Some Concepts and Results in Complex Geometry

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1 Some Concepts and Results in Complex Geometry Min Ru, University of Houston 1 Geometry on Hermitian and Kahler Manifolds A complex manifold is a differentiable manifold admitting an open cover {U α } and coordinate maps φ α : U α C n such that φ α φ 1 β is holomorphic on φ β (U α U β ) C n for all α, β. A function f on an open set U M is holomorphic if, for all α, f φ 1 α is holomorphic on φ α (U α U) C n. A map f : M N of complex manifolds is holomorphic if it is holomorphic given in terms of the local holomorphic coordinates of M and N. Examples of complex manifolds include: 1. One-dimensional complex manifolds, which are called the Riemann surfaces; P n (C), and complex torus C/Λ, where Λ is a discrete lattice. For p M, let (z 1,..., z n ) be a local holomorphic coordinates. Define z = 1 ( i 2 x 1 ) i y i and z = 1 ( i 2 x + 1 ), i y i = z i dzi, = z i d zi, and d = +. The complexified tangent space is { n } T C,p (M) =: C T p (M) = a i n x i p + b i y i p a i, b i C i=1 1 i=1

2 { } = C x,. i y i The holomorphic tangent space Tp 1,0 (M) and the antiholomorphic tangent space Tp 0,1 (M), for p M, are given by so that { } n Tp 1,0 (M) = C z i p i=1 { } n, Tp 0,1 (M) = C z i p, i=1 T C,p (M) = Tp 1,0 (M) Tp 0,1 (M). T (1,0) (M) = p M Tp 1,0 (M) is called the holomorphic tangent bundle. Γ(M, T (1,0)(M) ) is the set of smooth sections of T (1,0) (M), which is also called the smooth vector fields. A Hermitian metric on M, denoted by ds 2, is a set of Hermitian innerproduct {, p } p M on T p (1,0) (M) such that 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, we write then, if ξ = ξ i, η = η k z i ds 2 = g i, jdz i d z j,, z k ξ, η = g i kξ i η k. We also have g i, j = g j,ī. A complex manifold with a given Hermitian metric is said a Hermitian manifold. The metric form (associated to ds 2 ) is the form (1, 1)-form Φ = 1 2 g i, jdz i d z j. (M, ds 2 ) is said to be Kahler if dφ = 0. The connection is a linear operator D : Γ(M, T (1,0) ) Γ(M, Λ 1 (M) T (1,0) ), where Λ 1 (M) is set of (smooth) differential forms of degree 1 on U satisfies (1.24) D(fs) = df s + fds, for s Γ(M, T (1,0) ) and f is a smooth function on M. The connection gives a way to differentiate the vector fields. 2

3 The connection D is called the Hermitian connection if (i) D is compatible with the complex structure, i.e. if we split T C = T (1,0) T (0,1) and write D = D + D, then D =, (ii) D is compatible with the Hermitian metric, d < ξ, η >=< Dξ, η > + < ξ, Dη > where ξ, η Γ(M, T (1,0) ). Write D = dz i i, then i is called the covariant derivative. D also extends naturally to any type of tensor fields. From now on, the connection is always assumed to be the Hermitian connection. In terms of local coordinate (z 1,..., z n ), write ω j i j=1 D z = n i z j, where ω = (ω j i ) is a n n matrix whose entries are all 1-forms. ω is called the connection matrix. For the Hermitian connection D, its connection matrix satisfies ω = g g 1 (so the entries of ω are (1, 0)-forms), or equivalently, ω j i = g i t z k g tj dz k. Write n ω j i = Γ j ik dzk k=1 where the functions Γ k ji are called the Christoffel symbols. From above, Γ j ik = g i t z k g tj. M is Kahler if and only if Γ k ji = Γ k ij. Th other equivalent conditions for being Kahler is (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. 3

