h α } α I is a positive functions, such that h α = φ βα 2 h β, on U α U β ; its connection form is and the curvature form is θ = h α h α 1 = log h α, Θ = log h α = log h β, on U α U β, where the last equality holds because φ αβ are holomophic on U α U β Proposition 07 Let E be a Hermitian vector bundle on a complex manifold M For p M there exists a holomorphic local frame e such that (1) h(z) = I + O( z 2 ), (2) Ω(0) = h(0) Proof We first choose a local coordinates z 1,, z n such that z(p) = (z (p),, z n (p)) = 0 There is a non-singular matrix B, such that h(0) = B B t Take the new frame f = B 1 e, then h(0) = I with the respect to frame f, and h(z) = I + S(z) + O( z 2 ), where S(z) is a r r matrix, whose entries are linear functions of z 1,, z n, and z 1,, z n Since h = ht, S(z) = S(z) t Decomposing S(z) = S 1 (z)+s 2 ( z),

17 02 HERMITIAN MANIFOLDS AND HERMITIAN VECTOR BUNDLES17 the entries of S 1 (z) and S 2 ( z) are linear functions of z 1,, z n and z 1,, z n respectively Since S(z) t = S 1 (z) t + S 2 ( z) t = S 1 (z) + S 2 ( z), S 1 (z) = S 2 ( z) t and S 2 ( z) = S 1 (z) t We now take the new frame e = (I S 1 (z))f We use h to denote the metric matrix with respect to the frame e, then h = (I S 1 (z))(i + S 1 (z) + S 1 (z) t + O( z 2 ))(I S 1 (z) t ) = I + O( z 2 ), and it is easy to verify (h ) 1 = I + O( z 2 ) in an open neighborhood of p So especially Ω(z) = (h h 1 ) = h + O( z ) Ω(0) = h(0) As we consider the holomorphic Tangent bundle T 10 (M) with the Hermitian metric ds 2 = g i jdz i d z j, where z 1,, z n is a local coordinate system, z 1,, z n is a frame for T 10 (M) The curvature matrix under this frame is Ω = (Ω i j), where Ω i j is (11) forms, 1 i, j n so Ω i j = R i j kl dz k dz l and Ω īj := g sī Ω s j = R īj kl dz k dz l R īj hl is called the curvature tensors, and R kj := R īj klgīj is called the Ricci Tensors of the Hermitian manifold M, where gīj is the entries of the inverse matrix of metric matrix g Definition 08 Let M be a Hermitian manifold with the metric ds 2 = g i jdz i d z j If the Kähler form 1 Φ = 2 g i jdz i d z j is closed, ie, dφ = 0, then we call M is a Kähler manifold Proposition 09 For a Hermitian manifold M, the following condition is equivalent (1) M is Kähler; (2) If w i j = Γi jk dzk is local expression of connection forms, then Γ i jk = Γi kj ;

18 18 (3) For p M, there is a C function φ on an open neighborhood of p, such that Φ = 1 φ; (4) For p M, there exists a local holomorphic coordinate system z 1,, z n, such that g ij (p) = δ j, dg ij (p) = 0 Such a coordinate is said to be normal at p Proof (1) (2) dφ = 1 g i j 2 dz k dz i dz j + g i j z k 1 2 g i j 1 z k dzk dz i dz j = 4 i,j,k ( gi j z k g k j z i z k d z k dz i d z j, ) dz k dz i dz j, and g i j z k d zk dz i d z j = ( gi j z k g ) i k z j d z k dz i d z j i,j,k dφ = 0 is equivalent to (134) g i j z k = g k j z i, and g i j z k = g i k z j 1 i, j n Since ω j i = g i t g tj dz k, Γ j z k ik = g i t g tj = g z k k t g tj z = Γ j i ki ; 1 i, j r (2) (1) g j s Γ j ik = g i t g tj z h g j s = g i s z k = g j sγ j ki = g g k t j s g tj z i = g k s z i, so we have g i s z k = g h s z i ; the conjugate of above equality is g s l z h = g s k z i ; 1 i, j, s n 1 i, j, s n so (134) is valid, ie dφ = 0 (1) (3) since Φ is a real closed (11) form, by P orincaré theorem, there is a 1-form H defined in a neighborhood of p such that Φ = dh, H = H 01 + H 10 is its decomposition of (01) form and (10) form Since Φ is real, H 01 = H 10 Φ = dh = ( + )(H 01 + H 10 ) = H 01 + H 01 + H 10 + H 01 However, Φ is (11) form, so H 10 = H 01 = 0 Hence, according to the Dolbeault-Grodendick Lemma, there exists a C function F defined in a neighborhood of p, such that H 01 = F and H 10 = F

