Complex Differential Geometry

Roger Bielawski

July 27, 2009

Complex manifolds

A complex manifold of dimension m is a topological manifold (M,U), −1 such that the transition functions φU ◦ φV are holomorphic maps between open subsets of Cm for every intersecting U,V ∈ U. We have a holomorphic atlas (or “we have local complex coordinates on M.”) Remark: Obviously, a complex manifold of dimension m is a smooth (real) manifold of dimension 2m. We will denote the underlying real

manifold by MR. Example m m+1 Complex projective space CP - the set of (complex) lines in C , i.e. the set of equivalence classes of the relation

∗ (z0,...,zm) ∼ (αz0,...,αzm), ∀α ∈ C

m+1 on C − {0}. m m+1 In other words CP = C − {0} / ∼. The complex charts are deﬁned as for RPm:

Ui = {[z0,...,zm];zi 6= 0}, i = 1,...,m m z0 zi−1 zi+1 zm φi : Ui → C , φi ([z0,...,zm]) = ,..., , ,..., . zi zi zi zi

Example m 1. Complex Grassmanian Grp(C ) - the set of all p-dimensional vector subspaces of Cm. 2. The torus T 2 ' S1 × S1 is a complex manifold of dimension 1. 3. As for smooth manifolds one gets plenty of examples as level sets of m+1 submersions f : C → C. If f is holomorphic and df (the holomorphic differential) does not vanish at any point of f −1(c), then f −1(c) is a holomorphic manifold.For example Fermat hypersurfaces:

m ! di (z0,...,zm); ∑ zi = 1 , d0,...,dm ∈ N. i=0

4. Similarly, homogeneous f give complex submanifolds of CPm. 5. Complex Lie groups: GL(n,C), O(n,C), etc. Almost complex manifolds

A complex manifold of (complex) dimension m is also a smooth real manifold of (real) dimension 2m. Obviously, the converse is not true, but it turns out that there is a characterisation of complex manifolds among real ones, which is much simpler than the existence of a holomorphic atlas. Identifying R2n with Cn is equivalent to giving a linear map j : R2n → R2n satisfying j2 = −Id. Let x ∈ M and let (U,φU ) be a holomorphic chart around x. −1 Define an endomorphism JU of Tx MR by JU (X) = φU ◦ j ◦φU (X). JU does not depend on U, and so we have an endomorphism 2 Jx : Tx MR → Tx MR satisfying Jx = −Id. The collection of all Jx , x ∈ M, defines a tensor J (of type (1,1)), which satisfies J2 = −Id and which is called an almost complex

structure on MR. Deﬁnition An almost complex manifold is a pair (M,J), where M is a smooth real manifold and J : TM → TM is an almost complex structure.

Thus a complex manifold is an almost complex manifold. The converse is not true, but the existence of complex coordinates follows from vanishing of another tensor. Remark: Obviously, an almost complex manifold has an even dimension, but no every even-dimensional smooth manifold admits an almost complex structure (e.g. S4 does not). Remark: S6 admits an almost complex structure, but it is still an open problem, whether it can be made into a complex manifold.

Deﬁnition

A smooth map f :(M1,J1) → (M2,J2) between two complex manifolds −1 is called holomorphic if ψV ◦ f ◦ φU is a holomorphic map between n open subsets in C , for any charts (U,φU ) in M1 and (V ,ψV ) in M2. This is equivalent to the differential of f commuting with the complex structures, i.e. f∗ ◦ J1 = J2 ◦ f∗. The complexiﬁed tangent bundle

Let (M,J) be an almost complex manifold. Since J is linear, we can diagonalise it, but only after complexifying the tangent spaces. We deﬁne the complexiﬁed tangent bundle:

C T M = TM ⊗R C,

and we extend all linear endomorphisms and linear differential operators from TM to T CM by C-linearity. Let T 1,0M and T 0,1M denote the +i- and the −i-eigenbundle of J. It is easy to verify the following:

T 1,0M = {X − iJX; X ∈ TM}, T 0,1M = {X + iJX; X ∈ TM},

T CM = T 1,0M ⊕ T 0,1M.