4 For ξ Γ(M, T (1,0) ), in terms of local coordinate (z 1,..., z n ), write ξ = n ξ j. Then z j or j=1 n Dξ = dξ j j=1 z + n ξ j ω i j j i,j=1 z, i i ξ j = ξj z i + n k=1 Γ j ik ξk. Note that, for covariant tensor field {ξ j }, the resulting { i ξ j } (when i is fiexed) is still a covariant tensor field. The connection also extends to Γ(M, Λ k (M) T (1,0) ) (where Λ k (M) is the set of k-forms): D(ω ξ) = dω ξ + ( 1) k ω Dξ. In particular, if, for ω = n j=1 φ j dz j (contra-variant tensor field), then ( n ) φj i ω = z Γ k ijφ i k dz j, or simply k=1 i φ j = φ j z i n k=1 Γ k ijφ k. From now on, we always assume that D is the Hermitian connection associated to the given Hermitian metric(it uniquely exists). We also assume M is Kahler. In terms of local coordinate (z 1,..., z n ), we write D 2 z = n i Ω i j j=1 z, j where Ω = (Ω i j) is called the curvature matrix. We have Ω = dω ω ω = ( g g 1 ) = (ω). Write Ω G = (Ω i k), where G = (g i j). Write Ω i j = R i j kl d zk dz l = R i jl k dzl d z k and Ω īj := s g sī Ω s j = R īj kld z k dz l. R i jk l are called the curvature tensors, and R kl := R īj klgīj are called the Ricci tensors of the Hermitian manifold M, where gīj are the entries of the inverse of metric matrix g. 4

5 From Ω = (ω), Ω i j = ( Γ i jldz l ) = k Γ i kld z k dz l. Hence R i j kl = k Γ i kl. From Ω = ( g g 1 ), we have and We also have R j ka b = 2 g j k z a z + g p q g p k g j q b z a z b R īj = ī j (log det g). R īj kl = R kjīl = R klīj = R īlk j R īj kl = R ji lk. 2 Hermitian Line and vector bundles The above concepts can be extended from the tangent bundle T (1,0) (M) to a general vector bundle. A holomorphic vector bundle E over M is a topological space together with a continuous mapping π : E M such that (i) E p = π 1 (p); p M, is a linear space with rank r; (ii) There exists an open covering {U α } α I of M and biholomorphic maps φ α with and such that φ α : π 1 (U α ) U α C r, α I φ α : E p {p} C r C r, p U α is a C linear isomorphism between complex vector space. On U α U β φ, let φ αβ := φ α φ 1 β, then φ αβ : U α U β GL(r, C) is holomorphic. φ αβ is called transitive function of E; The φ αβ satisfies the compatible conditions: φ αβ (p)φ βγ (p) = φ αγ (p) and φ αβ (p) = φ βα (p) 1 ; p U α U β. 5

6 The rank 1 vector bundle L is called the line bundle. In this case, on U α U β φ, for p U α U β, let φ αβ (p) := φ α φ 1 β φ β(l p ) : φ α L p C, is a C-linear isomorphism, it is represented by g αβ (p) GL(1, C) = C, i.e. φ αβ (p, z) = (p, g αβ (p)z). Furthermore g αβ : U α U β GL(1, C) = C is holomorphic. g αβ is called transitive function of L. So L corresponds to {U α, φ αβ } where {U α } is a finite cover of M and g αβ is nowhere zero holomorphic functions with the compatible conditions. C φ 1 β Let e α (p) = φ 1 (p, 1). Then e α is a local frame of L on U α. It is called the canonical frame. It is easy to check that e β = g α,β e α. A (holomorphic) section of L is a holomorphic map s : M L such that φ s = id. Write s = s α e α on U α. Then s α = g αβ s β. Consider O P n( 1), tautological line bundle on P n (C) (which some books called it the univeral line bundle). It can be described as follows: The set O P n( 1) P n C n+1 that consists of all pairs (l, z) P n C n+1 with z l (i.e. l = [z]) forms a natural way a holomorphic line bundle over P n. In fact, let P n = n i=0u i be the standard open covering. A canonical trivialization of O P n( 1) over U i is given by ψ i : π 1 (U i ) U i C, (l, z) (l, z i ). Since ψj 1 (l, 1) = (l, z/z j ), we have ψ i ψj 1 (l, 1) = ψ i (l, z i /z j ) Hence transition maps ψ ij (l) : C C are given by w z i z j w, here we used the fact that ψ ij (l) is linear, where l = [z 0 : : z n ], i.e. ψ ij ([z]) = z i z j. Note that the fiber O P n( 1) over P n is naturally isomorphic to l. The line bundle of hyperplane of P n : The dual of O P n( 1), denoted by O P n(1) is called the hyperplane line bundle. Its transition functions are g αβ = zβ z. On U z α α, consider s α = a 1 z a α 1 z α α 1 + a z α α + z a α+1 z α a n z α n. Then s z α α = zβ s z α β. So sα defined a holomorphic section s = a 0 z 0 + a n z n. It zero is the hyperplane H = {[z 0,, z n ] P n n a α z α = 0} in P n. This is where the name of hyperplane line α=0 bundle of P n comes from. We sometimes also denoted it by [H]. E is called a Hermitian vector bundle if there is an Hermitian inner product on each fiber E p for p M. 6