19 02 HERMITIAN MANIFOLDS AND HERMITIAN VECTOR BUNDLES19 Then Φ = H 10 + H 01 = F + F = (F F ) = 1φ, where φ = 2ImF is a real C function (3) (1) is trivial (1) (4) By a constant linear change of coordinate if necessary, we may assume that the z i (p) = 0; 1 i n and g ij (p) = δ ij, 1 i, j n Now we define a new holomorphic coordinate ( z 1,, z n ) by z j = z j n i,k=1 g i j z k (p)zk z i We use g to denote the metric matrix under ( z 1,, z n ) Setting (135) b ij = zj z i = δ ji = δ ji ( k n h,s=1 g s j z k (p)(δ siz k + δ ik z s ) ) g ij z k zk + 1 g sj 2 z i zs s = δ ji + k g ij z k (p)zk and B = (b ij ) is the n n matrix, then g = B 1 gb 1t Since B(p) = B 1 (p) = g(p) = I, d g(p) = (db 1 )(p) + dg(p) + (db 1t )(p) = db(p) + dg(p) db t (p) = B(p) + g(p) + g t B t (p) = 0, the last equality holds because, by (135), g(p) = B(p) On the other hand, for p M, there exists a local holomorphic coordinate coordinates z 1,, z n, such that dg(p) = 0 Then dφ(p) = 1 2 d g i j(p)d z i d z j = 0 Remark: Similar to the Riemannian case, we can get Γ i lj = n k=1 g ki g l k z j = ( jdet(g)) det(g 1 ) Let M be a Hermitian manifold, M can be also regarded as a Riemannian manifold, we call such Riemannian manifold the underlying Riemannian

20 20 of the Hermitian manifold The Riemannian manifold has a Levi-civita connection, D 0 In general the Levi-civita connection is not coincide with the Hermitian connection D Let φ = (φ 1,, φ n ) t be a local frame of T 10 (M) and (φ 1,, φ n ) its dual coframe, ie, each φ i ; 1 i n is a (10)-form and φ i (e j ) = δ ij, 1 i, j n Then we have the Cartan structure equations (136) { dφ = ω t φ + τ Ω = dω ω ω, where τ = (τ 1,, τ n ) t is a column vector We call this τ a torsion form of the Hermitian manifold It is evident that τ is a column vector of 2-form Under a frame change ẽ = Ae, φ = (A t ) 1 φ, and the corresponding connection matrix ω, curvature matrix Ω, and torsion form τ under the frame ẽ satisfy ω = AωA 1 + daa 1, Ω = AΩA 1, τ = (A 1 ) t τ The type of τ is invariant under the frame changement, as e is an holomorphic frame, then dφ and θ t φ are consisted of (20) forms, so we know τ is always consisted by (20) forms We claim that τ = 0 is equivalent to that M is Kähler In fatc, let (z 1,, z n ) be a local holomorphic coordinates, ( z,, 1 z ) t be the n frame and its dual coframe is (dz 1,, dz n ) t, by the first equation in (136), τ i = ω ji dz j = (g j lg li ) dz j = g j l z k g li dz k dz j Hence τ i = 0; 1 i n is equivalent to g j l = g z k kī z ; 1 i, k, l n This proves j the claimation From proposition 09, we know that, at any point of a Kähler manifold, the local difference between the Kähler metric and Euclidean metric of C n is the 2 orders infinitsimal, so under the suitable local holomorphic coordinates z 1,, z n, p M, z i (p) = 0, 1 i n and g(p) = I, dg(p) = 0 ie g(p) = g(p) = g 1 (p) = g 1 (p) = 0, and Ω(p) = ( g)(0) By (3) in proposition 09, there is a real C function φ on the local neighborhood of p, such that Φ = i φ so that Therefore 2 φ g l k = 2 z k, 1 l, k n zl 4 φ (137) R īj kl = 2 z i z j z h z l (0) So for a Kähler manifold, we always have

