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Sasakian Geometry$

Charles Boyer and Krzysztof Galicki

Print publication date: 2007

Print ISBN-13: 9780198564959

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780198564959.001.0001

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Kähler Manifolds

Kähler Manifolds

Chapter:
(p.75) Chapter 3 Kähler Manifolds
Source:
Sasakian Geometry
Author(s):

Charles P. Boyer

Krzysztof Galicki

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198564959.003.0004

Abstract and Keywords

This chapter reviews some basic facts about Kähler manifolds with special emphasis on projective algebraic varieties. All standard material is covered: complex structures, curvature properties, Hodge theory, Chern classes, positivity and Fano varieties, line bundles and divisors. Of particular interest is Yau's famous proof of the Calabi conjecture which ends this chapter.

Keywords:   algebraic varieties, complex structures, Kähler manifolds, Calabi conjecture, Chern classes, Hodge theory, line bundles, divisors, canonical divisor, canonical bundle

Kähler metrics1 were introduced and studied by Erich Kähler in 1933 [Käh33] (recently reprinted in [Käh03]). But it was not until late the 1940s when the importance of Kähler manifolds in both Riemannian and algebraic geometry was finally realized. This was largely due to the fundamental work of Chern. Already in 1946 Chern introduces the notion of his Chern classes for Hermitian manifolds [Che46]. As a special case, it follows that the cohomology class of the Ricci 2-form ρ‎ω‎ on a Kähler manifold (M,J,g,ω‎g) does not depend on the metric g but only on the complex structure and it is a fixed multiple of the first Chern class of M

[ ρ g ] = 2 π c 1 ( M ) ,
(cf. Proposition 3.6.1 of the last section of this chapter). All of a sudden Kähler metrics dramatically gained in their importance. Throughout the 1950s they were studied by such giants as Borel, Chern, Hodge, Kobayashi, Kodaira, Lichnerowicz, Spencer, among many others, and culminating in Weil's well-known book [Wei58]. However, it was not until 10 years after Chern's article when Calabi grasped the real significance of this remarkable relation and what it means in the context of Kähler–Einstein metrics [Cal56, Cal57]. Calabi's work directly or indirectly resulted in several mesmerizing conjectures which captivated the mathematical world for years to come and stimulated the incredible interest in Kähler geometry that has remained until today.2 The relation
λ [ ω g ] = [ ρ g ] = 2 π c 1 ( M )
will be a focal point of Chapter 3 and even more so in Chapter 5. Since Kähler geometry and algebraic geometry from a differential viewpoint play a crucial rôle in this book, we take some care in setting things up inspite of the fact that there are several comprehensive treatments of this subject in a book form. We mention here the following books where excellent presentations of Kähler geometry can be found: [KN69, LB70, GH78b, Wel80, Zhe00, Voi02, Bal06] to name a few. We thus recall the basic properties of Kähler manifolds and Kähler metrics for the most part without proofs. We focus on the curvature properties, Chern classes and end with a discussion of the famous Calabi Conjecture. In Chapter 4 we describe (p.76) how Kähler and algebraic geometry generalize to the category of Kähler orbifolds. Finally, in Chapter 5 we focus primarily on Kähler–Einstein metrics of positive scalar curvature.

3.1. Complex Manifolds and Kähler Metrics

Let us start with a definition of a complex manifold. There are several frequently used approaches to define such a structure. Perhaps the most natural one is the one that imitates the usual definition of a smooth structure on a real manifold.

Definition 3.1.1: Let M be a real manifold of dimension 2n. A complex chart on M is a pair (U;φ‎) such that U⊂ M is open and φ‎: U→ C n is diffeomorphism between U and an open set φ‎(U)⊂ C n. We say that M is a complex manifold if it admits an atlas A = { U α , φ α } α Ω of complex charts whose transition functions

φ β φ α 1 : φ α ( U α U β ) φ β ( U α U β )
are all biholomorphic.

Recall from Example 1.4.9 that if M is a real manifold then a smooth section J of the bundle of endomorphisms End(TM) such that J 2 = −1 is called an almost complex structure. Note that we must have dim(M) = 2n. We can extend J to act on the complexified tangent bundle TMR C by C-linearity. Then J induces a splitting TMR C = T 1,0 MT 0,1 M, where T 1,0 M and T 0,1 M are eigenspaces with eigenvalues ± i, respectively. Note that TM is naturally isomorphic to T 1,0 M by the map X 1 2 ( X i J X ) .

An almost complex structure is said to be integrable if M admits an atlas of complex charts with holomorphic transition functions such that J corresponds to the induced complex multiplication on TMR C. Hence, a manifold with an integrable almost complex structure is (by definition) a complex manifold. Conversely, given a complex manifold M we can define J by

J z j = i z j ,
where (z 1,…,z n) is a local holomorphic coordinate system.

The theorem of Newlander and Nirenberg [NN57] asserts that integrability of J is equivalent of the vanishing of the Nijenhuis tensor of Equation (1.4.4). In view of this we have the following

Theorem/Definition 3.1.2: Let M be a real manifold and J an almost complex structure on M. The Nijenhuis tensor N J ≡ 0 on M if and only if J is integrable in which case we call J a complex structure and the pair (M,J) a complex manifold.

The second part of this statement is very often taken as an alternative definition of a complex manifold. As we will see it is perhaps the most geometric and indeed we will use it most often. A third way of describing a complex manifold as a real manifold with a torsion-free GL(n,C)-structure was described earlier in Example 1.4.9. For maps between complex manifolds we have

Definition 3.1.3: Let (M 1,J 1), (M 2, J 2) be two complex manifolds. A map f:M 1M 2 is called holomorphic if J 2(df(v)) = df(J 1(v)) for all v∈ Γ‎(TM). If, in addition, f −1 exists and is holomorphic, it is called a biholomorphism between M 1 and M 2.

(p.77) If M 2 = M 1 = M and J 2 = J 1 = J the set of biholomorphisms from M to itself form a group 𝔘𝔲 𝔱 (M,J). Generally it is not a Lie group; however, it is a Lie group when M is compact by Proposition 1.6.7. In fact, if M is compact it is a complex Lie group. Explicitly, [Kob72] we have

Proposition 3.1.4: Let M be a compact complex manifold. Then 𝔘𝔲 𝔱 (M,J) is a complex Lie transformation group and its Lie algebra 𝔞𝔲 𝔱 (M,J) consists of the holomorphic vector fields on M.

For a much more extended discussion of the complex automorphism group on Kähler manifolds we refer the reader to Chapter III of [Kob72].

One can extend J to act on the complexified cotangent bundle T*Mr C which then splits as T ( 1 , 0 ) M T ( 0 , 1 ) M . This in turn defines the splitting of the bundle of complex p-forms on M

Λ p T M = j = 0 p Λ j T ( 1 , 0 ) M Λ p j T ( 0 , 1 ) M = j = 0 p Λ j , p j M ,
where the last equality defines the space of forms of type (j,pj). Note that such a decomposition holds even if J is only almost complex.

Proposition 3.1.5: Let (M,J) be an almost complex manifold. The following conditions are equivalent:

  1. (i) [V,W]∈ Γ‎(T1,0M) for all V,W∈ Γ‎(T1,0M),

  2. (ii) [V,W]∈Γ‎(T0,1M) for all V,W∈ Γ‎(T0,1M),

  3. (iii) Im ( d ) | Γ ( 1 , 0 M ) Γ ( 2 , 0 M ) , Γ ( 1 , 1 M ) , Im ( d ) | Γ ( 0 , 1 M ) Γ ( 0 , 2 M ) Γ ( 1 , 1 M ) ,

  4. (iv) Im ( d ) | Γ ( p , q M ) Γ ( p + 1 , q M ) Γ ( p , q + 1 M ) , ,

  5. (v) NJ ≡ 0.

PROOF. We will only show the equivalence of (i), (iii), and (v) as the rest is evident. To establish equivalence of (i) and (iii) we use

2 d ω ( X , Y ) = X ( ω ( Y ) ) Y ( ω ( X ) ) ω ( [ X , Y ] ) ,
which, if ω‎ is a 1-form of type (1,0) and X,Y are vector fields of type (0,1), clearly vanishes. Hence, dω‎ does not have any (0,2)-part. To establish equivalence of (i) and (v) we consider B = [XiJX,YiJY] for any two real vector fields X,Y on M. It is easy to see that B+iJB = −N(X,Y)−iJ(N(X,Y)). But B+iJB = 0 if and only if B is a vector field of type (1,0) if and only if N J ≡ 0.   □

Thus, if (M,J) is a complex manifold, the composition of the exterior derivative acting on (k,l)-forms with projections on Γ‎(Λ‎k+1,l M) and Γ‎(Λ‎k,l+1 M), defines a natural splitting of d into two parts. They are denoted by ∂ and ∂̄ where d = ∂ + ∂̄. Since d 2 = 0 we get ∂2 = ∂̄2 = 0 and ∂∂̄ = –∂̄∂. We also denote by ε‎k,l the sheaf of germs of smooth sections of the bundles Λ‎k,l M, and use the same notation ∂,∂̄, etc. for these operators acting on local sections of ε‎k,l. We also let Ώ‎p denote the sheaf of germs of holomorphic sections of the bundle Λ‎p,0 M. Then, by the well-known ∂̄-Poincaré lemma [GR65], we have a resolution of this sheaf, namely

0 Ω p ε p , 0 ̄ ε p , 1 ̄ ε p , n 1 ̄ ε p , n 0.

(p.78) Moreover, since the sheaves ε‎k,l are fine sheaves, this resolution is acyclic. So if we define the Dolbeault cohomology groups [Dol53] of (M,J)

H ̄ k , l ( M ) = Ker ( ̄ : Γ ( Λ k , l ) Γ ( Λ k , l + 1 ) ) Im ( ̄ : Γ ( Λ k , l 1 ) Γ ( Λ k , l ) )
to be the derived groups of the cochain complex on global sections (1.1.7), the Abstract de Rham Theorem 1.1.16 gives the well-known Dolbeault Theorem [Dol53].

Theorem 3.1.6: Let M be a complex manifold. Then there is an isomorphism

H q ( M , Ω p ) H ̄ p , q ( M ) .

In Section 3.4 we shall give a more general version of this theorem by twisting with a vector bundle.

Definition 3.1.7: Let (M,J) be an almost complex manifold and let g be a Riemannian metric on M such that

g ( J X , J Y ) = g ( X , Y ) , X , Y Γ ( T M ) .
Then g is called an Hermitian metric and (M,J,g) is called an almost Hermitian manifold. Furthermore, given a Hermitian metric we define its fundamental 2-form or Kähler form ω‎g of g by
ω g ( X , Y ) = g ( X , J Y ) , X , Y Γ ( T M ) .
The triple (J,g,ω‎g) is called an almost Hermitian structure on M.

Note that an almost complex manifold always admits a Hermitian metric. An almost Hermitian structure (J,g,ω‎g) with an integrable J is called Hermitian. Given an Hermitian metric g we can extend it to a unique Hermitian scalar product h on the complexified tangent bundle TMR C satisfying

  1. (i) h ( Z ¯ , W ¯ ) = h ( Z , W ) ¯ for all Z,W∈ Γ‎(TM⊗R C),

  2. (ii) h(Z, Z̄) > 0 for all Z≠ 0∈ (TM⊗R C),

  3. (iii) h(Z, W̄) for Z∈ Γ‎(T1,0M) and W∈ Γ‎(T0,1M).