Remark −J is also an almost complex structure. T 1,0(M,−J) = T 0,1(M,J).

Holomorphic tangent vectors

Let M be a complex manifold. Recall that we have three notions of a R tangent space of M at a point p: Tp M = TM - the real tangent space, C 1,0 0,1 Tp M - the complexiﬁed tangent space, and Tp M (resp. Tp M) - the holomorphic tangent space (resp. antiholomorphic tangent space). Let us choose√ local complex coordinates z = (z1,...,zn) near z. If we write zi = xi + −1yi , then: ∂ ∂ TpM = R , , ∂xi ∂yi

∂ ∂ ∂ ∂ C Tp M = C , = C , , ∂xi ∂yi ∂zi ∂z¯i where

∂ 1 ∂ √ ∂ ∂ 1 ∂ √ ∂ = − −1 , = + −1 . ∂zi 2 ∂xi ∂yi ∂z¯i 2 ∂xi ∂yi Consequently: 1,0 ∂ 0,1 ∂ Tp M = C , Tp M = C , ∂zi ∂z¯i

1,0 0,1 and Tp M (resp. Tp M) is the space of derivations which vanish on anti-holomorphic functions (resp. holomorphic functions). Now observe that:

V ,W ∈ ΓT 1,0M =⇒ [V ,W ] ∈ ΓT 1,0M,

and V ,W ∈ ΓT 0,1M =⇒ [V ,W ] ∈ ΓT 0,1M.

The Newlander-Nirenberg Theorem

Theorem (Newlander-Nirenberg) Let (M,J) be an almost complex manifold. The almost complex structure J comes from a holomorphic atlas if and only if

V ,W ∈ ΓT 0,1M =⇒ [V ,W ] ∈ ΓT 0,1M.

An almost complex structure, which comes from complex coordinates is called a complex structure. Remark: An equivalent condition is the vanishing of the Nijenhuis tensor:

N(X,Y ) = [JX,JY ] − [X,Y ] − J[X,JY ] − J[JX,Y ]. As for the proof: we just have seen the ”only if” part. The ”if” part is very hard. See Kobayashi & Nomizu for a proof under an additional assumption that M and J are real-analytic, and Hormander’s¨ ”Introduction to Complex Analysis in Several Variables” for a proof in full generality.

Definition A section Z of T 1,0M is a holomorphic vector field if Z(f ) is holomorphic for every locally defined holomorphic function f .

n ∂ In local coordinates Z = ∑ gi , where all gi are holomorphic. i=1 ∂zi Definition A real vector field X ∈ Γ(TM) is called real holomorphic if its (1,0)-component X − iJX is a holomorphic vector field.

Lemma The following conditions are equivalent: (1) X is real holomorphic. (2) X is an inﬁnitesimal automorphism of the complex structure J, i.e.

LX J = 0 (LX - the Lie derivative). (3) The ﬂow of X consists of holomorphic transformations of M. The dual picture:

We can decompose the complexiﬁed cotangent bundle Λ1M ⊗ C into 1,0 1 0,1 Λ M = ω ∈ Λ M ⊗ C; ω(Z) = 0 ∀Z ∈ T M and 0,1 1 1,0 Λ M = ω ∈ Λ M ⊗ C; ω(Z) = 0 ∀Z ∈ T M . We have 1 1,0 0,1 Λ M ⊗ C = Λ M ⊕ Λ M. and, consequently, we can decompose the k-th exterior power of Λ1M ⊗ C as k k 1,0 0,1 M p 1,0 q 0,1 Λ M ⊕ C = Λ Λ M ⊕ Λ M = Λ Λ M ⊗ Λ Λ M . p+q=k We write Λp,qM = Λp Λ1,0M ⊗ Λq Λ0,1M, so that k M p,q Λ M ⊕ C = Λ M. p+q=k

Sections of Λp,qM are called forms of type (p,q) and their space is denoted by Ωp,qM. In local coordinates, forms of type (p,q) are generated by

dzi1 ∧ ··· ∧ dzip ∧ dz¯j1 ∧ dz¯jq .