21 02 HERMITIAN MANIFOLDS AND HERMITIAN VECTOR BUNDLES21 Proposition 010 R īj kl = R kjīl = R klīj = R īlk j R īj kl = R ji lk Proposition 010 can be proved by using (137) and the equality of tensors is independent on the choice of the frame Recall that the tensor field is called the Ricci curvature R īj = R īj klg kl = Proposition 011 R īj = ī j (log det g) Proof Let g = (g i j) We use A i j to denote the cofactor of g i j, then det g = n g i ja i j Hence i,j=1 det g g i j n l=1 R l l ji = A i j = det g g ji, where we regard det g as a function in variables g i j So j det g = det g g i k g i k z j = det g g ki g i k z j Therefore, Recall that Hence, j log det g = n Γ i ij = i=1 R īj = ī Γ i lj = n i,k=1 ( n i=1 n k=1 n i,k=1 g ki g l k z j g i k z j g ki g ki g i k z j = j log det g Γ i ij ) = ī j log det g Definition 012 For ξ, η Tp 10 (M), the holomorphic bisectional sectional curvatureis (139) R(ξ η) = R īj kl ξ i ξ j ξk ξ l / ξ, ξ p η, η p

22 22 where ξ = ξ i z, η = η i i z, and R i īj kl is the curvature tensors under the natural f rame z,, 1 z The holomorphic sectional curvatureis n (140) R(ξ) = R īj kl(p) ξ i ξ j η k η l / ξ, ξ 2 p The Ricci curvature is (141) Ric(ξ) = R īj (p) ξ i ξ j / ξ, ξ p, and the scalar curvature at p M is (142) R = R īj gīj T 10 (M) is a Hermitian holomorphic tangent bundle, whose geometry properties intrinsically related to the Hermitian manifold M Let (E, h) be a Hermitian vector bundle on the complex manifold M We can define the curvature form of E as follows: Suppose u, v Γ(M, E), define (11)-form Θ u v = Θ β αh β γ u β v γ where u = r λ=1 i,j,k=1 u α e λ, v = r v λ e λ, Θ β α is the Hermitian curvature forms of E 1 under the frame (e 1,, e r ) Trivially, Θ u v is independent of the choice of the frame e, and Θh = ( h h 1 + h h 1 h h 1 )h = h + h h 1 h Therefore Θh + hθ t = 0, so Θ u v = Θ vū For X, Y Γ(M, T 10 (M)) then R X Ȳ u v := Θ u v (X, Ȳ ) R X Ȳ u v = R Y Xvū We call R X Xuū the curvature of (E, h) in the direction of u and X, as X Tp 10 (M) and u E p and X, X p = 1 = {u, u} p = 1 (E, h) is called positive curved if and only if R X Xuū > 0 fo r p M, any u E p ; and u 0, and X Tp 10 ; x 0 We introduce an equivalent relation on the holomorphic vector bundles, which will be used in chapter 12 Definition 013 Let E and E be two holomorphic vector bundle on the complex manifold M, m : E E be a holomorphic mapping, such that m E E (1) The diagramme π π is commutative ie m E x : E x M id M E x (2) m E x : E x E x is the linear mapping Then we call m a holomorphism of holomorphic vectors bundle