Locally, in a complex coordinate chart (U; z 1,…,z n) we can write the metric h as
(3.1.1)
h = i , j = 1 n h i j ̄ d z i d z ̄ j ,
where hi,j̄(p) is a Hermitian n × n matrix. Then the Hermitian form h can be written in terms of the Riemannian metric g and fundamental 2-form3 as h = g−2iω‎g and ω‎g can be written as
(3.1.2)
ω g = i 2 i , j = 1 n h i j ̄ d z i d z ̄ j .
Also, note that the volume form satisfies
ω g n = n ! ( i 2 ) n  det ( h i j ̄ ) d z 1 d z ̄ 1 d z n d z ̄ n
and, hence, ω g n = n ! d vol g where dvolg is the Riemannian volume form of (M,g).

(p.79) Definition 3.1.8: A Hermitian manifold (M,J,g,ω‎g) is said to be Kähler if ω‎g is a closed 2-form. We call g a Kähler metric, ω‎g its Kähler form, and the triple (J,g,ω‎g) a Kähler structure on M.

When we want to emphasize a Kähler form, we shall use the notation (M,ω‎) to denote a Kähler manifold. A manifold with a Kähler metric is often called a manifold of Kähler type.

From the previous discussion we see that a Kähler form is a real (1,1)-form which is both ∂-closed and ∂̄-closed. Hence [ω‎g] defines a Dolbeault cohomology class in H ̄ 1 , 1 ( M ) . Next we state several propositions and lemmas which describes some basic local and global properties of Kähler manifolds.

Proposition 3.1.9: Let (M,J,g,ω‎g) be an almost Hermitian manifold of real dimension 2n and let ω‎g be the fundamental 2-form associated to g. Letbe the Levi-Civita connection of g. Then the following conditions are equivalent:

  1. (i) ∇ J = 0,

  2. (ii) ∇ω‎g = 0,

  3. (iii) the holonomy group Hol(M,g) is contained in U(n),

  4. (iv) (M,J,g,ω‎g) is 1-integrable,

  5. (v) (M,J,g,ω‎g) is Kähler.

Yet another characterization of the Kähler condition is

Proposition 3.1.10: A Hermitian metric g on a complex manifold (M,J) is Kähler if and only if for any point p∈ M there exists a local holomorphic chart (U; z 1,…,z n) such that h i,j̄(p) = δ‎i,j, dh i,j̄(p) = 0.

Such a metric is said to osculate to order 2 to the Euclidean metric, and the coordinate chart is said to be normal at p.

We consider some important examples of complex Kähler manifolds [KN69].

EXAMPLE 3.1.11: Let M = R 2n and let (x 1,…,x n,y 1,…, y n) be the global coordinate chart on M. We can define

J x j = y j , J y j = x j .
Clearly, J 2 = −1 and N J ≡ 0 as J is a constant matrix on M. We can define z 1,…,z n by z j = x j+iy j for j = 1,…,n. We have
J z j = i z j , J z ̄ j = i z ̄ j
so that (z 1,…,z n)∈C n is a (global) holomorphic chart compatible with J. Now let us write the Hermitian metric h and the fundamental 2-form in complex coordinates: Let (z 1,…,z n) be the global holomorphic chart with z i = x i+iy i. Since dzi = dxi + idyi, dz̄i = dxiidyi, i = 1,…,n and the Hermitian metric h reads
h = i , j = 1 n h i j ̄ d z i d z ̄ j = i = 1 n d z i d z ̄ i .
As the fundamental 2-form ω‎g is simply minus the imaginary part of h we can write
ω g = i 2 i , j = 1 n h i j ̄ d z i d z ̄ j = i 2 i = 1 n d z i d z ̄ i
which is clearly closed.

(p.80) The fiducial example of a compact complex manifold is given by our next example.

EXAMPLE 3.1.12: Complex Projective Space. C P n is defined as the set of complex lines through the origin in C n+1. Let z = (z 0,…,z n) be a point (vector) in C n+1∖{0}. We say that two non-zero vectors z and z′ are equivalent if there is a λ‎∈ C* such that z′ = λ‎z. Then C P n is the quotient space (C n+1∖{0})/∼. We let π‎:C n+1∖{0}→C P n denote the natural projection. Local coordinate charts are defined as follows. Let ūi be the open subspace of C n+1 such that z i≠ 0. Then C P n is covered by open sets { U i = π ( U ˜ i ) } i = 0 n , together with homeomorphisms φ‎i:U i→C n = R 2n defined by

φ i ( [ z 0 , , z n ] ) = ( z 0 z i , , z i 1 z i , z i + 1 z i , , z n z i ) .
The coordinates on the right hand side of this equation are called affine coordinates, and it is convenient to set z i = 1 and write | Z | 2 = | Z 1 | 2 + + | Z i | 2 ^ + + | Z n | 2 , where as usual the hat means remove that element. Then the Fubini-Study metric in U i⊂ C P n is defined by
g k j ̄ = 2 z k z ̄ j  ln ( 1 + | z | 2 ) ,
and the Kähler form is
(3.1.3)
ω g = i 2 ̄  ln ( 1 + | z | 2 ) = i 2 ( Σ κ d z k d z ̄ k 1 + | z | 2 ( Σ k z ̄ k d z k ) ( Σ j z j d z ̄ j ) ( 1 + | z | 2 ) 2 ) .
Clearly, ω‎g is closed and one can check that both the Riemannian metric g and Kähler form ω‎g are globally defined on C P n.

Notice that by restricting the map π‎ to the unit sphere in C n+1 gives the well-known Hopf fibration S 1S 2n+1→C P n which as should become apparent will also be considered as a fiducial example for this book.

This example has an interesting generalization, namely:

EXAMPLE 3.1.13: Complex Grassmannian. Let G r p(C q+p) be the set of p-dimensional complex subspaces of the complex vector space C q+p. We will define complex structure on the complex Grassmannian G r p(C q+p) by construction an atlas of holomorphic charts. Let (z 1,…,z p+q) be the natural coordinates on C q+p. We can think of z α‎ as the complex linear mapping z α‎: C q+p→ C. Now, consider the partition of the set {1,…,p+q} = α‎∪α‎c = {α‎1,…,α‎p}∪ {α‎p+1,…,α‎p}, where both sets are ordered in increasing order. Let U α‎⊂ G r p(C q+p) be the subset of p-planes W for which z α‎ 1 |W,…,z α‎ p |W are linearly independent. Since, for each WU α‎, the mappings z α‎ 1 |W,…,z α‎ p |W form a basis for the dual space of W, we can write

z α p + k | W = j = 1 p w k j ( z α j | W ) , k = 1 , , q .
The coefficients w k j uniquely define a p × q complex matrix which maybe thought of as an element of C pq. Setting
ψ α ( W ) = ( w k j )
for each partition α‎ defines a map ψ‎α‎: U α‎→C pq. It is a simple exercise to check that these maps are injective and onto. Hence, the family S = { U α ; ψ α } of ( P + 1 P ) (p.81) charts form a holomorphic atlas on G r p(C q+p). There is a generalization of Equation (3.1.3) which shows that G r p(C q+p) is Kähler [KN69]. When p = 1, q = n we recover the standard atlas of the (n+1) holomorphic charts on complex projective space described in Example 3.1.12. In particular, we have G r 1(C n+1) = C P n.

One can describe further interesting generalizations by considering the set of all nested subspaces C k 1 ⊂ C k 2 ⊂ ··· ⊂ C k l ⊂ C n with 0 < k 1 < k 2 < ··· < k l < n. This gives rise to the so-called flag manifolds F lk 1,…,(C n), cf. [Akh90, Akh95]. In the case l = 1 we recover the Grassmannian. Even further generalizations are obtained by considering the generalized flag manifolds G/P, where G is a complex semi-simple Lie group and P is a parabolic subgroup (the ordinary flag manifolds correspond to taking G = SL(n,C)). These are all homogeneous Kähler manifolds. In fact, a well-known theorem of Borel and Remmert [BR62] (See also [Akh95]) says that any compact homogeneous Kähler manifold M is of the form (G/P) × A(M), where A(M) is a complex torus known as the Albanese torus of M.

Now, let fC (M). As the i ̄ f = 1 2 ( + ̄ ) ( ̄ ) f is real we conclude that i∂∂̄f is a real closed 2-form of type (1,1). We have the following two lemmas

Lemma 3.1.14: [Local i∂∂̄-Lemma] Let ρ‎ ∈ Γ‎(Λ‎1,1 U) be a smooth, closed, real (1,1)−form on a unit disc U = D n ( 1 ) n . Then there exists fC (U) such that ρ‎ = i∂∂̄f.

Lemma 3.1.15: [Global i∂∂̄-Lemma] Let (M,J) be a compact complex manifold and let γ‎ be a real (1,1)−form on M satisfying γ‎ = dα‎ for some real 1−form α‎. Then there exists a smooth real function f such that γ‎ = i∂∂̄f. In particular, if g 1, g 2 are two different Kähler metrics on M such that [ω‎g2] = [ω‎g2] ∈ H 2(M,R). Then

ω g 2 = ω g 1 + + i ̄ f ,
and f is unique up to a constant.

Lemma 3.1.14 says that on a Kähler manifold locally one can always find a function which “generates” the Kähler form (and, hence, the Kähler metric) via the simple formula ω‎g = i∂∂̄f. Such a function is called a Kähler potential. However, if M is compact a Kähler potential cannot exists globally. This is a simple consequence of the following

Lemma 3.1.16:; Let (M,J,g,ω‎g) be a compact Kähler manifold with the Kähler form ω‎g. Then [ ω g k ] H 2 k ( M , ) is nontrivial for all k = 0,…, n.

PROOF. This is a simple consequence of the Stokes′ Theorem and the fact that ω g n = n ! d vol g . For let ω g k = d α be exact. Then

0 < n ! vol ( M ) = M ω g n = M ω g k ω g n k = M d α ω g n k = M d ( α ω g n k ) = 0 ,
so we get a contradiction. In particular, when k = 1 the lemma implies that there is no globally defined function on M such that ω‎g = i∂∂̄f.   □

Let (M,J be a complex manifold and let N be a submanifold of M. then, in particular at each point pN, the tangent space TpNTpM is a vector subspace. We say that N is a complex submanifold of M TpN is a complex subspace for each pN, i.e., J(TpN) = TpN. Now, if N is a complex submanifold then the restriction JN of J to TN is a complex structure on N and the inclusion map (p.82) map ι‎: NM has the property

J ( d p ι ( w ) ) = ( d p ι ) ( J N ( w ) ) , w T p N ,
i.e., ι‎ is a holomorphic map between two complex manifolds (N,JN) and (M,J).

Since we can pull back Kähler forms, we have a simple but important result.

Proposition 3.1.17: A complex submanifold of a Kähler manifold is Kähler.

EXAMPLE 3.1.18: Consider the case when M = C P n and let the inclusion map ι‎:N ↪ C P n be defined by setting

N = N f = { m n | f ( m ) = 0 } ,
where f = (f1,…,fk) and fk ⊂ C[z0,…,zn] are complex homogeneous polynomials in (n+1) variables for each k = 0,1,…,n. Any such set is called a projective algebraic variety. If, in addition N_\bff is smooth then it is a complex submanifold in the way described above; hence, it is a smooth projective algebraic variety which by Proposition 3.1.17 is always Kähler. For instance, if k = 1 then N is called a hypersurface. For example, let f = f, where f is
f ( [ z 0 , , z n ] ) = z 0 p + + z n + 1 p .
Then Nf is called a Fermat hypersurface. When p = 2 we get a complex quadric.

