Kähler Manifolds
Kähler Manifolds
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 metrics^{1} 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 2form ρ_{ω} on a Kähler manifold (M,J,g,ω_{g}) 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
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 $\mathcal{A}={\left\{{U}_{\alpha},{\phi}_{\alpha}\right\}}_{\alpha \in \Omega}$ of complex charts whose transition functions
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 TM⊗_{R} C by Clinearity. Then J induces a splitting TM⊗_{R} C = T ^{1,0} M⊕ T ^{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 $X\mapsto {\scriptscriptstyle \frac{1}{2}}\left(XiJX\right)$.
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 TM⊗_{R} 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
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 torsionfree 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 _{1}→ M _{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*M⊗_{r} C which then splits as ${T}^{\ast \left(1,0\right)}M\oplus {T}^{\ast \left(0,1\right)}M$. This in turn defines the splitting of the bundle of complex pforms on M
Proposition 3.1.5: Let (M,J) be an almost complex manifold. The following conditions are equivalent:

(i) [V,W]∈ Γ(T^{1,0}M) for all V,W∈ Γ(T^{1,0}M),

(ii) [V,W]∈Γ(T^{0,1}M) for all V,W∈ Γ(T^{0,1}M),

(iii) $\begin{array}{l}Im\text{}(d){}_{\Gamma ({\wedge}^{1,0}M)}\subset \Gamma ({\wedge}^{2,0}M),\oplus \Gamma ({\wedge}^{1,1}M),\\ Im\text{}(d){}_{\Gamma ({\wedge}^{0,1}M)}\subset \Gamma ({\wedge}^{0,2}M)\oplus \Gamma ({\wedge}^{1,1}M),\end{array}$

(iv) $Im\text{}(d){}_{\Gamma ({\wedge}^{p,qM)}}\subset \Gamma ({\wedge}^{p+1,q}M)\oplus \Gamma ({\wedge}^{p,q+1}M),$,

(v) N_{J} ≡ 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
Thus, if (M,J) is a complex manifold, the composition of the exterior derivative acting on (k,l)forms with projections on Γ(Λ^{k+1,l} M) and Γ(Λ^{k,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 ε^{k,l} the sheaf of germs of smooth sections of the bundles Λ^{k,l} M, and use the same notation ∂,∂̄, etc. for these operators acting on local sections of ε^{k,l}. We also let Ώ^{p} denote the sheaf of germs of holomorphic sections of the bundle Λ^{p,0} M. Then, by the wellknown ∂̄Poincaré lemma [GR65], we have a resolution of this sheaf, namely
(p.78) Moreover, since the sheaves ε^{k,l} are fine sheaves, this resolution is acyclic. So if we define the Dolbeault cohomology groups [Dol53] of (M,J)
Theorem 3.1.6: Let M be a complex manifold. Then there is an isomorphism
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
Note that an almost complex manifold always admits a Hermitian metric. An almost Hermitian structure (J,g,ω_{g}) 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 TM⊗_{R} C satisfying

(i) $h(\overline{Z},\text{}\overline{W})\text{}=\text{}\overline{h(Z,W)}$ for all Z,W∈ Γ(TM⊗_{R} C),

(ii) h(Z, Z̄) > 0 for all Z≠ 0∈ (TM⊗_{R} C),

(iii) h(Z, W̄) for Z∈ Γ(T^{1,0}M) and W∈ Γ(T^{0,1}M).
(p.79) Definition 3.1.8: A Hermitian manifold (M,J,g,ω_{g}) is said to be Kähler if ω_{g} is a closed 2form. We call g a Kähler metric, ω_{g} its Kähler form, and the triple (J,g,ω_{g}) 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 [ω_{g}] defines a Dolbeault cohomology class in ${H}_{\stackrel{\u0304}{\partial}}^{1,1}\left(M\right)$. 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,ω_{g}) be an almost Hermitian manifold of real dimension 2n and let ω_{g} be the fundamental 2form associated to g. Let ∇ be the LeviCivita connection of g. Then the following conditions are equivalent:

(i) ∇ J = 0,

(ii) ∇ω_{g} = 0,

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

(iv) (M,J,g,ω_{g}) is 1integrable,

(v) (M,J,g,ω_{g}) 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) = δ_{i,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
(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 nonzero 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 ū_{i} be the open subspace of C ^{n+1} such that z _{i}≠ 0. Then C P ^{n} is covered by open sets ${\left\{{U}_{i}=\pi \left({\tilde{U}}_{i}\right)\right\}}_{i=0}^{n},$ together with homeomorphisms φ_{i}:U _{i}→C ^{n} = R ^{2n} defined by
Notice that by restricting the map π to the unit sphere in C ^{n+1} gives the wellknown Hopf fibration S ^{1}→S ^{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 pdimensional 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∪α^{c} = {α_{1},…,α_{p}}∪ {α_{p+1},…,α_{p}}, where both sets are ordered in increasing order. Let U _{α}⊂ G r _{p}(C ^{q+p}) be the subset of pplanes W for which z _{α} _{1} _{W},…,z _{α} _{p} _{W} are linearly independent. Since, for each W∈ U _{α}, the mappings z _{α} _{1} _{W},…,z _{α} _{p} _{W} form a basis for the dual space of W, we can write
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 socalled flag manifolds F l_{k} _{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 semisimple 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 wellknown 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 f∈ C ^{∞}(M). As the $i\partial \stackrel{\u0304}{\partial}f=\frac{1}{2}\left(\partial +\stackrel{\u0304}{\partial}\right)\left(\stackrel{\u0304}{\partial}\partial \right)f$ is real we conclude that i∂∂̄f is a real closed 2form of type (1,1). We have the following two lemmas
Lemma 3.1.14: [Local i∂∂̄Lemma] Let ρ ∈ Γ(Λ^{1,1} U) be a smooth, closed, real (1,1)−form on a unit disc $U={D}_{\u2102}^{n}\left(1\right)\subset {\u2102}^{n}$. Then there exists f ∈ C ^{∞}(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 [ω_{g2}] = [ω_{g2}] ∈ H ^{2}(M,R). Then
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 ω_{g} = 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,ω_{g}) be a compact Kähler manifold with the Kähler form ω_{g}. Then $\left[{\omega}_{g}^{k}\right]\in {H}^{2k}\left(M,\mathbb{R}\right)$ is non−trivial for all k = 0,…, n.
PROOF. This is a simple consequence of the Stokes′ Theorem and the fact that ${\omega}_{g}^{n}=n!d{\text{vol}}_{g}$. For let ${\omega}_{g}^{k}=d\alpha $ be exact. Then
Let (M,J be a complex manifold and let N be a submanifold of M. then, in particular at each point p ∈ N, the tangent space T_{p}N ⊂ T_{p}M is a vector subspace. We say that N is a complex submanifold of M T_{p}N is a complex subspace for each p ∈ N, i.e., J(T_{p}N) = T_{p}N. Now, if N is a complex submanifold then the restriction J_{N} of J to TN is a complex structure on N and the inclusion map (p.82) map ι: N ↪ M has the property
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
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

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

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

(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} andV_{2} with V _{i} ≠ V.
3.2. Curvature of Kähler Manifolds
Let (M,J,g,ω_{g}) be a Kähler manifold and ∇ the Levi−Civita connection. We can extend ∇ in a C−linear way to Γ(TM⊗_{R} C). In a local chart (U;z_{1},…, z_{n}) we have $\left\{\frac{\partial}{\partial {z}_{1}},\dots ,\frac{\partial}{\partial {z}_{n}}\right\}$ and $\left\{\frac{\partial}{\partial {\stackrel{\u0304}{z}}_{1}},\dots ,\frac{\partial}{\partial {\stackrel{\u0304}{z}}_{n}}\right\}$ as bases for T ^{1,0} M and T ^{0,1} M, respectively. (p.83) Define the Christoffel symbols ${\Gamma}_{AB}^{C}$ as follows
In a local chart (U;z_{1},…,z_{n}) we can define the following matrix−valued (1,0)−form
Proposition 3.2.2: Let (M,J,g,ω_{g}) be a Kähler manifold and let ∇ be the Levi−Civita connection of g extended to TM⊗ _{R} C≃ T ^{1,0} M⊕ T ^{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
If (M,J,g) is complex then one can extend all these curvature tensors by C−linearity to TM ⊗_{R} C. In addition, on a Kähler manifold we have ∇J = 0 so that
Definition 3.2.3: Let (M,J,g,ω_{g}) be a Kähler manifold and let R be the curvature tensor of (M,g). Extending R to TM ⊗_{R} C we define the following:

(i) The Riemann curvature tensor R as a section of ⊗^{4}(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
$$\left(X,Y,Z,W\right)\mapsto R\left(X,\stackrel{\u0304}{Y},Z\stackrel{\u0304}{W}\right),$$ 
(ii) The Riemann curvature tensor R_{X,Ȳ} as a section of End(T^{1,0}M), i.e., the map R_{X,Ȳ} : Γ(T ^{1,0} M)→Γ(T ^{0,1} M), X,Y ∈ Γ(T ^{1,0} M) given by
$${R}_{X,\stackrel{\u0304}{Y}}\left(Z\right)=R\left(X,\stackrel{\u0304}{Y}\right)\stackrel{\u0304}{Z},$$ 
(iii) A real (1,1)−form Ω with values in Γ(End(T^{1,0}M)), i.e., a skew−Hermitian map Ω : Γ(T ^{1,0} M) × Γ(T ^{0,1} M)→Γ(End(T ^{1,0} M))
$$\begin{array}{cc}\Omega \left(X,\stackrel{\u0304}{Y}\right)={R}_{X,\stackrel{\u0304}{Y}},& X,Y\in \Gamma \left({T}^{1,0}M\right)\end{array},$$
(p.85)

(iv) The Riemann curvature operator r as a section of ⊙^{2} Γ(Λ^{1,1} M), i.e., the map r : Γ(Λ^{1,1} M) × Γ(Λ^{1,1} M)→ C ^{∞}(M,C) given by
$$\mathcal{R}\left(X\wedge \stackrel{\u0304}{Y},V\wedge \stackrel{\u0304}{W}\right)=R\left(X,\stackrel{\u0304}{Y},Z,\stackrel{\u0304}{W}\right).$$
Proposition 3.2.4: Let (U; z _{1},…,z_{n}) be a local coordinate chart on a Kähler manifold (M,J,g,ω_{g}). Let R ∈ Γ (⊗^{4}(T ^{*1,0} M)) be the Riemann curvature tensor. Then
PROOF . It remains to do the local computations. We have
Definition 3.2.5: Let R_{X,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,ω_{g}). The Ricci curvature tensor is the map Ric_{ω} : Γ(T ^{1,0} M) ⊗ Γ(T ^{1,0})→C ^{∞}(M,C) defined by the trace
A priori it is not clear that Ric_{ω}(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},…,e_{n}} of ${T}_{p}^{1,0}M$ we have
(p.86) Proposition 3.2.6:; The Ricci curvature tensor Ric_{ω} : T ^{1,0} M ⊗ T ^{1,0} M → C is a Hermitian symmetric form on ${T}_{p}^{1,0}M$ at each p ∈ M. In a local chart (U; z _{1},…,z_{n}) this form can be written as
PROOF By definition we have
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
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 ⊂ T_{p}M spanned at point p ∈ M by X_{p} and Y_{p} as
Definition 3.2.7: Let (M,J,g,ω_{g}) be a Kähler manifold and let X,Y ∈ Γ(T ^{1,0} M). Then
(p.87) is called the bisectional curvature of g in the direction of X,Y while
If we write $X=\frac{1}{\sqrt{2}}\left(U+iJU\right),Y=\frac{1}{\sqrt{2}}\left(V+iJV\right)$ for some real vector fields U,V ∈ Γ(TM) we have
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
It now follows that in order to compute the full Riemann curvature tensor R_{ījk̄l} on C P ^{n} it is sufficient to compute it at one point [1,0,…,0]. Using local expressions for R_{ījk̄l} and g_{īj} one can easily see that at z = 0 we have R_{ījk̄l} = δ_{īj}δ_{k̄l} + δ_{īl}δ_{k̄j}. In particular, as g_{īj} _{z = 0}= δ_{īj}, at any other point
EXAMPLE 3.2.9: Bergman metric on the complex ball. Consider $M={B}_{\u2102}^{n}\left(1\right)=\{z\in {\u2102}^{n}{\leftz\right}^{2}<1\}$ and let
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,ω_{g}) 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 ${B}_{\u2102}^{n}$(1).
3.3. Hodge Theory on Kähler Manifolds
Let (M ^{2n},J,g,ω_{g}) be a compact Hermitian manifold. We define the Hodge star operator to be the map ✶ :Γ(Λ^{k,l} M)→ Γ(Λ^{nk,nl})M defined by
(p.89) The Laplacian Δ_{̄∂} is a formally self−adjoint operator and
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,ω_{g}) the dimensions dim (h ^{k,l}(M))∞ for all 0≤ k,l≤ n. Furthermore, we have the decompositions
Let us introduce the projections
To show this let us introduce the operator
Lemma 3.3.3: [Kähler identities] Let (M,J,g,ω_{g}) be a Kähler manifold. Then
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,ω_{g}) be a compact Kähler manifold. Then