23 02 HERMITIAN MANIFOLDS AND HERMITIAN VECTOR BUNDLES23 Obviously, if E, E and E are holomorphic vector bundles on M, and m : E E, m 1 : E E are the holomorphism of holomorphic vector bundles from E to E and from E to E respectively, then m 1 m : E E is still a holomorphic vector bundle Definition 014 Let E and E be two holomorphic vector bundles over a complex manifold M, and m : E E and m : E E be holomorphisms of holomorphic vector bundles If m m : E E and m m : E E are identity mappings of E and E respectively, then we say that m (or m ) is an isomorphism of holomorphic vector bundles Theorem 015 m : E E is an isomorphic of holomorphic vector bundles if and only if rank E = ranke = r and q α : U α GL(r, C), α I, are holomorphic mappings, where {U α } α I is the covering of the common trivialization neighborhoods of E and E such that (143) φ αβ = q α φ αβq β, where {φ α } and {φ αβ } are the transitive functions of E and E respectively Proof If m : E E is a holomorphism of holomorphic vector bundles, then m : E x E x is linear isomorphism, so, x M, ranke = ranke = r Without loss of generality, we may choose a common trivialization neighborhoods {U α } i I such that U α = M Let e α 1,, e α r and e α 1,, e α 1 be the i I canonical local frames on U α of E and E respectively Then (144) m(e α ij) = j=1 q (α) ij eαα j and m (e α α,j) = j s (α) jk e α,k According the definition of isomorphism vector bundle m : E E, we have q (α) ij s(α) jk = δ ik Hence the matrix q α = (q (α) ij )(s α = s (α) ij ) is non-degenerate and, since the frames {e α α,i } 1 i r and {e α i } 1 i r are holomorphic, q α : U α GL(n, C) is holomorphic If U α U β φ, then we have e α 1 m e α r = q α e α 1 e lpha r = q α φ αβ e β 1 e β r

24 24 On the other hand e α 1 m = m e α r φ αβ e β 1 e β r = φ αβ m e β 1 e β r = φ αβ q β e β 1 e β r Comparing above two equations, we obtain φ αβ = q α φ αβq 1 β In the contrary direction, if there exist ranke = ranke = r, and q α : U α GL(r, C) is holomorphic, α I {U α } α I is an open covering of E and E, such that φ αβ = q α φ αβq 1 β Then we define the linear mapping m : E E, by e α,1 e α,1 m = q α e α,r e α,r Obviou sly m is a homomorphism of holomorphic vector bundles from E to E The equality (143) assures the definition in (144) is well defined Let m e α,1 e α,r = q 1 α e α,1 e α,r Then m is an homomorphism of holomorphic vector bundle, m m = ide = ide, and m m = ide are obvious This implies that m is an isomorphism When r = 1, let L 1, L 2 be holomorphic line bundles on a complex manifold M, and U = {U α } α I be the open covering of M, where every U α, α I, is a common trivialization neighborhood of L 1 and L 2 Let {φ (1) αβ } and {φ(2) αβ } be its transitive functions respectively, then φ (i) αβ Γ(U α U β, O ); i = 1, 2 and φ (i) αβ φ(i) βγ φ(i) γα = φ (i) αβ φ(i) βγ φ(i) γα = 1; i = 1, 2, on U α U β γ The above equality implies that φ (i) = (φ αβ ), i = 1, 2, is a 1-cocycle on C 1 (U, O ), ie, φ (i) Z 1 (U, O ), i = 1, 2 If L 1 is isomorphic to L 2, according the theorem, there exists a h α Γ(U α, O ), such that φ (1) αβ = h 1 α φ (2) αβ h β; on U α U β, ie, φ (1) 1 αβ φ(2) αβ = h 1 α h β ; on U α U β

25 03 CHERN CLASSES OF DIFFERENTIAL VECTOR BUNDLES 25 The equality implies h = (h α ) α I C 0 (U, O ), δh = φ (1) φ (2) 1, ie, if φ ( 1), φ are regarded as 1-cochains with coefficient O, relative to the covering U, then φ (1), φ (2) belongs to the same equivalent class in H 1 (U, O ) On the contrary, every element in H 1 (U, O ) is just an equivalent class of holomorphic line bundles on M For every complex manifold, the cohomology group H 1 (M, O ) is the direct limit of H 1 (U, O ), ie, for all open covering U of M, H 1 (M, O) = lim U H 1 (U, O ) So every element in H 1 (M, O ) has a representation in H 1 (U, O ) for a covering U of M Therefore every element in H 1 (M, O ) is also an equivalent class of some holomorphic line bundle on M 03 Chern Classes of Differential Vector Bundles The object of this section is to give a differential-geometric derivation of the Chern classes of Differential (comlex) Vector Bundles The Chern classes will turn out to be the primary obstruction to admitting global frames To begin, we need some milti-linear algebra Let M r be the set of r r matrices with complex entries A k-linear form is said to be invariant if Φ : M r M r C Φ(gA 1 g 1,, ga k g 1 ) = Φ(A 1,, A k ) If for any permutation σ of {1,, k}, we have Φ(A σ(1),, A σ(k) ) = Φ(A 1,, A k ), then we say that Φ is symmetric Let Ĩk(M r ) be the C-vector space of all invariant k-linear forms on M r Suppose that Φ Ĩk(M r ) Then Φ induces by setting Φ : M r C Φ(A) = Φ(A,, A) It is clear then that Φ is a homogeneous polynomila of degree k in the entries of A More over, for g GL(r, C), Φ(gAg 1 ) = φ(a)