The concept of variety can vary somewhat depending on the context. We now formalize this. An important point is that varieties allow for a certain type of singular behavior.

Definition 3.1.19: We have

  1. (i) An affine algebraic variety is the common zero locus of a collection of polynomials in C n.

  2. (ii) A projective algebraic variety is the common zero locus in C P n of a collection of homogeneous polynomials in C n+1.

  3. (iii) An analytic (sub)variety V of a complex manifold M is a closed subset that can written locally as the common zero locus of a finite collection of local holomorphic functions in M. In particular, a closed subset N ∈M of a complex manifold M is said to be a hypersurface if every point p ⊂ N has an open neighborhood U and a non−zero holomorphic function f:U → C such that N ∩ U = p ⊂ U | f(u) = 0.

By a variety we mean any of these three, and by an algebraic variety either of the first two. When working with algebraic varieties it is common to use the Zariski topology which is defined by the condition that its closed subsets are precisely the common zero loci of polynomials. So by a subvariety of an algebraic variety V we shall mean any Zariski closed subset of V.

Definition 3.1.20: A variety V is said to be irreducible if it cannot be written as the union of two subvarieties V 1 andV2 with V iV.

3.2. Curvature of Kähler Manifolds

Let (M,J,g,ω‎g) be a Kähler manifold and ∇ the Levi−Civita connection. We can extend ∇ in a C−linear way to Γ‎(TMR C). In a local chart (U;z1,…, zn) we have { z 1 , , z n } and { z ̄ 1 , , z ̄ n } as bases for T 1,0 M and T 0,1 M, respectively. (p.83) Define the Christoffel symbols Γ A B C as follows

/ z A z B = C = 1 n Γ A B C z C , A , B , C { 1 , , n , 1 ̄ , , n ̄ } .
It follows by C−linearity that
Γ A B C ¯ = Γ A ¯ B ¯ C ¯ , Γ A B C = Γ B A C .
Lemma 3.2.1: On a Kähler manifold the only non−vanishing Christoffel symbols are Γ i j k , and Γ i ̄ j ̄ k ̄ and>
Γ i j n = k ̄ g n k ̄ g j k ̄ z i .
PROOF . We have
/ z i z j = k Γ i j k z k + k ̄ Γ i j k ̄ z ̄ k , / z i z ̄ j = k Γ i j ̄ k z k + k ̄ Γ i j k ̄ z ̄ k .
Since, ∇ J = 0 and J acts by multiplication by i on the basis of T 1,0 M and -i on the basis of T 0,1 M, we get
/ Z i ( J z j ) = J / z i Z j .
By comparing two sides of this equation we get
i ( k Γ i j k z k + k ̄ Γ i j k ̄ z ̄ k ) = i ( k Γ i j k z k k ̄ Γ i j k ̄ z ̄ k ) ,
which means that Γ i j k ̄ = 0 . A similar argument applied to the second equation yields Γ i j ̄ k ̄ = 0 , , while Γ i j k ̄ = 0 follows by the symmetry two lower indices. Hence, the only non−zero Christoffel symbols are Γ i j k and their complex conjugate Γ i ̄ j ̄ k ̄ . Since Xg(Y,Z) = g(∇XY,Z) + g(Y,∇XZ) we obtain
z i g j k ̄ = z i g ( z j , z ̄ k ) = g ( l Γ i j l z l , z ̄ k ) = l Γ i j l g l k ̄ .
Contracting both sides of this equation with the inverse of the metric gives the formula.   □

In a local chart (U;z1,…,zn) we can define the following matrix−valued (1,0)−form

θ = g 1 g = i = 1 g 1 g z i d z i ,
where g = (gi). The above lemma implies that

Proposition 3.2.2: Let (M,J,g,ω‎g) be a Kähler manifold and let ∇ be the Levi−Civita connection of g extended to TM⊗ R C≃ T 1,0 MT 0,1 M. Accordingly, ∇ decomposes as ∇ = ∇ 1,0 + ∇ 0,1, and θ‎ is the connection 1−form of ∇ 1,0.

(p.84) Note that we have the following simple expression

j Γ i j j = j , k ̄ g j k ̄ g j k ̄ z i = z i ln G ,
where G = det(g) = det(gij̄). The determinant G ∈ C(U) is a smooth real function defined locally on U. The last equality follows from the well−known formula det(A) = exp{Tr ln A} as we have
z i   ln G = Tr ( g 1 g z i ) = n ̄ , k g n ̄ k g k n ̄ z i .
These simple formulas for the Christoffel symbols simplify the expression for the full Riemann curvature tensor. Recall that the Riemannian curvature tensor R of the metric g on M is defined by R(X,Y):Γ‎(TM) → Γ‎(TM), where
R ( X , Y ) Z = X Y Z Y X Z [ X , Y ] Z , X , Y , Z Γ ( T M ) .
Point−wise R(X,Y) is a skew−endomorphism of TpM at any pU. Since also R(X,Y) = –R(Y,X) one can think of R as a 2−from on M with values in the skew−endomorphisms of TM. It is sometimes convenient to think of R as a curvature operator, that is a section of ⊙2Λ‎2 TM defined by
( X Y , W Z ) ) = g ( R ( X , Y ) Z , W ) , X , Y , Z , W Γ ( T M ) .
Alternatively, one can introduce R as the section of ⊗4(T*M), i.e., the 4−linear map R : Γ‎(TM) × Γ‎(TM) × Γ‎(TM) × Γ‎(TM) → C (M), defined by
R ( X , Y , V , Z ) = g ( R ( X , Y ) Z , V ) , X , Y , V , Z Γ ( T M ) .
All these different pictures are useful.

If (M,J,g) is complex then one can extend all these curvature tensors by C−linearity to TMR C. In addition, on a Kähler manifold we have ∇J = 0 so that

R ( V , W ) J Z = J R ( V , W ) Z , V , W , Z Γ ( T M ) .
The above property has many important consequences. Since g is Hermitian we get R(U,V,JW,JZ) = R(U,V,W,Z). Hence, R(U,V,W,Z) = 0 unless W, Z are of different type. In particular, in the Kähler case one can define the following curvature tensors:

Definition 3.2.3: Let (M,J,g,ω‎g) be a Kähler manifold and let R be the curvature tensor of (M,g). Extending R to TMR C we define the following:

  1. (i) The Riemann curvature tensor R as a section of4(T *1,0 M), i.e., the map R : Γ‎(T 1,0 M) × Γ‎(T 0,1 M) × Γ‎(T 1,0 M) × Γ‎(T 0,1 M) → C(M, C), given by

    ( X , Y , Z , W ) R ( X , Y ̄ , Z W ̄ ) ,

  2. (ii) The Riemann curvature tensor RX,Ȳ as a section of End(T1,0M), i.e., the map RX,Ȳ : Γ‎(T 1,0 M)→Γ‎(T 0,1 M), X,Y ∈ Γ‎(T 1,0 M) given by

    R X , Y ̄ ( Z ) = R ( X , Y ̄ ) Z ̄ ,

  3. (iii) A real (1,1)−form Ω‎ with values in Γ‎(End(T1,0M)), i.e., a skew−Hermitian map Ω‎ : Γ‎(T 1,0 M) × Γ‎(T 0,1 M)→Γ‎(End(T 1,0 M))

    Ω ( X , Y ̄ ) = R X , Y ̄ , X , Y Γ ( T 1 , 0 M ) ,
    called the The Kähler–Riemann curvature form.

  4. (p.85)
  5. (iv) The Riemann curvature operator r as a section of2 Γ‎(Λ‎1,1 M), i.e., the map r : Γ‎(Λ‎1,1 M) × Γ‎(Λ‎1,1 M)→ C (M,C) given by

    ( X Y ̄ , V W ̄ ) = R ( X , Y ̄ , Z , W ̄ ) .

Proposition 3.2.4: Let (U; z 1,…,zn) be a local coordinate chart on a Kähler manifold (M,J,g,ω‎g). Let R ∈ Γ‎ (⊗4(T *1,0 M)) be the Riemann curvature tensor. Then

R i j ̄ k l ̄ = 2 g k l ̄ z i z ̄ j p , r ̄ , m g p r ̄ g k r ̄ z i g p l ̄ z ̄ j .

PROOF . It remains to do the local computations. We have

R i j ̄ k l ̄ = R ( z i , z ̄ j , z k , z ̄ l ) = ( R ( z i , z ̄ j ) z ̄ l , z k ) .
Since
R ( z i , z ̄ j ) z ̄ l / z i / z ̄ j z ̄ l / z ̄ j / z i z ̄ l = / z i ( n ̄ Γ j ̄ l ̄ n ̄ z ̄ n ) .
Hence
R i j ̄ k l ̄ = g ( / z i ( n ̄ Γ j ̄ l ̄ n ̄ z ̄ n ) , z k ) = z i g ( n ̄ Γ j ̄ l ̄ n ̄ z ̄ n , z k ) g ( n ̄ Γ j ̄ l ̄ n ̄ z ̄ n , m Γ i k m z m ) = z i ( n ̄ Γ j ̄ l ̄ n ̄ g n ̄ k ) n ̄ , m Γ j ̄ l ̄ n ̄ Γ i k m g n ̄ m = = z i ( n ̄ , p g n ̄ p g l ̄ p z ̄ j g n ̄ k ) n ̄ , p , r ̄ , m g n ̄ p g l ̄ p z ̄ j g m r ̄ g k r ̄ z i g n ̄ m = 2 g k l ̄ z i z ̄ j p , r ̄ g p r ̄ g k r ̄ z i g p l ̄ z ̄ j .
   □

Definition 3.2.5: Let RX,Ȳ : Γ‎(T 1,0 M)→Γ‎(T 1,0 M), X,Y ∈ Γ‎(T 1,0 M) be the Riemann curvature tensor on a Kähler manifold (M,J,g,ω‎g). The Ricci curvature tensor is the map Ricω‎ : Γ‎(T 1,0 M) ⊗ Γ‎(T 1,0)→C (M,C) defined by the trace

Ric ω ( X , Y ̄ ) = Tr ( R X , Y ̄ ) .

A priori it is not clear that Ricω‎(X,Ȳ) is the Ricci curvature tensor in the usual sense. However, it is clear that Ric(X,Ȳ) defines a Hermitian symmetric form on T p 1 , 0 M . For with respect to any unitary basis {e 1,…,en} of T p 1 , 0 M we have

Ric ω ( X , Y ̄ ) = i = 1 n R ( X , Y ̄ , e i , e ̄ i ) ,
so that
Ric ω ( X , Y ¯ ) ¯ = i = 1 n R ( X ¯ , Y , e ¯ i , e i ) = i = 1 n R ( Y , X ¯ , e i , e ¯ i ) = Ric ω ( Y , X ¯ ) .

(p.86) Proposition 3.2.6:; The Ricci curvature tensor Ricω‎ : T 1,0 MT 1,0 M → C is a Hermitian symmetric form on T p 1 , 0 M at each pM. In a local chart (U; z 1,…,zn) this form can be written as

Ric ω = i , j ̄ = 1 n R i j ̄ d z i d z ̄ j ,
where
R i j ̄ = 2 z i z ̄ j ( ln G ) .
The real part of this Hermitian symmetric form is the Ricci curvature tensor of the metric g while the imaginary part is a real (1,1)−form on M called the Ricci form and denoted by ρ‎ω‎ ≡ ρ‎g. With respect to the local chart (U; z 1,…,zn) we have
ρ ω = i 2 i , j ̄ R i j ̄ d z i d z ̄ j = ̄ ( ln G ) .