(i) [L*,Δ_{d}] = [L*,Δ_{d}] = 0

(ii) Δ_{d} = 2Δ_{∂} = 2Δ_{̄}∂

(iii) [Δ_{d},π^{p,q}] = 0.
(p.90) PROOF. We have
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
Theorem 3.3.5: For a compact Kähler manifold we have
Since [⋆,Δ_{̄∂}] = 0, by Kodaira–Serre duality^{4} the map
Corollary 3.3.6: Let (M ^{2n},J,g,ω_{g}) be a compact Kähler manifold, h ^{k,l}(M) be the Hodge numbers, and b _{r}(M) = ∑_{k+l = r}h^{k,l}(M) the Betti numbers of M. Then for all 0≤ k,l≤ N we have

(i) h ^{k,l}(M) < ∞,

(ii) h ^{k,k}(M)≥1 and h ^{n,n} = h ^{0,0} = 1,

(iii) h ^{k,l}(M) = h ^{l,k}(M) = h ^{nk,nl}(M),

(iv) b _{2r+1}(M) are even and ${h}^{1,0}\left(M\right)={h}^{0,1}\left(M\right)=\frac{1}{2}{b}_{1}\left(M\right)$ 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
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 $h={\sum}_{r=0}^{2n}(nr){\pi}^{r}.$ Show that we have the following relations
Definition 3.3.7: Let (M,J,g,ω_{g}) be a compact Kähler manifold. We define the primitive cohomology groups of M as kernels of the L*−operator, i.e.,
Since {L,L*,h} all commute with the Kähler Laplacian we have
Theorem 3.3.8: (Hard Lefschetz) On a compact Kähler manifold M the map L^{k}:H^{nk}(M)→ H^{n+k}(M) is an isomorphism for 1 ≤ k ≤ n. Furthermore,
(p.92) 3.4. Complex Vector Bundles and Chern Classes
Let (M,J) be a smooth manifold and let E→M 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
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
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 $\left[{c}_{i}\left(E,\nabla \right)\right]\in {H}_{DR}^{2i}\left(M\right)$ 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 $c\left(E\right)={\displaystyle {\sum}_{i=0}^{\infty}{c}_{i}}\left(E\right)$, 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(E⊕ E') = c(E)c(E'). If ̄E denotes the complex conjugate bundle to E, then c_{i}(̄E) = (1)^{i}c_{i}(E) and since ̄E is isomorphic to the dual complex vector bundle c_{i}(E*) = (1)^{i}c_{i}(E). There are also certain “multiplicative sequences″ of interest [Hir66,MS74]. We define the Chern roots x_{i} of a rank k complex vector bundle by $c\left(E\right)=1+{c}_{1}\left(E\right)+\cdot \cdot \cdot +{c}_{k}\left(E\right)={\displaystyle {\prod}_{i=1}^{k}\left(1+{x}_{i}\right)}$. Then the Chern character ch(E) and the Todd class Td(E) are defined by
Theorem 3.4.3: Let E be a holomorphic vector bundle on a compact complex manifold M ^{n}. Then
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} M⊕ T ^{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, c_{i}(M) = c_{i}(T ^{1,0} M), called the i^{th} Chern class of the complex manifold M, and denoted simply by c_{i} 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 c_{n}(M) = e(M) is the Euler class. Given any partition I = i_{1},…,i_{r} 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., c_{I}[M] = < c_{i} _{1}⋯ c_{i} _{r},[M]>. Note that the top Chern number c_{n}[M] is just the Euler–Poincar′e characteristic c_{n}[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,$\left(M,\mathcal{O}\right)=1q+{p}_{g}=\frac{1}{12}\left({c}_{1}^{2}\left[M\right]+{c}_{2}\left[M\right]\right)$
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⊗ Λ^{k,l} M, and we let e^{k,l}(E) denote the sheaf of germs of smooth sections of E⊗ Λ^{k,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
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 connection ∇ in E such that ∇^{0,1} = ∂̄.
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 E_{x} of E that varies smoothly with x. A connection ∇ in E is called an Hermitian connection if ∇ h = 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 connection ∇ such that ∇^{0,1} = ∂̄.
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
Now consider the cochain complex
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 Ω^{p}(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
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
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
To study the holomorphic line bundles on M we consider the exact sequence
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 $\frac{i}{2\pi}\Omega $. Notice that when L is (p.96) positive, the form $\frac{i}{2\pi}\Omega $ 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, $\omega =\frac{i}{2\pi}\Omega $. 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 Λ^{n}T*(1,0)M = Λ^{n},0M is a holomorphic line bundle, called the canonical bundle and denoted by k _{M}. Its dual or inverse line bundle ${\mathcal{K}}_{M}^{1}$ 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
There are important discrete invariants associated to k _{M} and, hence, to M. On any complex manifold we define the plurigenera by
Consider the commutative ring
Definition 3.5.5: Let M be a compact complex manifold. The Kodaira dimension of M is defined by
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