Remark: The above decomposition of Λk M ⊗ C is valid on any almost complex manifold. The difference between ”complex” and ”almost complex” lies in the behaviour of the exterior derivative d. Theorem Let (M,J) be an almost complex manifold of dimension 2n. The following conditions are equivalent: (i) J is a complex structure. (ii) dΩ1,0M ⊂ Ω2,0M ⊕ Ω1,1M. (iii) dΩp,qM ⊂ Ωp+1,qM ⊕ Ωp,q+1M for all 0 ≤ p,q ≤ m. Proof. The only non-trivial bit is (ii) =⇒ (i). Use the following elementary formula for the exterior derivative of a 1-form:

2dω(Z,W ) = Zω(W ) − W ω(Z) − ω([Z,W ]).

Using statement (iii), we decompose the exterior derivative d :Ωk M → Ωk+1M as d = ∂ + ¯∂, where ∂ :Ωp,qM → Ωp+1,qM, ¯∂ :Ωp,qM → Ωp,q+1M. Lemma

∂2 = 0,,¯∂2 = 0, ∂¯∂ + ¯∂∂ = 0.

Proof. 0 = d 2 = (∂ + ¯∂)2 = ∂2 + ¯∂2 + (∂¯∂ + ¯∂∂) and the three operators in the ∗ last term take values in different subbundles of Λ M ⊗ C.

The Dolbeault operator

Deﬁnition The operator ¯∂ :Ωp,qM → Ωp,q+1M is called the Dolbeault operator.A p-form ω of type (p,0) is called holomorphic if ¯∂ω = 0.

One uses ¯∂ to deﬁne Dolbeault cohomology groups of a complex manifold, analogously to the de Rham cohomology:

p,q Z (M) = {ω ∈ Ωp,q(M); ¯∂ω = 0} − ¯∂-closed forms. ¯∂ p,q Z (M) p,q ¯∂ H¯ (M) = . ∂ ¯∂Ωp,q−1(M) A holomorphic map f : M → N between complex manifolds induces a map p,q p,q f ∗ : H (N) → H (M). ¯∂ ¯∂ Warning: Dolbeault cohomology is not a topological invariant: it depends on the complex structure. The ordinary Poincare´ lemma says that every closed form on Rn is exact, and, hence, that the de Rham cohomology of Rn or a ball vanishes. Analogously:

Lemma (Dolbeault Lemma) p,q For B a ball in n,H (B) = 0, if p + q > 0. C ¯∂

For a proof, see Grifﬁths and Harris, p. 25.

Example (Dolbeault cohomology of P1=CP1) 0,2 1 2,0 1 1 First of all Ω (P ) = Ω (P ) = 0, since dimC P = 1. Hence 0,2 2,0 0,0 H ( 1) = H ( 1) = 0. Also H ( 1) = . ¯∂ P ¯∂ P ¯∂ P C 1,1 Secondly, ¯∂Ω1,0 = dΩ1, and, hence H ( 1) = . ¯∂ P C 1,0 Thirdly, if ω ∈ Z ( 1) = 0, then dω = 0 and using the vanishing of ¯∂ P 1,0 H1(S2), we conclude that H ( 1) = 0. This also shows that there ¯∂ P 1,0 are no global holomorphic forms on 1, i.e. Z ( 1) = 0. P ¯∂ P 0,1 Finally, we compute H ( 1) = 0: ¯∂ P

Complex and holomorphic vector bundles

Let M be a smooth manifold. A (smooth) complex vector bundle (of

rank k) on M consists of a family of {Ex }x∈M of (k-dimensional) complex vector spaces parameterised by M, together with a C∞ S manifold structure on E = x∈M Ex , such that: ∞ (1) The projection map π : E → M, π(Ex ) = x, is C , and

(2) Any point x0 ∈ M has an open neighbourhood, such that there exists a diffeomorphism

−1 k φU : π (U) −→ U × C

k taking the vector space Ex isomorphically onto x × C for each x ∈ U.