PROOF By definition we have

R i j ̄ = k l ̄ g k l ̄ R i j ̄ k l ̄ = 2 z i z ̄ j ( ln G ) .
A priori it is not obvious that Ric(X,Ȳ) coincides with the usual definition of the Ricci curvature. A simple calculation shows that this is indeed the case.   □

Note that the Kähler–Riemann curvature 2−form Ω‎ = dθ‎ + θ Λ θ‎ is simply the curvature 2−form of the ∇1,0 part of the Levi−Civita connection. In local coordinates we can write

Ω k l = i , j ̄ , p ̄ R i j ̄ g l p ̄ d z i d z ̄ j
and, hence, the Ricci form ρ‎ω‎ is a closed 2−form which is an invariant under the complex linear group GL(n,C), namely,
(3.2.1)
ρ ω = Tr ( Ω ) .

There are several more curvatures typically considered in the context of complex and Kähler manifolds. The usual notion of sectional curvature is one of them. Recall that for any X,Y ∈ Γ‎(TM) we define the sectional curvature of the 2−plane σ‎ ⊂ TpM spanned at point pM by Xp and Yp as

K ( σ , p ) = K ( X , Y ) = R ( X , Y , Y , X ) | X Y | 2 ,
where |X Λ‎ Y|2 = |X|2|Y|2 – [g(X,Y)]2 = g(X,X)g(Y,Y) – [g(X,Y)]2 is the square of the area of the parallelogram spanned by X,Y. For real manifolds, if (M,g) is complete, simply connected and of constant sectional curvature then (depending on the sign) (M,g) is isometric to Sn, R n, or the real hyperbolic ball B n ( 1 ) . Now, in complex dimension greater than 1 the only Kähler manifolds of constant sectional curvature are flat. To obtain a Kähler analog of the real space forms we introduce the following

Definition 3.2.7: Let (M,J,g,ω‎g) be a Kähler manifold and let X,Y ∈ Γ‎(T 1,0 M). Then

B ( X , Y ) = R ( X , X ̄ , Y , Y ̄ ) g ( X , X ̄ ) g ( Y , Y ̄ )

(p.87) is called the bisectional curvature of g in the direction of X,Y while

H ( X ) = B ( X , X ) = R ( X , X ̄ , X , X ̄ ) [ g ( X , X ̄ ) ] 2
is called the holomorphic sectional curvature of g in the direction of X.

If we write X = 1 2 ( U + i J U ) , Y = 1 2 ( V + i J V ) for some real vector fields U,V ∈ Γ‎(TM) we have

B ( X , Y ) = | J U V | 2 | U | 2 | V | 2 K ( J U , V ) + | U V | 2 | U | 2 | V | 2 K ( U , V ) ,
and H(X) = K(U,JU), where K the sectional curvature of g. In particular, if g is of positive (negative, non−positive, non−negative) sectional curvature then g is of positive (negative, non−positive, non−negative) bisectional curvature. Clearly, the holomorphic sectional curvature is the curvature of all J−invariant planes. It is not hard to show that on a Kähler manifold the holomorphic sectional curvature H determines the Riemann curvature tensor completely.

EXAMPLE 3.2.8: This is a continuation of the discussion of complex projective space as described in Example 3.1.12 An easy computation gives

ω g n = ( i / 2 ) 2 n ! ( 1 + | z | 2 ) n + 1 d z 1 d z ̄ 1 d z n d z ̄ n ,
which means that G = det(gīj) = (1 + |z|2)–(n+1). Hence, the Ricci curvature tensor is
(3.2.2)
R i j ̄ = 2 z i z ̄ j ln G = ( n + 1 ) g i j ̄ .
EXERCISE 3.1: Show that the Fubini−Study metric is U(n + 1)−invariant and that U(n + 1) acts on M transitively.

It now follows that in order to compute the full Riemann curvature tensor Rījk̄l on C P n it is sufficient to compute it at one point [1,0,…,0]. Using local expressions for Rījk̄l and gīj one can easily see that at z = 0 we have Rījk̄l = δ‎ījδ‎k̄l + δ‎īlδ‎k̄j. In particular, as gīj |z = 0= δ‎īj, at any other point

R i j ̄ k l ̄ = g i j ̄ g k l ̄ + g i l ̄ g k j ̄
and M is of constant holomorphic sectional (and bisectional) curvature c = +1.

EXAMPLE 3.2.9: Bergman metric on the complex ball. Consider M = B n ( 1 ) = { z n | | z | 2 < 1 } and let

ω g = i 2 ̄ ln ( 1 | z | 2 ) .
Likewise, one can show that Bergman metric g is U(n,1)−invariant and that U(n,1) acts on M = B n ( 1 ) transitively. Repeating the calculations of the previous example we can easily see that Rīj = –(n + 1)gīj and (M,g) is of constant holomorphic sectional curvature c = –1.

It can be shown that any two simply connected complete Kähler manifolds with constant sectional curvature c are holomorphically isometric. Hence, the above examples together with the flat metric on C n show that

(p.88) Theorem 3.2.10: Let (M 2n,J,g,ω‎g) be a complete Kähler manifold of constant holomorphic sectional curvature c. Depending on the sign of c and up to scaling, the universal cover M̃ iis holomorphically isometric to C P n, C n, or B n (1).

3.3. Hodge Theory on Kähler Manifolds

Let (M 2n,J,g,ω‎g) be a compact Hermitian manifold. We define the Hodge star operator to be the map ✶ :Γ‎(Λ‎k,l M)→ Γ‎(Λ‎n-k,n-l)M defined by

(3.3.1)
φ ψ = g ( φ , ψ ̄ ) d vol g = g ( φ , ψ ̄ ) ω g n n ! , φ , ψ Γ ( Λ k , l M ) .
Note that this map is linear over R but conjugate−linear over C. In addition, we have ⋆2φ‎ = (-1)k+lφ‎. The action of the Hodge star operator on any complex k−form φ‎ ∈ Γ‎(Λ‎k M⊗ C) is defined by the splitting of φ‎ as a sum of forms of type (kj,j). Locally, in a holomorphic chart (U; z1,…,zn), N = dimC(M) one can write
Γ ( Λ k , l M ) = { | I | = k , | J | = l ω I J ̄ d z I d z ̄ J , ω I J ̄ C ( U , ) } ,
where
I = ( i 1 , , i k ) , 1 i 1 < < i k n , d z I = d z i 1 d z i k ,
J = ( j 1 , , j l ) , 1 j 1 < < j k n , d z ̄ J = d z ̄ k 1 d z ̄ j l .
Let us assume that (U; z1,…,zn) are normal coordinates, i.e., {dz1,…,dzn} is a unitary frame of Γ‎(Λ‎1,0 M). Denote by Î,Ĵ the likewise ordered complements of I,J in {1,…,n} and define σ‎(I) by
d z I d z I ^ = ( 1 ) σ ( I ) d z 1 d z n .
Then we have an explicit formula
( ω I J ̄ d z I d z ̄ J ) = 2 k + l m i n ε I J ω I J ̄ ̄ d z I ̄ d z ̄ J ^ ,
where the sign ∊IJ = (-1)n(n-1)/2+(n-k)l+σ‎(I)+σ‎(J). For compact M we can define an L 2 Hermitian inner product on each Γ‎(Λ‎k,l M) by
(3.3.2)
< φ , ψ > = M φ ψ , φ , ψ Γ ( Λ k , l M ) .
One can check that ⋆ is an isometry with respect to this inner product, i.e., <⋆φ‎,⋆ψ‎> = <φ‎, ψ‎>. We proceed to define
(3.3.3)
̄ = ̄ : Γ ( Λ k , l M ) Γ ( Λ k , l 1 M ) .
It follows that this is a formal adjoint of the ̄∂ operator as for any φ‎ ⊂ Γ‎(Λ‎k,l-1 M) and any ψ‎ ∈ Γ‎(Λ‎k,l M) we have
̄ φ ψ φ ̄ = ̄ φ ψ + ( 1 ) k + l 1 φ ̄ ψ = ̄ ( φ ψ ) = d ( φ ψ ) ,
so that, by Stokes′ Theorem, < ̄ ∂ φ‎,ψ‎ > = < φ‎,̄ ∂ * ψ‎ >  . Similarly, one can define
= : Γ ( Λ k , l M ) Γ ( Λ k 1 , l M )
which is an adjoint of the ∂ operator. Hence, we have 6 different operators acting on forms d,d*,∂,∂*, ∂,̄ ∂̄* and 3 natural Laplacians
Δ d = d d + d d , Δ = + , Δ ̄ = ̄ ̄ + ̄ ̄ .

(p.89) The Laplacian Δ‎̄∂ is a formally self−adjoint operator and

< Δ ̄ φ , φ > = < ̄ φ , ̄ φ > + < ̄ φ , ̄ φ > 0.
Definition 3.3.1: The space h k,l(M)∈Γ‎(Λ‎k,l M)
k , l ( M ) = { φ Γ ( Λ p , q M ) | Δ ̄ φ = 0 } .
is called the space of harmonic (k,l)−forms. Furthermore, the numbers dim (h k,l(M)) = h k,l(M) are called the Hodge numbers of M.

The famous Hodge Theorem implies that the Hodge numbers are well−defined integers, viz.

Theorem 3.3.2: On a compact Hermitian manifold (M 2n,J,g,ω‎g) the dimensions dim (h k,l(M))∞ for all 0≤ k,l≤ n. Furthermore, we have the decompositions

Γ ( Λ k , l M ) = k , l ( M ) ̄ ( Γ ( Λ k , l 1 M ) ) ̄ ( Γ ( Λ k , l 1 M ) ) .
In particular, any class in H ̄ k , l ( M ) has a unique harmonic representative, i.e., we have a complex vector space isomorphism
k , l ( M ) H ̄ k , l ( M ) .

Let us introduce the projections

π k , l : Γ ( Λ T M ) Γ ( Λ k , l M ) .
π r = k + l = r π k , l : Γ ( Λ T M ) Γ ( Λ r M ) .
Since the metric G is Hermitian we have π‎k,l⋆ = ⋆π‎k,l. In general, given a complex Δ‎d−harmonic r−form ψ‎ the projection π‎k,l need not be Δ‎̄∂ −harmonic. However, on a Kähler manifold we have some additional structure.

To show this let us introduce the operator

L : Γ ( Λ k , l M ) Γ ( Λ k + 1 , l + 1 M )
defined by wedging with the Kähler form ω‎g, i.e., L(φ‎) = φ Λ ω‎g. The adjoint of L in the L 2−norm is given by
L = ( 1 ) p + q L : Γ ( Λ k , l M ) Γ ( Λ k + 1 , l 1 M ) .
We have the following

Lemma 3.3.3: [Kähler identities] Let (M,J,g,ω‎g) be a Kähler manifold. Then

[ L , ] = i ̄ , [ L , ̄ ] = i , [ L , ] = i ̄ , [ L ̄ ] = i .