26 26 and we say that Φ is invariant Let I k (M r ) be the set of invariant homogeneous polynomila of degree kas above Since the isopmorphism of the symmetric tensor algebra S(M r) and the polynomials on M r preserves degrees, one obtain form Φ I k (M r ) an element Φ Ĩk(M r ) (this process is called the polarization) The usal determinant of an r r matrix is clearly a membet of I k (M r ) and is symmetric More over, for A M r, let ( ) 1 det I + 2π A = r! k!(r k)! P k(a), 1 k r where P k (A) is a homogeneous polynomial of elements of A of order k Then P k (gag 1 ) = P k (A), so each P k is an invariant polynmial Now suppose that we have a connection D : Γ(M, E) Γ(M, ε(e)) defined on E M Then we have the curvature Θ If φ I k (M r ), φ M (Θ) is a global 2k-form on M We can now state the following theorem due to A Weil Theorem 016 Let E M be a differential vector bundle, let D be a connection on E and suppose that φ I k (M r ) Then (i) φ M (Θ) is closed (ii) The image of φ M (Θ) in the de Rham group H 2k (M, C) is independent of the connection D We now in the position to defind the Chern classes of a differential vector bundle E M Let ( ) 1 det I + 2π A = r! k!(r k)! P k(a), 1 k r Then the k-th Chern form of E related to the connection D is defined to be c k (E, D) = P k (Θ(D)) ε 2k (M) The (total) Chern form of E related to the connection D is defined to be c(e, D) = c k (E, D) k=0 The k-th Chern class of E related to the connection D, denoted by c k (E), is defined to be the cohomology class of c k (E, D) in the de-rham group H 2k (M, C), and the (total) Chern class of E related to the connection D, denoted by c(e), is the cohomology class of c(e, D) in H (M, C) Theorem 017 Let D be a connection on a Hermitian vector bundle E compatibale with the Hermitian metric h Then the Chern form c(e, D) is a real differetial form, and it follows that c(e) H (M, R) H (M, C)

27 04 HODGE THEOREM 27 The proof of the above theorem use the following observation: From the compatibilties for the Hermitian connection, we get Exteriorly differentiaing it we get Hence det Hence ( I + ) ( 1 2π Ω = det I dh = ωh + h ω Ωh + h Ω t = 0 1 2π ) ( ) ( ) Ω 1 1 = det I + 2π h 1 Ωh = det I + 2π Ω P k (Ω) = P k (Ω) In the case of holomorphic line bundle L, the Hermitian matrix is locally a positive function h Let {U α } α I be trivialization neighborhoods of L Then the Hermitian metric {h α } α I is a positive functions, such that its connection form is and the curvature form is h α = φ βα 2 h β, on U α U β ; θ = h α h α 1 = log h α, Θ = log h α = log h β, on U α U β, Then the first Chern form in this case is c 1 (L, h) = sqrt 1 2π Θ = sqrt 1 log h α = dd c log h α 2π where d c = 1 4πsqrt 1 ( ) so dd c = 1 2πsqrt 1 04 HODGE THEOREM The main content of this section is to prove Hodge theorem We will introduce the inner-product and sobolev norms on the E-valued differential forms and whose completions Then use the functional method to prove the Hodge theorem At the end of the chapter, we give the proof of Gärding inequality, Sobolev lemma and Rellich theorem, which play an important role in the proof of the Hodge theorem Let M be a n-dimensional complex manifold with the Hermitian metric ds 2 = g ij dz i d z j The associated Kähler form is Φ = 1 2 g ijdz i d z j, a real (11)-form The volume form is (111) 1 n! Φn = 1 n! ( 1 2 )n g i1, j 1,, g in, j n dz i1 d z j1 dz in d z jn