7 Similar to above, with the given Hermitian metric, there is a canonical connection (called Hermitian connection) D : Γ(M, E) Γ(M, Λ 1 (M) E) which is compatible with the complex structure and with the Hermitian metric on E. For simplicity, we only focus on the line bundle E = L, i.e. r = 1. Let {U α } α I be trivialization neighborhoods of L. Then the Hermitian metric {h α := h(e α, e α )} α I is a set of positive functions with h α = g βα 2 h β on U α U β. where g βα are transition functions. Its connection form is θ = h α h α 1 = log h α, and the curvature form is Θ = log h α = log h β, on U α U β. So Θ is a global (1.1)-form on M. Define the first Chern form of the Hermitian line bundle (L, h) as c 1 (L, h) = 1 Θ = 1 log h 2π 2π α. (L, h) is said to be positive (or ample) if c 1 (L, h) is positive. On the hyperplane line bundle of hyperplane line bundle of P n. We endow with a Hermitian metric h on line bundle [H], h = (h α ) 0 α n, where h α is the local expression of h on U α. h α = zα 2 z = 1 2 m. zβ z α 1 c 1 ([H]) = 2π 1 log h α = 2π log z 2 > 0. so [H] is positive line bundle. It is easy to see that [H] is, in fact, independent of the choice of H, so we denote it by O P n(1). α β A divisor D on M is a formal linear combination D = n i [Y i ] where Y i M irreducible hypersurfaces and n i are integers. A divisor D is called effective if n i 0 for all i. 7

8 Any divisor D induces O(D), the line bundle associated to D, in a canonical way: If D is a hypersurface locally defined by f α = 0 on U α, then φ αβ = f α /f β are the transition functions for O(D). If D = H is a hyperplane, then O(H) = O P n(1). Canonical line bundle on M: Let {U α } α I be a holomorphic coordinate covering of M, (z(α) 1,, zn (α)) be a local coordinate system of U α. The canonical line bundle K M is the line bundle with the transitive functions φ αβ = det (z1 (β),,zn (β) ) (z 1 (α),,zn ). Sections of K M are (n, 0)-forms (α) ω = a α dz(α) 1 zn (α). With Hermitian metric ds 2 = g (α) ij dz(α) i dzj (α) on M, det g (α) = det (g (α) ij ) is an Hermitian metric of det(t 1.0 (M)), thus det g (α) 1 is the Hermitian metric of K M. The connection form of K M is thus θ (α) = det g 1 (α) det g (α) = log detg (α), and the curvature form is Ω (α) = log detg (α) = R jidz i d z j. 3 Sheaves and Cohomology Origins. The Mittage-Leffler Problem: Let S be a Riemann surfac, not necessarily compact, p Swith local coordinate z centered at p. A principal part at p is the polar part n k=1 a k z k of Laurent series. If O p is the local ring of holomorphic functions around p, M p the field of meromorphic unctions around p, a principal is just an element of the quotient group O p /M p. The Mittage-Leffler question is, given a discrete set {p n } of points in S and a principal part at p n for each n, does there exist a meromorphic function f on S, holomorphic outside {p n }, whose principal part at each p n is the one specified? The question isclearly trivial locally, and so the problem is one of passage from local to global data. Here are two approaches, both lead to cohomology theories. Cech: Take a covering {U α } of S by open sets such that each U α contains at most one point p n, and let f α be a meromorphic function on U α solving the problem in U α. Set f αβ = f α f β O(U α U β ). 8