PROOF. Since both sides of all these inequalities are first order differential operators it is enough to check them for the Euclidean space C n with the standard Hermitian metric. The general statement follows then from the fact that the Kähler condition is equivalent to the existence of normal coordinates. □

EXERCISE 3.2: Verify the identities of Lemma 3.3.3 in normal coordinates.

Proposition 3.3.4: Let (M,J,g,ω‎g) be a compact Kähler manifold. Then

  1. (i) [L*,Δ‎d] = [L*,Δ‎d] = 0

  2. (ii) Δ‎d = 2Δ‎ = 2Δ‎̄

  3. (iii) [Δ‎d,π‎p,q] = 0.

(p.90) PROOF. We have

Δ d = ( + ̄ ) ( + ̄ ) + ( + ̄ ) ( + ̄ ) = Δ + Δ ̄ + ( ̄ + ̄ ) + ( ̄ + ̄ ) .
Using the Kähler identities we can replace ̄ ∂* = -i[L*,∂] which gives
( ̄ + ̄ ) = i ( [ L , ] + [ L , ] ) = 0.
On the other hand, using [L*,∂̄] = -i∂*, we have   □
i Δ = ( i ) + ( i ) = ( L ̄ ̄ L ) + ( L ̄ ̄ L ) = = L ̄ L ̄ + L ̄ + ̄ L ̄ L = [ , L ] ̄ + ̄ [ , L ] = i ( ̄ ̄ + ̄ ̄ ) = i Δ ̄ .
 □

There are two important consequences of the above propositions. The first is the so−called Hodge decomposition. The second is the so−called hard Lefschetz Theorem. Let us first discuss the Hodge structure. We define

H k , l ( M , ) = Z d k , l ( M ) d Γ ( Λ M ) Z d k , l ( M ) ,
to be the space of d−closed forms of type (k,l) modulo d−exact forms and let d r ( M ) be the space of Δ‎d−harmonic r−forms. Since the usual Laplacian Δ‎d is real, commutes with π‎k,l and equals 1/2 of the ∂̄−Laplacian we have
r ( M ) = k + l = r k , l ( M ) ,
H ̄ k , l ( M ) = k , l ( M ) = k , l ( M ) ̄ = H k , l ( M , ) .
Using the usual Hodge theorem H D R ( M ) = ( M ) for the Laplacian Δ‎d we get

Theorem 3.3.5: For a compact Kähler manifold we have

r ( M , ) = k + l = r H k , l ( M , ) , H k , l ( M , ) ̄ = H l , k ( M , ) .

Since [⋆,Δ‎̄∂] = 0, by Kodaira–Serre duality4 the map

: k , l ( M ) n k , n l ( M )
is an isomorphism. Summarizing, we have the following properties of the Hodge numbers

Corollary 3.3.6: Let (M 2n,J,g,ω‎g) be a compact Kähler manifold, h k,l(M) be the Hodge numbers, and b r(M) = ∑k+l = rhk,l(M) the Betti numbers of M. Then for all 0≤ k,l≤ N we have

  1. (i) h k,l(M) < ∞,

  2. (ii) h k,k(M)≥1 and h n,n = h 0,0 = 1,

  3. (iii) h k,l(M) = h l,k(M) = h n-k,n-l(M),

  4. (iv) b 2r+1(M) are even and h 1 , 0 ( M ) = h 0 , 1 ( M ) = 1 2 b 1 ( M ) is a topological invariant.

(p.91) For any compact Kähler manifold one can consider the Hodge diamond, i.e., the arrangement of the Hodge numbers in a diamond−shape array. For instance, when n = 3 we can have

                   Kähler Manifolds

The above corollary implies that every Hodge diamond has two symmetries: conjugation gives the symmetry through the central vertical axis while Hodge star yields the symmetry through the center of the diamond. There are more relations between the Hodge numbers. They are due to the Lefschetz decomposition which we will discuss below.

EXERCISE 3.3: Let us define an operator h = r = 0 2 n ( n r ) π r . Show that we have the following relations

[ L , L ] = h , [ h , L ] = 2 L , [ h , L ] = 2 L
and, hence, {L,L*,h} are generators of the Lie algebra 𝔰 𝔩(2,R).

Definition 3.3.7: Let (M,J,g,ω‎g) be a compact Kähler manifold. We define the primitive cohomology groups of M as kernels of the L*−operator, i.e.,

P r ( M , ) = ker ( L : H r ( M , ) H r 2 ( M , ) ) ,
P k , l ( M , ) = ker ( L : H k , l ( M , ) H k 1 , l 1 ( M , ) ) .
The dimension of Pr(M,C) is called primitive r th Betti number of M.

Since {L,L*,h} all commute with the Kähler Laplacian we have

P r ( M , ) = k + l = r P k , l ( M , ) .
The generators {L,L*,h} give a finite−dimensional representation of the Lie algebra 𝔰 𝔩(2,R) acting on H*(M,C), where H r(M,C) is the eigenspace of the operator h with eigenvalue (nr). The next theorem follows directly from the theory of finite−dimensional representations of 𝔰 𝔩(2,R):

Theorem 3.3.8: (Hard Lefschetz) On a compact Kähler manifold M the map Lk:Hn-k(M)→ Hn+k(M) is an isomorphism for 1 ≤ k ≤ n. Furthermore,

H r ( M ) = k 0 L k P r 2 k ( M ) .

(p.92) 3.4. Complex Vector Bundles and Chern Classes

Let (M,J) be a smooth manifold and let EM be a complex vector bundle of complex rank r over M. Let ∇ be a complex Koszul connection (cf. Definition1.3.5) on E with curvature form Ω‎. Consider the space m r× r(C) of complex r× r matrices. For any A m r× r(C) we define

det ( A + λ I ) = f r ( A ) + λ f r 1 ( A ) + + λ r 1 f 1 ( A ) + λ r .
Clearly, for any i = 1,…,r the function fi:m r× r(C)→C is a GL(r,C)−invariant, complex homogeneous polynomial of deg(fi) = i. Note that fi is the i th elementary symmetric function of the eigenvalues of A. In particular, fr(A) = det(A) and f1(A) = Tr(A).

Definition 3.4.1: Let E→ M be a rank r complex vector bundle over M, and let ∇ be a complex connection on E with curvature 2−form Ω‎. For each i = 1,…,r we define the 2i−form

f i ( i 2 π Ω ) = c i ( E , )
and call it the i th Chern form of E.

We have the following

Proposition/Definition 3.4.2: Given (E,∇) and any 1≤i≤ r, the i th Chern form c i(E,∇) is closed. Furthermore, if ∇̃ is another complex connection on E the difference c i(E,∇)-c i(E,∇̃) is exact, i.e., the cohomology class [ c i ( E , ) ] H D R 2 i ( M ) is independent of ∇. The resulting cohomology class is called the i th Chern class of E and is denoted by c i(E).

When working with Chern classes it is convenient to consider the total Chern class c ( E ) = i = 0 c i ( E ) , where c 0(E) = 1 and the sum is always finite for a finite rank vector bundle, so c(E) ∈ H*(M,Z). For Whitney sums the total Chern class satisfies c(EE') = c(E)c(E'). If ̄E denotes the complex conjugate bundle to E, then ciE) = (-1)ici(E) and since ̄E is isomorphic to the dual complex vector bundle ci(E*) = (-1)ici(E). There are also certain “multiplicative sequences″ of interest [Hir66,MS74]. We define the Chern roots xi of a rank k complex vector bundle by c ( E ) = 1 + c 1 ( E ) + + c k ( E ) = i = 1 k ( 1 + x i ) . Then the Chern character ch(E) and the Todd class Td(E) are defined by

(3.4.1)
ch ( E ) = i = 1 k e x i , Td ( E ) = i = 1 k x i 1 e x i
Generally, these are elements in the rational cohomology ring H*(M,Q). Here is the celebrated Hirzebruch–Riemann–Roch Theorem:

Theorem 3.4.3: Let E be a holomorphic vector bundle on a compact complex manifold M n. Then

χ ( M , E ) = t 2 n ( Td ( M ) ch ( E ) ) ,
where ( M , E ) = i = 1 n h i ( M , O ( E ) ) is the holomorphic Euler characteristic of E.

When E is the trivial bundle we get an invariant of the complex structure, namely the holomorphic Euler characteristic χ‎(M,𝖪) and Theorem 3.4.3 gives the “Todd–Hirzebruch formula″ χ‎(M,𝖪) = t 2n(Td(M)), the later being known as the Todd genus.

(p.93) EXAMPLE 3.4.4: The complexified tangent bundle. Consider the case where E = TM⊗ C is the complexified tangent bundle of a complex manifold which splits as TM⊗ C = T 1,0 MT 0,1 M. Now T 1,0 M is a holomorphic vector bundle, called the holomorphic tangent bundle, which is isomorphic as a real vector bundle to TM. Now T 1,0 M has a complex connection ∇1,0 which can be used to compute its Chern classes, ci(M) = ci(T 1,0 M), called the ith Chern class of the complex manifold M, and denoted simply by ci when the complex manifold M is understood. These Chern classes depend only on the homotopy class of the complex structure J on M. Note that the top class cn(M) = e(M) is the Euler class. Given any partition I = i1,…,ir of the integer n, we can define the I th Chern number c I[M] by evaluation on the fundamental homology class [M], i.e., cI[M] = < ci 1ci r,[M]>. Note that the top Chern number cn[M] is just the Euler–Poincar′e characteristic cn[M] = χ‎(M) = ∑i(-1)i b i(M).

In low dimensions the Todd–Hirzebruch formula reduces to well−known classical formulae. Recall that h 0,1 = q is called the irregularity. For n = 1 (compact Riemann surfaces) we get h 0,1 = g, where g is the genus of the Riemann surface. For n = 2 we get Noether′s formula for compact complex surfaces, ( M , O ) = 1 q + p g = 1 12 ( c 1 2 [ M ] + c 2 [ M ] )

Now suppose that (M,J) is a complex manifold and E is a complex vector bundle on M. We can consider the tensor product bundle E⊗ Λ‎k,l M, and we let ek,l(E) denote the sheaf of germs of smooth sections of E⊗ Λ‎k,l M. Smooth sections of this sheaf are (k,l)−forms with coefficients in E, the set of which we denote by A k,l(E). The connection ∇ in E induces a connection, also written as ∇, in A k,l(E). This connection splits as ∇ = ∇1,0 + ∇0,1 giving maps

1 , 0 : A k , l ( E ) A k + 1 , l ( E ) , 0 , 1 : A k , l ( E ) A k , l + 1 ( E )
We are interested in the case when E admits a holomorphic structure. Then similarly, we let Ω‎p(E) denote the sheaf of germs of holomorphic sections of the bundle Λ‎p,0E. So we shall analyze the structure of connections in holomorphic vector bundles following [Kob87].

Theorem 3.4.5: A smooth complex vector bundle E over a complex manifold admits a holomorphic structure if and only if there is a connectionin E such that0,1 = ∂̄.

The holomorphic structure in E is uniquely determined by the condition ∇0,1 = ̄∂, and this condition says that the (0,2) component ∇0,1∘ ∇0,1 of the curvature of ∇ vanishes. It is straightforward to generalize Definition 3.1.7 to an arbitrary complex vector bundle.

Definition 3.4.6: An Hermitian metric h on a complex vector bundle E is an assignment of an Hermitian inner product to each fiber Ex of E that varies smoothly with x. A connectionin E is called an Hermitian connection ifh = 0.