28 28 = 1 1 n! ( 2 )n g i1, j 1,, g in, j n δ i1,,i n δ j1,,j n dz 1 d z 1 dz n d z n 1 = ( 2 )n g 1, j 1,, g n, j n δ j1,,j n dz 1 d z 1 dz n d z n 1 = ( 2 )n det gdz 1 d z 1 dz n d z n = ( 1) n(n 1) 1 2 ( 2 )n det gdz 1 dz n d z d z 1 n = det gdx 1 dy 1 dx n dy n = ( 1) n(n 1) 2 det gdx 1 dx 2 dy n 1 dy n It is easy to verify that the volume form 1 n! Φn is scalar, which is independent of 1,,n the choose of the local holomorphic coordinates In (111), δ i1,,i n = δi 1,,i m is the multi-generalized Kronecker symbol, ie 1 as l 1,, l k is even permutation of i 1,, i k δ l1,,l k i 1,,i k = 1 as l 1,, l k is odd permutation of i 1,, i k 0 others We now extend the inner-product to the ε p,q (M) Let ξ, η ε p,q (M) and and ξ = 1 p!q! a i 1 i p j 1 j q dz i1 dz ip d z j1 d z jq = a i1 i p j 1 j q dz i1 dz ip d z j1 d z jq η = i 1 < <ip j 1 < <jq i 1 < <ip j 1 < <jq b i1 i p j 1 j q dz i1 dz ip d z j1 d z jq = 1 p!q! b i 1 i p j 1 j q dz i1 dz ip d z j1 d z jq be the expression of ω and η under the coordinates z 1,, z n for x M We define the Hermitian inner-product at x (112) ξ, η x = 1 p!q! a i 1 i p j 1 j q (x)g j 1s 1 (x) g j qs q (x) (x)g t 1i 1 (x) g t pi p (x) b t1,,t p, s 1,, s q (x), where g js are the entries of the g 1, the inverse matrix of metric g For simplicity, we use a Ip Jq to denote a i1 i p j 1 j q and g J qs q to denote g j 1s1 g j qs q, so rewrite (112) to (113) ξ, η x = 1 p!q! a I p Jq (x)g J qs q (x)g T pi p (x)b Tp Sq (x)

29 04 HODGE THEOREM 29 Sometimes, for simplicity, we use ξ Ip Jq to denote the co efficient a Ip Jq and ξīpjq to denote gīpsp g T qj q a Sp Jq Form (112) it is easy to verify and if ξ 1, ξ 2, η ε p,q (M) ω, η x = η, ω x and λ C ξ 1 + ξ 2, η x = ξ 1, η x + ξ 2, η x λξ, η x = λ ξ, η x (112) shows that ξ, η x is a scalar, so it is independent of the choice of the local holomorphic coordinates We may choose a holomorphic coordinates such that g ij (x) = δ ij, then g ji (x) = δ ij so especially, ξ, η x = 1 p!q! a I p Jq (x)b Ip Jq (x) ξ, ξ x = 1 p!q! I p,j q a Ip Jq (x) 2 0 and the equ ality is only valid in the case ξ(x) = 0, so (112)is really a Hermitian inner-product on ε p,q x (M) Let M be a Hermitian manifold with metric ds 2 = g i jdz i d z j, E be a rank r Hermitian vector bundle with the Hermitian matric h, and {U α } α I be the trivialization neighborhoods of E Then there is a holomorphic frame {e α,λ } 1 λ r of E on U α, such that any ξ Γ(M, ε p,q (E)) has the local expression on U α ξ = ξ λ e α,λ λ=1 where ξ λ Γ(M, ε p,q (M)), 1 λ r Let ξ, η Γ(M, ε p,q (M)) and their local expression are ξ = r λ=1 ξ λ e α,λ and η = we may introduce the inner-produce (114) ξ, η x := r µ=1 ξ λ, η µ x {e α,λ, e α,µ } x = λ,µ=1 η µ e α,µ on U α, for forallx U α, ξ λ, η µ x h (α) λ µ (x) λ,µ=1 We may directly verify that ξ, η x is a Hermitian inner-product Furthermore, we can define the global inner-product (115) (ξ, η) = ξ, η dv = M M n ξ λ, η µ h (α) λ,µ=1 λ µ dv