9 In U α U β U γ, we have f αβ + f βγ + f γα = 0. Solving the problem globally is equivalent to finding {g α O(U α )} such that f αβ = g β g α in U α U β : given that g α, f = f α + g α is globally defined function satisfying the conditions, and conversely. In the Cech theory, Z 1 ({U α }, O) = {{f αβ } : f αβ + f βγ + f γα = 0} δc 0 ({U α }, O) = {{f αβ } : f αβ = g β g α, some {g α }} and the first Cech cohomology group H 1 ({U α }, O) = Z 1 ({U α }, O)/δC 0 ({U α }, O) is th obstruction to solving the problem. Dolbault: As before, take f α be a meromorphic function on U α solving the problem in U α, and let ρ α be a bump function, 1 in a neighborhood of p n U α and having compact support in U α. Then φ = α (ρ α f α ) is a -closed c -(0,1)-form on S (φ 0 in a neighborhood of p n ). If φ = η for η C (S), then the function f = α ρ α f α η satisfies the conditions of the problem: thus the obstruction to solving the problem is in H 0,1 Dol (S). As the theorem of Dolbault, these two co-homology groups are the same (isomorphism). A Sheaf F over a complex manifold X consists of, for each open set U X, an abelian group (or vector spaces, rings, or any desired object) F(U) (also denoted Γ(F, U) and called the set of sections over U), 9

10 and a collection of restriction maps such that for each U V X, ρ V,U : F(V ) F(U), and satisfy: (1) Identity: ρ U,U = id F(U), (2) Compatibility: If U V W X, then ρ V,U ρ W,V = ρ W,U ; (3) Sheaf axiom (gluing): Let U = Up α U α and σ α Uα U β = σ β Uα U β for all α, β, then there exists a (unique) σ F(U) such that σ α = σ α for all α. If only (1) and (2) are satisfied, then F is call a presheaf. Examples: Let s us consider a few familiar examples, each with the natural sections and restriction maps: O X, the sheaf of holomorphic functions on X; Ω p X, the sheaf of holomorphic p-forms on X; A n X, the sheaf of n-forms on X; A a,b X, the sheaf of (a, b)-forms on X; The skyscraper sheaf C p given by C p (U) = C if p U, and C p (U) = 0 if p U along with the natural restriction maps. Cech cohomology: Let F be an abelian group sheaf over a complex manifold X. Let U = {U i } i I be an open covering of topological space X. For nonnegative integer p, consider f : { σ = U i0 U ip } {Γ( σ, F)} : U i0 U ip f io i p. If U i0 U ip, we take f io i p = 0. Such f = {f i0 i p } is called a p-cochain of U with coefficients in the sheaf F. We use C p (U, F) to denote the set of all p-cochains of U with coefficients in the sheaf F. For {f i0 i p }, {g i0 i p } C p (U, F), defining the addition operation {f i0 i p } + {g i0 i p } = {f i0 i p + g i0 i p } then C p (U, F) becomes an abelian group, we called C p (U, F) p-dimensional cochains group of U with coefficients in sheaf F. Now we define the operator where δ p : C p (U, F) C p+1 (U, F) : f δ p f p+1 (1) (δ p f) i0 i p+1 = ( 1) k f i0 î k i p k=0