A vector bundle equipped with a Hermitian metric is often called a Hermitian vector bundle. Using partitions of unity one easily sees that Hermitian metrics exists on complex vector bundles.

Proposition 3.4.7: Let E be a holomorphic vector bundle with an Hermitian metrich. Then there exists a unique Hermitian connectionsuch that0,1 = ∂̄.

Again for a proof see [Kob87]. The unique connection of Proposition 3.4.7 is called the Hermitian connection. There is a version of Hodge theory tensored with (p.94) holomorphic vector bundles with an Hermitian metric. Such a choice of Hermitian metric gives an isomorphism τ‎ between a holomorphic vector bundle E and its dual E*. Then we can define the Hodge star isomorphism

E = τ : A k , l ( E ) A n k , n l ( E )
or, alternatively, on the sheaf level
E = τ : ε k , l ( E ) ε n k , n l ( E ) .
We can now extend the Hodge inner product defined by Equation(3.3.2), to an inner product on A k,l(E) by
(3.4.2)
< φ , ψ > = M φ E ψ , φ , ψ A k , l ( E ) ,
where we write φ‎,ψ‎A k,l(E) = Γ‎(Λ‎k,l ME) as φ‎ = φ‎⊗ e, and ψ‎ = ψ‎⊗ f, respectively, in which case we have φ‎Λ‎⋆E ψ‎ = (φΛ‎ ⋆ψ‎) < e,f>E, where < e,f>E is the pairing between E and E*.

Now consider the cochain complex

A k , l ( E ) ̄ A k , l + 1 ( E )
and let H ̄ p , q ( M , E ) denote the derived cohomology groups of this complex. The operator ∂̄ has a (formal) adjoint ∂̄E* = -⋆E∂̄⋆E with respect to the Hodge inner product (3.4.2), and we have complex Laplacian □̄E = ∂̄ ∂̄E*+∂̄E*∂̄ acting on A*,*(E). As in Definition 3.3.1 we define the space h k,l(M,E) of harmonic E−valued (k,l)−forms by
(3.4.3)
k , l ( M , E ) = { φ A k , l ( E ) | ̄ E φ = 0 }

Combining the resulting vector bundle version of the Hodge isomorphism Theorem 3.3.2 with the Abstract de Rham Theorem 1.1.16, and using the acyclic Ω‎∂̄−resolution of the sheaf Ω‎p(E), one gets Serre′s generalized Dolbeault Theorem for holomorphic vector bundles [Ser55]:

Theorem 3.4.8: Let M be a complex manifold, and let E be a holomorphic vector bundle on M. Then there are isomorphisms

H q ( M , Ω p ( E ) ) H ̄ p , q ( M , E ) p , q ( M , E ) .

This theorem can be used to prove the celebrated Kodaira–Serre Duality Theorem [Ser55]:

Theorem 3.4.9: Let M be a compact complex manifold of complex dimension n, and let E be a holomorphic vector bundle over X. There there is a conjugate−linear isomorphism

ψ : H q ( M , Ω p ( E ) ) H n q ( M , Ω n p ( E ) ) .

3.5. Line Bundles and Divisors

The purpose of this section is to give a brief review of the fundamental concepts employed in complex manifold theory, namely, line bundles and divisors as well as to discuss their interrelationship. We refer to the literature [GH78b,Wel80,Voi02,Laz04a] for complete treatments of these important subjects.

(p.95) 3.5.1. Line Bundles

We begin by considering the set of smooth complex line bundles on a complex manifold M. These are determined by their transition functions which have values in the group GL(1,C) = C*. The isomorphism classes of complex line bundles form a group under tensor product, and by Theorem 1.2.2 isomorphism classes of C*−bundles over M are in one−to−one correspondence with elements of the sheaf cohomology group H 1(M,ε‎*). Thus, we consider the exponential short exact sequence of sheaves on M

(3.5.1)
0 ι ε exp ε 0 ,
where the map ι‎ is ι‎(k) = 2π‎ ik and the exponential map sends the germ f of any holomorphic function to exp(f). Since e is a fine sheaf, Propositions 1.1.12 and 1.1.13 imply H q(M,e) = 0 for all q0. So the induced long exact cohomology sequence (1.1.6) gives an isomorphism H1(M,e*)≈ H 2(M,Z) which says that the topological invariant H 2(M,Z) can be thought of as the group of complex line bundles on M. This isomorphism is realized by associating to a complex line bundle l its first Chern class c 1().

To study the holomorphic line bundles on M we consider the exact sequence

(3.5.2)
0 ι O exp O 0.
This induces a long exact sequence in cohomology,
(3.5.3)
H 1 ( M , O ) H 1 ( M , O ) δ H 2 ( M , ) H 2 ( M , O ) .
The group H 1(M,o*) represents the group of holomorphic line bundles on M with group multiplication being the tensor product, and the inverse bundle being the dual bundle. This group is called the Picard group of M and often denoted by Pic(M). As seen above the connecting homomorphism δ‎ takes a holomorphic line bundle to its first Chern class c (l), and the group H 2(M,Z) is isomorphic to the group of topological complex line bundles on M. So if H 2(M,o)≠ 0 we see that not every complex line bundle gives rise to a holomorphic line bundle. Similarly, if H 1(M,o)≠ 0, there can be inequivalent holomorphic bundles associated to the same complex line bundle. The kernel of the map δ‎ is denoted by Pic0(M) and represents the subgroup of holomorphic line bundles that are trivial topologically. The quotient group Pic(M)/Pic0(M) is known as the Neron–Severi group denoted by NS(M). The rank of NS(M) is called the Picard number of M and denoted by ρ‎(M). When M is a smooth projective algebraic variety, we have 1≤ ρ‎(M)≤ b 2(M), and it follows from the well−known Lefschetz Theorem on (1,1) classes that there is an isomorphism NS(M) ≈ H 1,1(M,C)∩ H 2(M,Z). So NS(M) is a free Abelian group of rank ρ‎(M).

The notion of positivity is fundamental in the study of holomorphic line bundles.

Definition 3.5.1: Let M be a complex manifold and L a holomorphic line bundle on M. We say that L is positive (negative) if it carries an Hermitian metric whose curvature form ω‎ with respect to the Hermitian connection is a positive (negative) (1,1)−form on M.

Since Chern classes can be computed with respect to any connection, we see that the first Chern class c 1(L) can be represented by i 2 π Ω . Notice that when L is (p.96) positive, the form i 2 π Ω defines a Kähler metric ω‎ on M with an integral cohomology class. This leads to

Definition 3.5.2: The pair (M,L) consisting of a complex manifold M with a positive line bundle L is called a polarized Kähler manifold.

Of course, the complex manifold M is necessarily a manifold of Kähler type, and the Kähler structure is that determined by the curvature of L, namely, ω = i 2 π Ω . So we often refer to the pair (M,[ω‎]) as a polarized Käahler manifold. If M is a Kähler manifold with an integral Kähler class [ω‎] or equivalently a positive line bundle L, we also say that M admits a polarization or is polarized by [ω‎] or L.

There is a complex line bundle canonically associated with every complex manifold, called the canonical bundle.

Definition 3.5.3 Let M be a complex manifold of complex dimension n. The n th exterior power Λ‎nT*(1,0)M = Λ‎n,0M is a holomorphic line bundle, called the canonical bundle and denoted by k M. Its dual or inverse line bundle K M 1 is called the anticanonical bundle.

When the underlying manifold M is understood we often write just k for k M. It is easy to see that

Proposition 3.5.4: The first Chern class of k M satisfies

c 1 ( K M ) = c 1 ( M ) .

There are important discrete invariants associated to k M and, hence, to M. On any complex manifold we define the plurigenera by

(3.5.4)
P m ( M ) = dim H 0 ( M , O ( K m ) ) = h 0 ( M , O ( K m ) ) .
Recall that P 1(M) is called the geometric genus and is usually denoted by p g. Note that by the well−known Dolbeault Theorem 3.4.8,p g = hn,0. Another important invariant is the holomorphic Euler characteristic χ‎(M,𝒪) = ∑ih i(M,𝒪).

Consider the commutative ring

(3.5.5)
R ( M ) = m 1 H 0 ( M , O ( K m ) )
. R(M), called the canonical ring, has finite transcendence degree tr(R(M)) over C.

Definition 3.5.5: Let M be a compact complex manifold. The Kodaira dimension of M is defined by

Kod ( M ) = { i f R ( M ) = tr ( R ( M ) ) 1 o t h e r w i s e ,
where tr(R) denotes the transcendence degree of the ring R.

More generally one can define (cf. [Laz04a]) the Iataka dimension κ‎(M,l) of any line bundle l by simply replacing k by l in Equation (3.5.5). Then Kod(M) = κ‎(M,k). The Iataka dimension and, therefore, the Kodaira dimension can be defined on any normal projective algebraic variety. We shall make use of the Iataka dimension when discussing the “orbifold Kodaira dimension″ in Chapter 4. Recall that the transcendence degree of the field of meromorphic functions on a compact complex manifold is called the algebraic dimension and is denoted by a(M). Regarding the Kodaira dimension we have Kod(M)≤ a(M)≤ N = dim (M). The Kodaira dimension gives a measure of the asymptotic growth of the plurigenera.

(p.97) Theorem 3.5.6: Let M be compact complex manifold. Then

  1. (i) Kod (M) =-∞ if and only if P m(M = 0 for all m;

  2. (ii) Kod (M) = 0 if and only if Pm(M) = 0 or 1, but not 0 for all m;

  3. (iii) Kod(M) = k if and only if there exist constants α‎, β‎ such tha α‎mk ≥ Pm(M) ≥ β‎mk.

The importance of the Kodaira dimension is that it is a birational invariant. Recall [Har77] that a rational map (it is not a map in the usual sense) is defined as follows: Let X,Y be varieties (either both affine or both projective) and consider pairs (U,φ‎U), where U is a non−empty open subset of X, and φ‎U:U→ Y is a holomorphic map. Two such pairs (U,φ‎ U) and (V,φ‎V) are equivalent if φ‎U|UV = φ‎V|UV. Then a rational “map″ φ‎:X→ Y is an equivalence class of such pairs. φ‎ is said to be dominant of if the image φ‎U is dense in Y. A birational map is a rational map that admits an inverse, namely, a rational map ψ‎:Y→ X such that ψ‎^ φ‎ = 1 X and φ‎^ ψ‎ = 1 Y. For a thorough treatment of rational maps we recommend the recent book [Har92].

The most important example of a rational map is the “blowing−up map″ which we now describe.

EXAMPLE 3.5.7: Blowing−up. Let B r(0) be a ball of radius r centered at 0 in C n, where n≥ 2, and let z = (z1,…,zn) be the standard coordinates in C n. Then blowing−up B r(0) at 0 is the complex manifold defined by

B ˜ = { ( z,y ) B × n 1 | z i y j = z j y i , 1 i < j n } ,
where y = (y 1,…,y n) are homogeneous coordinates for C P n-1. There is a natural surjective holomorphic map π‎:B→ B defined as the restriction to B of the projection map B× C P n-1 onto the first factor. For z≠ 0 the fiber π‎-1(z) is the single point (z,[z]); whereas, at z = 0 we have π‎-1(0)≈C P n-1. So B is B with the origin 0 replaced by a projective space C P n-1. The fiber π‎-1(0) = C P n-1 is called the exceptional divisor, and often denoted by E. Away from the exceptional divisor π‎ is a biholomorphism. Replacing by B is called blowing−down. One can show that B is diffeomorphic to B#̄C P n, where # denotes the connected sum operation and ̄C P n is C P n with the reverse orientation.