30 30 where dv is the volume form of Hermitian manifold For simplicity, we take dv = 2 n! Φn = ( 1) n(n 1) n 2 1 det g dz 1 dz n d z 1 d z n Of cour se (115) is only have the real sense in the case (ξ, η) < +, for example, when M is compact or when ξ and η have compact supports From now on, our discussion will be restrict to the case that M is a compact complex manifold and p = 0 Γ(M, ε (0,q) (E)) endowed with inner-product (15) becomes an inner-product space To apply the theory of Hilbert space, we have to completion Γ(M, ε (0,q) (E)), ie we consider the E-valued L 2 (0, q)-forms on M We use Γ(M, L (0,q) 2 (E)) to denote the space of E-valued L 2 (0, q)-forms For every η Γ(M, L (0,q) 2 (E)), η has a local expression under the frame {e α,λ } 1 λ r η = η λ e αλ, λ=1 where η λ, 1 λ r, are L 2 (0, q) form on M, ie η λ, η λ dv < + ; 1 λ r M For simplicity, we use L (0,q) 2 (E) to denote Γ(M, L (0,q) 2 (E)) We now consider (116) L (0,q 1) 2 (E) L (0,q) 2 (E) L (0,q+1) 2 (E) The above three space are all Hilbert spaces and Γ(M, ε (0,k) (E)), k = q 1, q, q + 1 are the dense subspaces in L (0,k) 2 (E), k = q 1, q, q + 1 respectively : Γ(M, ε (0,k) (E)) Γ(M, ε (0,k+1) (E)) is well defined, and we consider the closed extension of from L (0k) 2 (E) to L (0k+1) 2 (E) We still use to denote the operation after the closed extension Therefore, is a densely defined closed operator in (116) : L (0,k) (E) L (0,k 1) (E) is the adjoint operator of According its definition, if η L (0k) (E), and ξ L (0,k 1) (E) such that (117) ( ψ, η) = (ψ, ξ), for ψ Γ(M, ε 0k 1 (E)) then η Dom and η = ξ In fact, we only need that (117) is valid on a dense set of Dom The adjiont operator is also a closed operator, so is a densely defined closed linear operator Definition 018 The Harmonic operator is Obviously = + : L (0,q) (E) L (0,q) (E) ( η, η) = ( η + η, η) = ( η, η) + ( η, η)

31 04 HODGE THEOREM 31 Definition 019 A Harmonic Form η L 0,q (E) if and only if η = 0 From ( η, η) = ( η, η)+( η, η), so η = 0 is equivalent to η = 0 = η The meaning of Hodge Theorem is that every Harmonic form is smooth, ie every Harmonic form belongs to Γ(M, ε (0 ) (E)) and if we use H (0,q) (M, E) to denote the space of all E-valued harmonic (0,q) forms, then H (0,q) (M, E) = H (0,q) (M, E) Since M is compact, every open covering of M possess a finite subcovering, so there is always a finite coordinates covering and a partition of unity of this finite coordinates covering Moreover, we may assume every open set of the finite coordinates is contained in a trivialization neighborhood of E Applying partition of unity, we may assume that all the forms, functions, etc which we considered, have the compact supports in a coordinate neighborhood So we will pr oceed the discussion only in a coordinates neighborhood G, a bounded domain in R 2n except for the special explaination We first introduce some notations Let α = (α 1,, α 2n ), α i are all nonnegative integers α := 2n α i i=1 α = α 1 α 1, i = x i ; 1 i 2n W (oq) s = {φ = φ i1 i q d z i1 d z iq φ i1 i q possess the partial derivatives in the distribution sence until s 0} Let A (0,q) be the space consists of all C E-valued (0, q)-forms with the compact support in G For φ A (0,q), (118) φ 2 s = α φ 2 0 α s where 0 is the ordinary L 2 module Then according the fundamental fact of theory of the Sobolev space, W s (0,q) is the completion of A (0,q) by means of s In (118), α φ is not the tensor, or, more precisely, the coefficients of α φ are not the tensors In general Dφ = n i=1 φ dz i + φ d z i + X φ, z i z i where D is the Hermitian connection of M, and X is a multiple operator which includes quantities related the connection Since φ with the compact support,