The map φU is called a trivialisation of E over U. The vector spaces Ex are called the ﬁbres of E. A vector bundle of rank 1 is called a line bundle. Examples: The complexiﬁed tangent bundle T CM, T 1,0M, T 0,1M, bundles Λp,qM. ∞ For any pair φU ,φV of trivialisations, we have the C -map gUV : U ∩ V → GL(k,C) given by −1 gUV (x) = φU ◦ φ k . V |{x}×C These transition functions satisfy

gUV (x)gVU (x) = I, gUV (x)gVW (x)gWU (x) = I. ∞ Conversely, given an open cover U = {Uα} of M and C -maps gαβ : Uα ∩ Uβ → GL(k,C) satisfying these identities, there is a unique complex vector bundle E → M with transition functions {gαβ}. All operations on vector spaces induce operations on vector bundles: dual bundle E∗, direct sum E ⊕ F, tensor product E ⊗ F, exterior powers Λr E. The transition functions of these are easy to write down, e.g. if E,F are vector bundles of rank k and l with transition functions {gαβ} and {hαβ}, respectively, then the transition functions of E ⊕ F and E ⊗ F are gαβ(x) 0 k l k l ∈ GL C ⊕C , gαβ(x)⊗hαβ(x) ∈ GL C ⊗C . 0 hαβ(x)

An important example is the determinant bundle Λk E of E (k = rankE). It is a line bundle with transition functions ∗ detgαβ(x) ∈ GL(1,C) = C .

A subbundle F ⊂ E of a vector bundle E is a collection {Fx ⊂ Ex }x∈M S of subspaces of the ﬁbres Ex such that F = x∈M Fx is a submanifold of E. This means that there are trivialisations of E, relative to which the transition functions look like hUV (x) kUV (x) gUV (x) = . 0 jUV (x)

The bundle F has transition functions hUV , while jUV are transition functions of the quotient bundle E/F, given by Ex /Fx . A homomorphism between vector bundles E and F on M is given by a ∞ C -map f : E → F, such that fx = f|Ex : Ex → Fx and fx is linear. We have the obvious notions of ker(f ) - a subbundle of E, and Im(f ) - a subbundle of F. Also, f is an isomorphism, if each fx is an isomorphism. A vector bundle E on M is trivial, if E is isomorphic to the product bundle M × Ck . π Given a C∞-map f : M → N and a vector bundle E → N, we deﬁne the pullback bundle f ∗E on M by

f ∗E = {(x,e) ∈ M × E; f (x) = π(e)},

∗ i.e. (f E)x = Ef (x). Finally, a section s of a vector bundle E → M is a C∞-map s : M → E such that s(x) ∈ Ex for all x ∈ M (just like a vector ﬁeld). The space of sections is denoted by Γ(E). Observe that trivialising a rankk bundle E over U is equivalent to giving k sections s1,...,sk , which are linearly independent at every point of U. Such a collection (s1,...,sk ) is called a frame for E over U.

Now, let M be a complex manifold. A holomorphic vector bundle π E → M is deﬁned as a complex vector bundle, except that C∞ is replaced by “holomorphic” everywhere. Examples: T 1,0M, Λp,0M (but not Λp,qM if q 6= 0). The line bundle n,0 Λ M, n = dimC M is called the canonical bundle of M, and is denoted by KM . Its sections are holomorphic n-forms. The dual ∗ n 1,0 KM = Λ T M is the anti-canonical bundle.

Example (The tautological and canonical bundles on CPn) n The tautological line bundle J → CP is defined by setting JA = A, n+1 n where A is a line in C defining a point of CP . n+1 We have KCPn = J . For every holomorphic bundle E → M we define the bundles Λp,qE = Λp,qM ⊗ E of E-valued forms of type (p,q). The space of sections is denoted by Ωp,qE. If we choose a trivialisation of E, i.e. a local frame, then an element σ of Ωp,qE is, in this trivialisation, σ = (ω1,...,ωk ), ωi - a local (p,q)-form on M.Now observe that we have a well-defined operator ¯∂ :Ωp,qE → Ωp,q+1E, given by: ¯ ¯ ¯ ∂σ = (∂ω1,...,∂ωk ) in any trivialisation. Note that ¯∂ satisfies ¯∂2 = 0 and the Leibniz rule ¯∂(f σ) = ¯∂f ⊗ σ + f (¯∂σ), for any f ∈ C∞(M) and σ ∈ Ωp,qE. The existence of a such a natural operator ∂ on sections of E = Λ0,0E is a remarkable property of holomorphic vector bundles. On a general complex vector bundle there is no canonical way of differentiating sections. It has to be introduced ad hoc, via the concept of connection: Connections and their curvature

Deﬁnition A connection (or covariant derivative) on a complex vector bundle E → M is a map D : Γ(E) → Ω1E satisfying the Leibniz rule:

D(fs) = (df ) ⊗ s + f (Ds),

for any f ∈ C∞(M) and s ∈ Γ(E).