It is easy to transfer the blowing−up process to an arbitrary complex manifold M. Let pM and V a neighborhood containing p that is biholomorphic to B with p mapping to 0. Identifying V with B, we let M̃ be the manifold obtained from M by replacing V = B by B. So M̃ is a complex manifold that is diffeomorphic to M#̄C P n. Analytically, π‎ is a birational map, so we have the following birational invariants: a(M̃) = a(M), ,P m() = P m(M), and Kod(M̃) = Kod(M). We also have an isomorphism H i(M,o M)≈ H i(M̃, 𝒪,) for all i≥ 0, and k = π‎*k Mo((n-1)E), where 𝒪(D) is the line bundle associated to the divisor D as described in subsection 3.5.2 below.

Blowing−up can be applied to singular points of algebraic varieties to obtain smooth manifolds. This procedure has led to the celebrated Hironaka Resolution of Singularities Theorem which says that a singular algebraic variety can be desingularized after a finite sequence of blowing−ups.

Now for certain complex manifolds there is a “vanishing theorem″ H q(M,𝒪) = 0 for all q > 0. When this happens we get an isomorphism between the Picard group H 1(M,o*) of holomorphic line bundles on M and the topological invariant H 2(M,Z). Such vanishing theorems are of great importance in complex geometry. (p.98) Here we only give the well−known Kodaira–Nakano vanishing Theorem, and we refer to Chapter 4 of [Laz04a] and [SS85] for further development.

Theorem 3.5.8: Let M be a compact complex manifold of complex dimension n, and let l be a holomorphic line bundle on M.

  1. (i) If l⊗ k -1 is positive, then H q(M,𝒪(l)) = 0 for all q>0. In particular, if the anticanonical bundle k -1 is positive, H q(M,o) = 0 for all q0.

  2. (i) If ℒ is negative, then H q(M,ω‎p(l)) = 0 for all p+qn.

Compact manifolds with positive anticanonical bundles are called Fano manifolds, and it follows immediately from the long exact sequence (3.5.3) and (i) of Theorem 3.5.8 that

Corollary For any Fano manifold M there are isomorphisms Pic(M)≈ NS(M)≈ H 2(M,Z). So for Fano manifolds ρ‎(M) = b 2(M).

EXAMPLE 3.5.10:A simple example of a Fano manifold is the complex projective space C P n as discussed in Examples 3.1.12 and 3.2.8. It follows from Equation (3.2.2) and Proposition 3.6.1 that c 1(C P n) is positive. It is known that H 2(C P n,Z) = Z, and it follows from Proposition 3.5.4 and Theorem 3.5.8 that Pic(C P n)≈ NS(C P n)≈ Z. The positive generator of Pic(C P n) is known as the hyperplane bundle and denoted by H or as we do using the invertible sheaf notation 𝒪(1). So every holomorphic line bundle on C P n is 𝒪(n) for some n∈Z. The line bundle 𝒪(-1) is called the tautological bundle since its total space is the C n+1 from which C P n is constructed. It is easy to compute the canonical bundle of C P n finding k C P n = 𝒪(-n-1).

EXAMPLE 3.5.11: Low−dimensional Fano manifolds. The smooth Fano varieties have been classified for N = 1,2,3. There is a unique 1−dimensional smooth Fano variety up to isomorphism; it is the complex projective plane C P 1. For n = 2 they are classical and known as del Pezzo surfaces. A smooth del Pezzo surface is, up to isomorphism, one of the following:

  1. (i) C P 2,

  2. (ii) a smooth quadric Q = C P 1×C P 1,

  3. (iii) a double cover F 1 of a quadric cone Q C P 3 ramified along a smooth curve of degree 6 not passing through the vertex of the cone,

  4. (iv) a double cover F 2 of C P 2 ramified along a smooth curve of degree 4,

  5. (v) a surface F d⊂C P d, d = K F d 2 where 3≤ d≤7,

  6. (vi) a geometrically ruled surface F 1 with the exceptional section s, s 2 = -1.

One can show that surfaces F d, d = 1,…,7 can be obtained by blowing−up of 9-d points on C P 2 which are in general position, i.e., no two of these points lie on a line nor any three lie on a conic. Also, F 1 can be obtained by blowing−up C P 2 at 1 point. Thus, as a smooth manifold a del Pezzo surface must be diffeomorphic to 2 , Q = 1 × 1 , F 1 = 2 # ¯ 2 , or F d = ¯ 2 # ( 9 d ) ¯ 2 , where 1≤ d≤ 7.

The classification for N = 3 was begun by Fano and almost completed by Iskovskikh in [Isk77,Isk78,Isk79]. However, an additional 3−fold was found a bit later by Mukai and Umemura [MU83], and it was shown by Prokhorov [Pro90] to complete the classification. We refer to [Ip99] for a survey including the complete list of Fano 3−folds. Generally, Fano manifolds and orbifolds are of much importance for us in this book. For recent treatments of Fano varieties see [Kol96,IP99].

(p.99) More generally, any generalized flag manifold G/P is Fano. Fano manifolds have the important property that they can always be embedded into a complex projective space C P N for some N, i.e., they are examples of projective algebraic varieties as discussed in Example3.1.18. The beautiful Kodaira Embedding Theorem gives precise conditions to have a projective algebraic variety.

Theorem 3.5.12: Let M be a compact complex manifold and L a holomorphic line bundle on M. Then L is positive if and only if there is a holomorphic embedding φ‎: M → C P N for some N such that φ‎* o(1) = L m for some m > 0.

The condition of the holomorphic embedding gives rise to

Definition 3.5.13: A holomorphic line bundle L on a complex manifold M is said to be ample if there is an embedding φ‎: M → C P N for some N such that φ‎* o (1) = Lm for some m > 0. If we can take m = 1 then L is said to be very ample.

The Kodaira Embedding Theorem can now be reformulated in two more equivalent ways.

Theorem 3.5.14: Let M be a compact complex manifold. Then

  1. (i) A holomorphic line bundle L on M is positive if and only if it is ample in which case M is a projective algebraic variety.

  2. (ii) M admits polarization if and only if it is projective algebraic.

So a Fano manifold could be defined by the condition that the anticanonical line bundle K M 1 is ample. In this case the corresponding embedding is said to be an anticanonical embedding. Similarly, if the canonical line bundle k M is ample, then the manifold M is projective algebraic and the corresponding embedding is called a canonical embedding.

A very important criterion for the ampleness of a line bundle was discovered by Nakai in the case of complex surfaces and generalized to arbitrary smooth algebraic varieties by Moishezon. It was generalized further to complete schemes by Kleiman. Here we simply state the version for smooth algebraic varieties and refer to Lazarsfeld′s recent book [Laz04a] for further discussion and proof of the scheme theoretic version.

Theorem 3.5.15: Let M be a compact manifold of complex dimension n, and L a holomorphic line bundle on M. Then L is ample if and only if

V c 1 ( L ) k > 0
for every irreducible subvariety VM of dimension k.

In the case of surfaces Nakai′s criterion can be stated as L is ample if and only if c 1 2 ( L ) > 0 and

D c 1 ( L ) > 0
for every effective divisor D on M.

If the Kähler class [ω‎] is a rational class, i.e., it lies in H 2 (M,Q), then k[ω‎] is an integral class for some positive integer k. So k[ω‎] defines a positive holomorphic line bundle L on M, and Theorem 3.5.14 implies that (M,ω‎) is a projective algebraic variety. A Kähler manifold (M,ω‎) with [ω‎] ⊂ H 2(M,Q) is called a Hodge manifold.

(p.100) EXAMPLE 3.5.16; A complex torus is the quotient manifold Tn = C n/Λ‎, where Λ‎ is a lattice, i.e., a discrete subgroup of C n of rank 2n. A flat Kähler structure on C n induces a Kähler structure ω‎ on Tn. However, for a generic complex structure, the Kähler class [ω‎] ⊂ H 1,1 (M,C)∩ H 2(M,R) is not a rational class. So generally a complex torus Tn is not a projective algebraic variety. When one can find a compatible Kähler form ω‎ such that [ω‎] is a rational class, Tn is projective algebraic by the Kodaira Embedding Theorem 3.5.12 in which case it is called an Abelian variety. Alternatively, a complex torus Tn which admits a positive line bundle L is an Abelian variety. So the pair (Tn,L) is called a polarized Abelian variety. For a complete treatment of Abelian varieties we refer the reader to [LB92].

3.5.2. Divisors

There are two equivalent way to describe divisors on smooth complex manifolds. Since they are not equivalent for singular algebraic varieties or more generally for complex spaces, we discuss both of these here. The singular case is treated in Section 4.4 below.

Definition 3.5.17: A Weil divisor D on a complex manifold M is a locally finite formal linear combination of irreducible analytic hypersurfaces V i

D = i a i V i , a i ,
where locally finite means that every point p ⊂ M has a neighborhood intersecting only finitely many of the V is. D is said to be effective if a i0 for all i with not all a i equal to zero.

Under the formal sum operation Weil divisors form a group, called the divisor group and denoted by Div(M). Now from Definition 3.1.19 a divisor is described locally by the zero set of holomorphic functions.

Recall that a meromorphic function on an open set UM is a ratio f/g of relatively prime holomorphic f,g on U. We let m denote the sheaf of holomorphic functions and * the subsheaf of not identically zero meromorphic functions. Consider the short exact sequence

(3.5.6)
0 O / O 0.
We have

Definition 3.5.18: Let X be a complex manifold or an algebraic variety. Elements of the group

H 0 ( X , / O )
are called Cartier divisors on X. A Cartier divisor is principal if it is the divisor of a global meromorphic function, i.e., it is in the image of the natural quotient map H 0(X,m*) → H0(X,m*/o*).

Since on a smooth manifold the local rings ox are unique factorization domains (UFD) Weil divisors and Cartier divisors coincide.

Theorem 3.5.19: Let M be a smooth complex manifold. Then there is an isomorphism

Div ( M ) H 0 ( M , / O ) .
This isomorphism does not hold on singular complex spaces. For example the complex spaces discussed in Section 4.4.1 below are not “locally factorial.″ On smooth complex manifolds we shall often identify Weil divisors and Cartier divisors (p.101) by just referring to a divisor. Two divisors D and D′ on M are said to be linearly equivalent, written D′D, if D′ = D +(f), where (f) denotes the divisor defined by the global meromorphic function f as follows: write f locally as f = g/h as (f) = ord(g)Zg–ord(h)Zh. Here Zg denotes the zero set of the holomorphic function G and ord(g) denotes its order of vanishing. We denote by |D| the set of all divisors on M that are linearly equivalent to D. It is called the linear system of divisors defined by D. The common intersection ∩D′∈|D| D′ is called the base locus of linear system |D|.

We now describe the relationship between line bundles and divisors. From the short exact sequence (3.5.6) one has

(3.5.7)
0 H 0 ( M , / O ) / H 0 ( M , ) H 1 ( M , O ) H 1 ( M , ) .
This says that every divisor D on M determines a holomorphic line bundle o(D), and the line bundle o(D) is holomorphically trivial if and only if D is the divisor of a meromorphic function. The quotient H0(M,*/o*)/H 0(M,*) is called the Cartier divisor class group and is denoted by CaCl(M). Furthermore, H1(M,*) = 0 if and only if every holomorphic line bundle on M has a global meromorphic section. In this case we get an isomorphism between the divisor class group and the Picard group. This happens, for example, for smooth projective algebraic varieties, i.e.,

Proposition 3.5.20: Let X be a smooth projective algebraic variety. Then Cl(X)≈ Pic(X).