11 In the right hand side of (1), each f i0 î k i p+1 restricts to U i0 U ip+1 and proceeds the addition operation in Γ(U i0 U ip+1, F). It is easy to verify δ p is a homeomorphism of group, and δ p+1 δ p = 0; p 1. Z p (U, F) := Ker δ p C p (U, F), p 0, is called the p-dimensional cocycles group of U with coefficients in sheaf F, and B p (U, F) = Im δ p 1, p 1, is called the p-dimensional coboundaries group of U with coefficients in sheaf F, and B 0 (U, F) 0. From δ p+1 δ p 0, B p (U, F) Z p (U, F). Define { Z H p (U, F) = p (U, F) / B p (U, F), p 1 Z 0 (U, F), p = 0, (2) H p (U, F) is called the p-dimensional cohomology gy group of U with coefficients in the sheaf F. Define H p (X, F) = lim U H p (U, F). Let C p be the skyscraper sheaf. Then (i) H 0 (M, C p ) = C, (ii) H 1 (M, p ) = 0. The assertion of (i) is trivial. As for (ii), consider a cohomology class ξ H 1 (M, C p ), which is represented by a cocycle in Z(U, C p ). The covering U has a refinement B = {V α } such that the point p is contained in only one V α. But then Z(U, C p ) = 0 and hence ξ = 0. This finishes the proof. The Dolbeault Theorem. Recall the define of the Dolbeault cohomology H a,b Dol (M) = ker a,b /im a,b 1. Let U be an open subset of M, denote by Ω p M(U) := Γ(U, Ω p M) := {ω A p,0 (U), ω = 0}, the set of holomorphic p-forms on U. Ω p M is the sheaf of holomorphic p-forms. Also let Z p,q (U) ::= {ω Ap,q (U), ω = 0}, the set of holomorphic (p, q)-forms on U. is the sheaf of holomorphic (p, q)-forms. The following is the statement of the Dolbeault Theorem: Let X be a compact complex manifold. Then Z p,q H p,q Dol (M) = Hq (M, Ω p M), 11

12 where Ω p M is the sheaf of holomorphic p-forms. The proof of Dolbeault Theorem uses the following fact that, by the -Poincare lemma, the sequence, under, 0 Ω p A p,0 A p,1 A p,2 is exact on any complex manifold, as well as the theorem about that the short exact sequence about sheaves induces the long exact sequence about the co-homologies. 4 Hodge Theory Suppose M is compact complex manifold of complex dimension n and E is a holomorphic bundles of rank r. From the Dolbeault isomorphism we know that the cohomology group H q (M, E) is isomorphic to the set of all -closed smooth E-valued (0, q)-forms on M modulo the set of all -exact smooth E-valued (0, q)-forms on M (if you are not familiar with the sheaf theory, you can regard it as the definition of H q (M, E)). Though this gives us a more practical way of computing H q (M, E) by using differential forms, it would be more convenient for computational purpose if a cohomology class is represented by a unique differential form rather than an equivalence class of differential forms. The Hodge theorem states that such is case if M is compact Kahler, i.e. every equivalence class of differential forms is uniquely represented by the harmonic differential form (which is unique). Here the concept of harmonic is defined as follows: The Kahler metric g on M and Hermitian metric h on E together induce a (canonical) inner produce on the space of smooth E-valued (p, q)-forms as follows: φ, ψ = φ I Jψ K Lhg i 1 k1 g ip k p g l 1 j1 g lq j q. It is easy to check that it is independent of all the frames that were chosen. When we want to indicate the dependence on the metrics g and h, we shall write φ, η g,h. The (global) inner product is defined as (φ, ψ) = 1 φ, ψ g,h dv, p!q! M 12

13 where dv is the volume form. Denote by the adjoint of with respect to the global inner product defined above. := + is called the complex Laplacian. An E-valued form φ is called harmonic if φ = 0. 5 Kodaria s vanishing theorem The vanishing theorem of Kodaira says that the cohomology groups H q (M, L K M ) vanishes for q 1 when L is a holomorphic line bundle over a compact complex Kahler manifold M and L admits a Hermitian metric whose curvature form is positive definite. Here K M is the canonical line bundle of M. The proof of Kodaria s vanishing theorem relies on the following Bochner- Kodaira s formula: for any smooth E-valued (0, q)-form φ, (7.5) ( φ, φ) = φ M (Ricφ, φ) M + (Ωφ, φ) M, where φ M = (Ricφ, φ) M = (Ωφ, φ) M = M q k=1 M q k=1 where Ω is the curvature form of E. g i j j φ ᾱ J q i φ j 1 j q ᾱ, M R s j k φ ᾱ j 1 ( s) k j q φ j 1 j q ᾱ, Ω l j k φ ᾱ j 1 ( l) k j q φ j 1 j q ᾱ, Note that the curvature form on K M is given by R jidz i d z j. Take E = L K M If L is positive, i.e. the Hermitian matrix Ω i j R ji is positive definite at each point, then we will have ( φ, φ) > 0. This implies that φ = 0 if φ is harmonic. Thus H q (M, L K M ) = 0 for q 1 by the Hodge theorem. Assuming E is trivial. Then the calculation of depends on the following local formula for and : for φ = φ Ip Jq dz Ip d z Jq, q ( φ) Ip j 0 j 1 j q = ( 1) p ( 1) k jk φ Ip j 0 j 1 ˆ j k j q. 13 k=0