If we choose a frame (local basis) (e1,...,ek ) for E over U, then

Dei = ∑θij ej , j where the matrix of 1-forms is called the connection matrix with respect to e. The data e and θ determine D. The connection matrix depends on the 0 0 0 choice of e: if (e1,...,ek ) is another frame with e (z) = g(z)e(z), g(z) ∈ GL(k,C), then: −1 −1 θe0 = gθeg + dgg .

Curvature

One can extend any connection D to act on ΩpE, i.e. on E-valued p-forms, by forcing the Leibniz rule: D(ω ⊗ σ) = dω ⊗ σ + (−1)pω ∧ Dσ, ∀ω ∈ ΩpM, σ ∈ Γ(E). In particular, we have the operator D2 : Γ(E) → Ω2E. It is tensorial, i.e. linear over C∞(M): D2(f σ) = Ddf ⊗ σ + f (Dσ) = −df ∧ Dσ + df ∧ Dσ + fD2σ = fD2σ. Consequently, D2 is induced by a bundle map E → Λ2M ⊗ E, i.e. by a section of Λ2M ⊗ Hom(E,E). This Hom(E,E)-valued 2-form RD is called the curvature of the connection D.

The curvature matrix. In a local frame, we had Dei = ∑j θij ej , and, hence: 2 D ei = D ∑θij ej = ∑ dθij − ∑θip ∧ θpj ⊗ ej . Therefore, we get the Cartan structure equations for the curvature matrix with respect to a frame e:

Θe = dθe − θe ∧ θe. Hermitian metrics

Deﬁnition Let E → M be a complex vector bundle. A hermitian metric on E is a

smoothly varying hermitian inner product on each ﬁbre Ex , i.e. if σ = (s1,...,sk ) is a frame for E, then the functions ∞ hij (x) = hsi (x),sj (x)i are C .

A complex vector bundle equipped with a hermitian metric is called a hermitian vector bundle. Example: E = T 1,0M - a hermitian metric on M. On a hermitian vector bundle over a complex manifold, we have a canonical connection D by requiring: D is compatible with the metric, i.e.

dhs1,s2i = hDs1,s2i + hs1,Ds2i, i.e. the metric is parallel for D, and D is compatible with the complex structure, i.e. if we write D = D1,0 + D0,1 with D1,0 : Γ(E) → Ω1,0E and D0,1 : Γ(E) → Ω0,1E, then D0,1 = ¯∂.

Theorem If E → M is a hermitian vector bundle over a complex manifold, then there is unique connection D (called the Chern connection) compatible with both the metric and the complex structure.

Proof.

Let (e1,...,ek ) be a holomorphic frame for E and put hij = hei ,ej i. If D is compatible with the complex structure, then Dei is of type (1,0), and, as D is compatible with the metric:

¯ dhij = hDei ,ej i + hei ,Dej i = ∑θiphpj + ∑θpj hip. p p

The ﬁrst term is of type (1,0) and the second one of type (0,1) and, hence: ¯ ∂hij = ∑θiphpj , ∂hij = ∑θpj hip. p p

Therefore ∂h = θh and ¯∂h = hθ¯T and θ = ∂hh−1 is the unique solution to both equations. Remark on the curvature of the Chern connection. If D is compatible with the complex structure, then the (0,2)-component of the curvature vanishes:

R0,2 = D0,1 ◦ D0,1 = ¯∂2 = 0.