(i) Kod (M) =∞ if and only if P _{m}(M = 0 for all m;

(ii) Kod (M) = 0 if and only if P_{m}(M) = 0 or 1, but not 0 for all m;

(iii) Kod(M) = k if and only if there exist constants α, β such tha αm^{k} ≥ P_{m}(M) ≥ βm^{k}.
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,φ_{U}), where U is a non−empty open subset of X, and φ_{U}:U→ Y is a holomorphic map. Two such pairs (U,_{φ} _{U}) and (V,φ_{V}) are equivalent if φ_{U}_{U}∩ V = φ_{V}_{U}∩ V. Then a rational “map″ φ:X→ Y is an equivalence class of such pairs. φ is said to be dominant of if the image φ_{U} 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 = (z_{1},…,z_{n}) be the standard coordinates in C ^{n}. Then blowing−up B _{r}(0) at 0 is the complex manifold defined by
It is easy to transfer the blowing−up process to an arbitrary complex manifold M. Let p ∈ M 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}(M̃) = P _{m}(M), and Kod(M̃) = Kod(M). We also have an isomorphism H ^{i}(M,o _{M})≈ H ^{i}(M̃, 𝒪_{M̃},) for all i≥ 0, and k _{M̃} = π*k _{M}⊗ o((n1)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.

(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.

(i) If ℒ is negative, then H ^{q}(M,ω^{p}(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(n1).
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:

(i) C P ^{2},

(ii) a smooth quadric Q = C P ^{1}×C P ^{1},

(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,

(iv) a double cover F _{2} of C P ^{2} ramified along a smooth curve of degree 4,

(v) a surface F _{d}⊂C P ^{d}, $d={K}_{{F}_{d}}^{2}$ where 3≤ d≤7,

(vi) a geometrically ruled surface F _{1} with the exceptional section s, s ^{2} = 1.
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) = L^{m} 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

(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.