14 and ( φ) Ip j 1 j q 1 = ( 1) p+1 g ji i φ Ip j j 1 j q 1, as well as the computation of [ i, j] for (1, 0) a k dz k and (0, 1)-forms b kd z k as follows: [ i, j ]a k = R t k ji a t, [ i, j ]b k = R k t jib t. In general (when E is not trivial with metric h α ), then the connection induced by h α is D E : Γ(M, ε p,q (E)) Γ(M, ε p,q+1 (E)), which is given by D E ω α = ω α + ( log h α ) ω α. So if we write D E = dz i L i then E i ω is still a E-valued (p, q)-form. We call E i ; 1 i n, the covariant derivatives with respect to Hermitian line bundle E. Similar to above, for η Γ(M, ε p,q+1 (E)), ( Eη α ) i1 i p j 1 j q = ( 1) p+1 g ji ( i + i log h α )η αi1 i p j j 1 j q. 6 Singular metric, currents, Nadel s multiplier ideal sheave and Nadel s vanishing theorem Kodaria s vanishing theorem has been extended by Nadel to line bundles with singular metric (i.e. h = {h α }, where h α may be singular). We write h α = e κα, here we usually write κ := κ α if no risk of confusion, then the curvature Θ h := κ is not a smooth differential form anymore if the metrics singular(it is in fact is called current). We say that e κ has non-negative (reps. positive) curvature current if Θ h is a non-negative (reps. (1,1)- current, or equivalently, the local representatives κ are plurisubharmonic. Currents: Recall that if f, g C 0 [0, 1], then f g if and only if 1 0 f(x)φ(x) = 1 0 g(x)φ(x) for every φ C 0 [0, 1]. Also for closed intervals A, B R, A = B if and only if A φ(x) = B φ(x). Here functions and subsets can be regarded as linear functional forms on 14

15 C 0 [0, 1]. These concepts are unified by a general concept of currents: Let M be a real differentiable manifold with dim M = m. A current of degree q = m p (or dimension p) is a real linear map T : D p (M) R, where D p (M) is the set of smooth p-forms on M with compact support. Typical examples are smooth or L 1 loc q-forms β with T = [β] is defined by T (φ) = M β φ for φ Dp (M) with q = m p, as well as p-dimensional oriented submanifold S M with T = [S] defined as T (φ) = S φ. Poincare-Lelong formula: Let f O(M) be a holomorphic function. Then 1 2π log f 2 = [f = 0] holds as currents. Example of singular metric: Let L M be a holomorphic line bundle. Let m be a positive integer and s 1,..., s N be sections of ml. Write s = s α e α, and define κ α = 1 m log( s1 α s N α 2 ). This singular metric blows up exactly on the common zeros of the sections s 1,..., s N. Let U M be an open subset, and let φ be a locally integrable function on U. We define I(U) := {f O M (U) : f 2 e φ L 1 loc(u)}. The corresponding sheaf of germs I φ is called the multiplier ideal sheave associated to φ. Nadel proved that if φ is a plurisubharmonic, then the multiplier ideal sheave I φ is a coherent sheaf of ideals. Nadel s Vanishing Theorem: Let X be a compact Kahler manifold with Kahler form ω, and let F be a line bundle with singular Hermitian metric h = e φ such that 1 φ ɛω for some continuous function 15