If D is also compatible with the hermitian metric, then the (2,0)-component vanishes as well, since

2 2 2 0 = d hei ,ej i = hD ei ,ej i + 2hDei ,Dej i + hei ,D ej i

and so the (2,0)-component of D2 is the negative hermitian transpose of the (0,2)-component. Thus, the curvature of the Chern connection is of type (1,1).

Curvature of a Chern connection in a holomorphic frame

Recall that the curvature matrix of a connection D with respect to any

frame e = (e1,...,en) is given by

Θe = dθe − θe ∧ θe = dθe − [θe,θe],

where θe is the connection matrix w.r.t. e. If D is the Chern connection of a hermitian holomorphic vector bundle, −1 and e is a holomorphic frame, then we computed: θe = ∂hh , where hij = hei ,ej i. We compute dθe: ¯ ¯ −1 ¯ −1 ¯ −1 −1 (∂+∂)θe = ∂θe +∂(∂hh ) = ∂θe −∂h∧∂(h ) = ∂θe +∂hh ∧∂hh .

Therefore, the curvature matrix of the Chern connection w.r.t. a holomorphic frame is given by Θ = ¯∂θ.

In the case of a line bundle, a local non-vanishing holomorphic section s and h = hs,si: θ = ∂logh, Θ = ¯∂∂logh. Example (Curvature of the tautological bundle of CPn) n n+1 J is a subbundle of the trivial bundle P × C :

n n+1 J = {l × z ∈ P × C ; z ∈ l}.

n n+1 We have a hermitian metric on P × C given by the standard n+1 hermitian inner product on C . It induces a hermitian metric on the line bundle J. We compute the curvature of the associated Chern

connection in the patch U0 (z0 6= 0), in which J is trivialised via φ : Cn × C → J: (z1,...,zn),u 7−→ u(1,z1,...,zn).

We have a non-vaninshing section s(z1,...,zn) = (1,z1,...,zn), and, n 2 consequently, h = hs,si = 1 + ∑i=1 |zi | . Thus, the curvature is

n 2 ¯ 1 + ∑r6=i |zr | ∂∂logh. = dz¯i ∧ dzi . ∑ n 2 2 i=1 (1 + ∑r=1 |zr | )

Hermitian metrics on complex manifolds

We now look at the particular case E = T 1,0M (M - a complex manifold). A choice of holomorphic frame is given by a choice of local ∂ holomorphic coordinates z1,...,zn (ei = ), and a hermitian metric ∂zi on T 1,0M can be written as

g = ∑hij dzi ⊗ dz¯j . i,j ∗ where hij is hermitian matrix, h = h. Notice that this also deﬁnes a Hermitian metric on T CM, and by restriction a C-valued hermitian metric on TM. Its real part is a real-valued symmetric bilinear form, which we also denote by g, and refer to as hermitian metric on M. Observe that

g(JX,JY ) = g(X,Y ) ∀X,Y ∈ Tx M. The negative of the imaginary part is a skew-symmetric bilinear form ω called the fundamental form of g. We have ω(X,Y ) = g(JX,Y ), and in local coordinates: 1 g = Re∑hij dzi dz¯j , 2 i,j √ −1 ω = ∑hij dzi ∧ dz¯j . 2 i,j Observe that ω ∈ Ω1,1M. Example (Connection and curvature in dimension one) ∂ z = x + iy - a local coordinate, ∂z - a local holomorphic frame, and a hermitian metric on T 1,0M is written as hdz ⊗ dz¯, for a local function −1 ∂logh h > 0. The connection matrix of the Chern matrix is ∂hh = ∂z dz, and the curvature matrix is

∂2 logh 1 Θ = ¯∂∂logh = dz¯ ∧ dz = − ∆loghdz ∧ dz¯. ∂z¯∂z 4 √ −1 Now, the fundamental form on M is 2 hdz ∧ dz¯ and hence, √ Θ = − −1K ω, where K = (−∆logh)/2h is the usual Gaussian curvature of a surface.

Recall the different behaviour of positively and negatively curved surfaces. Similarly, we say that the curvature 1,1 RE ∈ Γ Λ M ⊗ Hom(E,E) of a Chern connection on a hermitian vector bundle is positive at x ∈ M (notation RE (x) > 0), if the hermitian matrix RE (x)(v,v¯) ∈ Hom(Ex ,Ex ) is positive deﬁnite. In local coordinates, this means that Θ(x)(v,v¯) is positive deﬁnite ∀v, e.g. for surfaces with positive Gaussian curvature.