We now briefly describe the intersection theory for divisors. The general theory is laid out in [Ful84], but our treatment follows closely the abbreviated version in [Laz04a]. Here we work in the category of varieties (either affine or projective). Let X be an irreducible variety. To each Cartier divisor DH 0(X,*/o*) we associate a line bundle o(D), and to this line bundle we can associate a Chern class c1(o(D)) ⊂ H 2(X,Z). Now let C⊂X be an irreducible curve, i.e., a 1−dimensional irreducible complex subvariety of X, and denote by [C] its homology class in H2(X,Z). Then

Definition 3.5.21: The intersection numberC ⊂ Z of a Cartier divisor D with an irreducible curve C is defined by

D C = C D = < c 1 ( O ( D ) ) , [ C ] > ,
where <ċ,…> denotes the Kronecker pairing.

This definition can easily be generalized as follows: let V be a k−dimensional subvariety of X, and let D 1,…, D k be Cartier divisors on X. Then we define the intersection number by

(3.5.8)
D 1 D k V = < c 1 ( O ( D 1 ) ) c 1 ( O ( D k ) ) , [ V ] >
Definition 3.5.22: We say that two Cartier divisors D,D' are numerically equivalent, denoted by D'≡ n D if D'· C = D· C for all irreducible curves C. A divisor D is numerically trivial if it is numerically equivalent to 0. We denote by Num(X) the subgroup of H 0(X,*/o*) consisting of numerically trivial Cartier divisors.

(p.102) We now have another characterization of the Neron–Severi group, namely as the quotient group of numerical classes of Cartier divisors,

N S ( X ) = H 0 ( X , / O ) / Num ( X ) .
We shall also make use of the following terminology [Laz04a].

Definition 3.5.23: Let X be a complete variety of dimension n. A (Cartier) divisor D is said to be numerically effective or nef if for every irreducible curve C we have D· C≥ 0. The divisor D is big if the Iataka dimension κ‎(X,𝒪(D)) = n.

An important criterion for the bigness of a nef divisor is: a nef divisor D is big if and only if Dn>0.

3.6. The Calabi Conjecture and the Calabi–Yau Theorem

The Ricci form ρ‎ ω‎ defined in Proposition 3.2.6 plays a special role in Kähler geometry. Indeed, from Equation(3.2.1) and Proposition/Definition(3.4.2) we have

Proposition 3.6.1: Let (M,J,g,ω‎g) be a Kähler manifold and ρ‎ω‎ the Ricci form of g. Then

[ ρ ω ] = 2 π c 1 ( M ) H D R 2 ( M ) H 1 , 1 ( M , ) .
This proposition says that on a Kähler manifold the Ricci curvature 2−form ρ‎ω‎ of any Kähler metric represents the cohomology class 2π‎ c1(M). The well−known Calabi Conjecture [Cal56,Cal57] is the question whether or not the converse is also true, i.e., the assertion that on a compact complex manifold any real closed (1,1)−form that represents 2π‎ c1(M) is the Ricci form of a unique Kähler metric G on M. This conjecture was proven in the celebrated work of Yau which was announced in [Yau77] and the details of which appeared in [Yau78]. (Uniqueness was actually proved by Calabi [Cal57] before the conjecture was settled). By now there are several excellent treatments in book form [Joy00,Tia00,Aub82,Siu87,Aub98]. While these proofs all assume one is working with compact Kähler manifolds, they readily extend to the case of compact Kähler orbifolds since the computations are local in nature and one needs only to consider local functions and sections on the local uniformizing neighborhoods that are invariant under the local uniformizing groups

To be more specific we begin with a couple of definitions

Definition 3.6.2: Let (M,J,g,ω‎g) be a compact Kähler manifold. The Kähler cone of M

K ( M ) = { [ ω ] H 1 , 1 ( M , ) H 2 ( M , ) | ω = ω h  for some kahler metric  h } ,
i.e., it is the set of all possible Kähler classes on M.

It is easy to show that K(M) is a convex open set in H1,1(M,C)∩ H 2(M,R). Of particular interest to us is the Kähler; lattice KL(M) which is the intersection of K(M) with the Neron–Severi group, i.e., KL(M) = K(M)∩ NS(M). Beyond surfaces, not much is known about the Kähler cone. However, recent progress has been made by Demailly and Paun [DP04] who characterize K(M) as a connected component of the set of (1,1) cohomology classes which are numerically positive on all analytic cycles. This generalizes the well−known Nakai–Moishezon criterion for ample line bundles.

(p.103) Definition 3.6.3: Let (M,J,g,ω‎g) be a compact Kähler manifold and K(M) its Kähler cone. For any fixed Kähler class [ω‎] ⊂ K(M) we define

K [ ω ] = { h Γ ( 2 ( T M ) ) | h  is a Kahler metric and  [ ω ] = [ ω h ] }
to be the space of all Kähler metrics in a given cohomology class.

The Global i∂∂̄−Lemma(3.1.15) provides for a simple description of the space of Kähler metrics K[ω‎]. Suppose we have a Kähler metric G with Kähler class [ω‎g] = [ω‎] ⊂ K(M). If h ⊂ K[ω‎] is another Kähler metric then, up to a constant, there exists a global function φ‎ ∈ C (M,R) such that ω‎h -ω‎g = i∂∂̄φ‎. We can fix the constant by requiring, for example, that ∫Mφ‎ dvolg = 0. Hence, we have

Coroary 3.6.4: Let (M,J,g,ω‎g) be a compact Kähler manifold with [ω‎ g] = [ω‎] ∈ K(M). Then, relative to the metric G the space of all Kähler metric is K [ω‎] can be described as

K [ ω ] = { φ C ( M , ) | ω h = ω g + i ̄ φ > 0 , M φ d vol g = 0 } ,
where the 2−form ω‎h>0 means that ω‎h(X,JY) is a Hermitian metric on M.

The following theorem is the statement of the famous conjecture made by Calabi and proved by Yau in 1977–78 [Yau77,Yau78].

Theorem 3.6.5: Let (M,J,g,ω‎g) be a compact Kähler manifold. Then any real (1,1)−form ρ‎ on M which represents the cohomology class 2π‎c 1(M) is the Ricci form of a unique Kähler metric h such that [ω‎h] = [ω‎g].

Let us reformulate the problem using the Global i∂∂̄−Lemma. We start with a given Kähler metric G on M in Kähler class [ω‎ g] = [ω‎]. Since ρ‎ g also represents 2π‎ c 1(M) there exists a globally defined function f ∈ C (M,R) such that

ρ g ρ = i ̄ f .
Appropriately, f is called a discrepancy potential function for the Calabi problem and again we can fix the constant by asking that ∫M(ef-1)dvolg = 0.

Now suppose that the desired solution of the problem is a metric hK [ω‎]. We know that the Kähler form of h can be written as

ω h = ω g + i ̄ Φ ,
for some smooth function φ‎ ∈ C(M,R). We normalize φ‎ as in previous corollary. Combining these two equations we see that
ρ h ρ g = i ̄ f .
If we define a smooth function F ∈ C (M,R) relating the volume forms of the two metrics dvolh = eFdvolg then the left−hand side of the above equation takes the following form
i ̄ F = ρ h ρ g = i ̄ φ ,
and, hence, simply i∂∂̄(F-f) = 0. Hence, F = f+c. But since we normalized ∫M(ef-1)dvolg = 0 we must have c = 0. Hence, F = f, or dvolh = efdvolg. We can now give two more equivalent formulations of the Calabi Problem.

Theorem 3.6.6: Let (M,J,g,ω‎g) be a compact Kähler manifold, [ω‎g] = ω‎ ⊂ K(M) the corresponding Kähler class and ρ‎g the Ricci form. Consider any positive (1,1)−form Ω‎ on M such that [Ω‎] = 2π‎c1(M). Let ρ‎ g-Ω‎ = i∂∂̄ f, with ∫ M(ef-1)dvolg = 0.

  1. (i) There exists a unique Kähler metric h ⊂ K[ω‎] whose volume form satisfies dvolh = efdvolg.

  2. (p.104)
  3. (ii) Let (U;z1,…,zn) be a local complex chart on M with respect to which the metric g = (gīj). Then, up to a constant, there exists a unique smooth function φ‎ in K[ω‎], which satisfies the following equation

    det ( g i j ̄ + 2 φ z i z ̄ j ) det ( g i j ̄ ) = e f .

The equation in (ii) is called the Monge–Ampere equation. Part (i) gives a simple geometric characterization of the Calabi–Yau theorem. On a compact Kähler manifold one can always find a metric with arbitrarily prescribed volume form. The uniqueness part of this theorem was already proved by Calabi. This part involves only the Maximum Principle. The existence proof uses the continuity method and it involves several difficult a priori estimates. These were found by Yau [Yau78] in 1978. For further discussion we refer the reader to the books mentioned previously. Here we only list some immediate consequences of this theorem:

Corollary 3.6.7: Let (M,J,g,ω‎g) be a compact Kähler manifold with c1(M) = 0. Then M admits a unique Kähler–Ricci flat metric.

Such a metric has holonomy group inside SU(n). Kähler manifolds with this property are called Calabi–Yau manifolds.

Corollary 3.6.8: Let (M,J,g,ω‎g) be a compact Kähler manifold with c 1(M)>0. Then M admits a Kähler metric of positive Ricci curvature.

We can combine this corollary with an older result of Kobayashi [Kob61] to give a beautiful proof of

Theorem 3.6.9: Any Fano manifold is simply connected.

PROOF . Let M be a Fano manifold. Then its anticanonical line bundle K M 1 is positive, so c 1 ( M ) = c 1 ( K M 1 ) > 0. Thus, by Corollary3.6.8 M has a Kähler metric of positive Ricci curvature. By Myers′ Theorem M has finite fundamental group. Let M̃ be the universal cover of M. It is compact, Fano and a finite, say d−fold, cover of M. By the Kodaira–Nakano Vanishing Theorem 3.5.8 Hq(M̃,o) = 0 for all q>0 which implies that the holomorphic Euler characteristic χ‎(M̃,o) = 1. On the other hand, χ‎(M̃,o) = dχ‎(M,o) = D implying that D = 1. So M is simply connected. □

We shall return to important consequences of Yau's Theorem in Chapter 5.

Notes:

(1) Schouten considered Kähler manifolds 4 years earlier in the article Über unitäre Geometrie but never got much credit for his work [Sch29,SD30]. Curiously, neither Schouten nor Kähler appeared very interested in their own inventions. The term Kähler manifolds became standard after the World War II in the late 1940s. Interestingly, Schouten and von Dantzig arrive at the Kähler condition while investigating parallel transport of the Levi-Civita connection associated to the a Hermitian metric.

(2) We refer the reader interested in the historical development of Kähler geometry from its birth to the volume devoted to the mathematical works of Kähler [Käh03] and to Jean-Pierre Bourguignon's excellent article in that volume.

(3) The factor -2 multiplying ω‎g is chosen so that the volume form satisfies ω g n = n ! d vol g .

(4) The general version of the Kodaira–Serre Duality Theorem is given in Theorem 3.4.9 below.