16 ɛ > 0 (in the sense of distribution). Then, for q 1, H q (X, O X (K X + F) I φ ) = where K X is the canonical line bundle of X. The point of the proof is that any plurisubharmonic function is the limit of a decreasing sequence of smooth plurisubharmonic functions, so eventually it can be reduced to the smooth case. Lelong numbers of plurisubharmonic functions: The zero of the ideal sheaf I φ is then the set of points where e φ is not locally integrable. Such points only occur where φ has poles, but the poles need to have a sufficiently high order. If φ = 1 m log( s1 α s N α 2 ) as in the example earlier, then one has a notion of (log)-pole order. In general, the pole orders are defined using the so-called Lelong numbers: Let X be a complex manifold and φ a plurisubharmonic function in a neighborhood U of x X. Fix a coordinate chart U near x, and let zbe a local coordinates vanishing on x. The Lelong number of φ is defined to be the number We also set υ(φ, x) := lim inf z x φ(z) log x z 2. E c (φ) := {x X; υ(φ, x) c}. A famous paper of Siu showed that E c (φ) is a complex analytic set. The Lelong number information υ(φ, x) gives the information about the vanishing order of f at x for f I φ,x which is stated as the lemma of Skoda: Let φ a plurisubharmonic function on an open set U of X containing x. Then (1). If υ(φ, x) < 1, then e φ is integrable in a neighborhood of x. In particular, I φ,x = O U,x ; ( 2). If υ(φ, x) n + s for some positive integer, then the estimate e φ C z x 2(n+s) holds in a neighborhood of x, In particular, one obtains that I φ,x m s+1 U,x, where m U,x is the maximal ideal of O G,x ; 3. The zero variety V (I φ ) of I φ satisfies E 2n (φ) V (I φ ) E 2 (φ). Nadel s vanishing theorem plus Skoda s lemma gives a new proof (without using blow-ups) of Kodaira s embedding theorem: Let X be a compact Kähler manifold. Assume there exists a positive line bundle L over X, then X can be embedded in projective space P N. 16

17 To prove the embedding theorem, it gets down to construct holomorphic sections. Consider the long exact sequence of cohomology associated to the short exact sequence 0 I φ O X O X /I φ 0 twisted by O(K x L), and apply Nadel s vanishing theorem of the first H 1 group, we ll have: Let X be a weakly pseudo-convex Kahler manifold with Kahler form ω, and let F be a line bundle with singular Hermitian metric h = e φ such that 1 φ ɛω for some continuous function ɛ > 0. Let x 1,..., x N be isolated points in the zero variety V (I φ ). Then there is a surjective map H 0 (X, K X L) 1 j N O(K X L) xj (O X /I φ ) xj. Exercise: Assume that X is compact and L is a positive line bundle. Let {x 1,..., x N } be a finite set. Show that there are constants a, b 0 depending only on L and N such that H 0 (X, L m ) generates jets of any order s at all points x j for m as + b, Hint. Apply the above Corollary to L = KX 1 L m, with a singular metric on L of the form h = h 0 e ɛψ, where h 0 is smooth of positive curvature, ɛ > 0 small and ψ(z) = χ j (z)(n + s 1) log w (j) (z) 2 with respect to coordinate systems (w (j) k (z) 1 k n centered at x j. The cut-off functions χ j can be taken of a fixed radius (bounded away from 0) with respect to a finite collection of coordinate patches covering X. It is easy to see such h serves our purposes. Taking s = 2 and m with m 2a + b as in the Exercise, then the sections of H 0 (X, L m ) generates any pair of L x L y for distinct points x y in X, as well as 1-jets of L at any point x X. The existence of the section of H 0 (X, L m ) which generates any pair of L x L y for distinct points x y in X implies that F is injective. Now, we use the 17

18 fact that there is a section s of H 0 (X, L m ) which generates 1-jets of L at any point x X, ie. the section s vanishes to the second order. Choosing sections s 1,, s n such that the function s 1 /s,, s n /s have independent differential at x, then the holomorphic map ( s 1 s,, sn s defined in a neighborhood of x is an immersion near x. This complete the proof of Kodaira s imbedding theorem. ) 18