(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 ${\mathcal{K}}_{M}^{1}$ 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
In the case of surfaces Nakai′s criterion can be stated as L is ample if and only if ${c}_{1}^{2}\left(L\right)>0$ and
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 T^{n} = 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 T^{n}. 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 T^{n} is not a projective algebraic variety. When one can find a compatible Kähler form ω such that [ω] is a rational class, T^{n} is projective algebraic by the Kodaira Embedding Theorem 3.5.12 in which case it is called an Abelian variety. Alternatively, a complex torus T^{n} which admits a positive line bundle L is an Abelian variety. So the pair (T^{n},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}
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 U ⊂ M is a ratio f/g of relatively prime holomorphic f,g on U. We let m denote the sheaf of holomorphic functions and ℳ* the subsheaf of not identically zero meromorphic functions. Consider the short exact sequence
Definition 3.5.18: Let X be a complex manifold or an algebraic variety. Elements of the group
Since on a smooth manifold the local rings o_{x} 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
We now describe the relationship between line bundles and divisors. From the short exact sequence (3.5.6) one has
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 D ∈ H ^{0}(X,ℳ*/o*) we associate a line bundle o(D), and to this line bundle we can associate a Chern class c_{1}(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 H_{2}(X,Z). Then
Definition 3.5.21: The intersection number Dċ C ⊂ Z of a Cartier divisor D with an irreducible curve C is defined by
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
(p.102) We now have another characterization of the Neron–Severi group, namely as the quotient group of numerical classes of Cartier divisors,
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 D^{n}>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},ω_{g}) be a Kähler manifold and ρ_{ω} the Ricci form of g. Then
To be more specific we begin with a couple of definitions
Definition 3.6.2: Let (M,J,_{g},ω_{g}) be a compact Kähler manifold. The Kähler cone of M
It is easy to show that K(M) is a convex open set in H^{1},1(M,C)∩ H ^{2}(M,R). Of particular interest to us is the Kähler; lattice K_{L}(M) which is the intersection of K(M) with the Neron–Severi group, i.e., K_{L}(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},ω_{g}) be a compact Kähler manifold and K(M) its Kähler cone. For any fixed Kähler class [ω] ⊂ K(M) we define
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 ω_{h} ω_{g} = i∂∂̄φ. We can fix the constant by requiring, for example, that ∫_{M}φ dvol_{g} = 0. Hence, we have
Coroary 3.6.4: Let (M,J,_{g},ω_{g}) 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
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},ω_{g}) 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 [ω_{h}] = [ω_{g}].
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
Now suppose that the desired solution of the problem is a metric h ∈ K _{[ω]}. We know that the Kähler form of h can be written as
Theorem 3.6.6: Let (M,J,_{g},ω_{g}) be a compact Kähler manifold, [ω_{g}] = ω ⊂ K(M) the corresponding Kähler class and ρ_{g} the Ricci form. Consider any positive (1,1)−form Ω on M such that [Ω] = 2πc_{1}(M). Let ρ _{g}Ω = i∂∂̄ f, with ∫ _{M}(e^{f}1)dvol_{g} = 0.

(i) There exists a unique Kähler metric h ⊂ K_{[ω]} whose volume form satisfies dvol_{h} = e^{f}dvol_{g}.
(p.104)

(ii) Let (U;z_{1},…,z_{n}) be a local complex chart on M with respect to which the metric g = (g_{īj}). Then, up to a constant, there exists a unique smooth function φ in K_{[ω]}, which satisfies the following equation
$$\frac{\text{det}\left({g}_{i\stackrel{\u0304}{j}}+\frac{{\partial}^{2}\phi}{\partial {z}_{i}\partial {\stackrel{\u0304}{z}}_{j}}\right)}{\text{det}\left({g}_{i\stackrel{\u0304}{j}}\right)}={e}^{f}.$$
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},ω_{g}) be a compact Kähler manifold with c_{1}(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},ω_{g}) 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 ${\mathcal{K}}_{M}^{1}$ is positive, so ${c}_{1}\left(M\right)={c}_{1}\left({\mathcal{K}}_{M}^{1}\right)>0.$ 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 H^{q}(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 LeviCivita 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 JeanPierre Bourguignon's excellent article in that volume.
(3) The factor 2 multiplying ω_{g} is chosen so that the volume form satisfies ${\omega}_{g}^{n}=n!d{\text{vol}}_{g}$.
(4) The general version of the Kodaira–Serre Duality Theorem is given in Theorem 3.4.9 below.