Similarly RE (x) ≥ 0,RE (x) ≤ 0,RE (x) < 0. We write RE > 0 etc. if the curvature is positive everywhere. Observe that the curvature of the tautological bundle JPn (with induced metric) is negative. Let us compute the curvature RF of the Chern connection of a holomorphic subbundle F ⊂ E (with the induced hermitian metric). Let N = F ⊥. This is a C∞ complex subbundle of E. If s ∈ Γ(F) and t ∈ Γ(N), then:

0 = dhs,ti = hDs,ti + hs,Dti, so that in a frame for E consisting of frames for F and N, the connection matrix of D = DE is θF A θE = ∗ . −A θN Moreover, if the frame on F is holomorphic, then A (matrix of 1-forms) is of type (1,0). Now, the curvature matrix is

dθ − θ ∧ θ + A ∧ A∗ ∗ Θ = dθ − θ ∧ θ = F F F , E E E E ∗ ∗

∗ and so ΘF = ΘE |F − A ∧ A . Therefore

RF ≤ RE |F , so that the curvature decreases in holomorphic subbundles. In particular, if E is a trivial bundle with Euclidean metric (so that RD ≡ 0) and F ⊂ E is a holomorphic subbundle with the induced metric, then D RF ≤ 0. Applying this to a submanifold M of n and F = T 1,0M ⊂ T 1,0 n C C M with the induced hermitian metric, we see that the curvature of T 1,0M is always non-positive. In particular, if M is a Riemann surface, then its Gaussian curvature K ≤ 0.

Observe that the same calculation for the quotient bundle Q = E/F and conclude that

RQ ≥ RE |F , i.e. the curvature increases in holomorphic quotient bundles. As an application, consider a holomorphic vector bundle E → M “spanned by its sections”, i.e. there exist holomorphic sections s1,...,sk ∈ Γ(E), such that s1(x),...,sk (x), k ≥ rankE, generate Ex for every x ∈ M. Then we have a surjective map M × Ck → E, k (x,λ) 7→ ∑i=1 λi si (x). Thus, E is a quotient bundle of a trivial bundle and, if we give E the k metric induced from the Euclidean metric on M × C , then RQ ≥ 0. The ﬁrst Chern class

We go back to a non-holomorphic setting and consider complex vector bundles over smooth manifolds. We equip such a bundle E → M with a connection D. Recall that the curvature RD of D is a section of Λ2M ⊗ Hom(E,E), so basically a matrix of 2-forms. We can speak of the trace of the curvature: trRD, which is a 2-form. Recall the formula for the curvature matrix in a local frame: Θ = dθ − [θ,θ]. It follows that trΘ = trdθ = d trθ and, hence, trRD is a closed 2-form. It is called the Ricci form of D. Lemma D 2 The cohomology class trR ∈ HDR(M) does not depend on D.

Proof. 0 A = D − D0 is a section of Λ1M ⊗ Hom(E,E), so trRD − trRD = d trA, which is an exact form. √ −1 D 2 The cohomology class c1(E) = 2π trR ∈ HDR(M) is called the 1st Chern class of E. It is a topological invariant.

Example

We compute c (J 1 ). Recall that for the Chern connection induced 1 P 1 2 from P × C , we computed the curvature matrix in the chart U0 with holomorphic coordinate z as

1 dz¯ ∧ dz. (1 + |z|2)2

2 1 R Now, H ( ) is identiﬁed with via integration: ω 7→ 1 ω. Thus DR P C P √ −1 Z 1 1 Z r c1(J 1 ) = dz¯∧dz = dθ∧dr, P 2 2 2 2 2π C (1 + |z| ) 2π [0,2π]×[0,∞) (1 + r )

where z = reiθ. Taking the orientation into account we obtain

−1 Z +∞ Z 2π r c (J 1 ) = d dr = −1. 1 P 2 θ 2π 0 0 (1 + r 2)

It follows from this that c1(JPn ) = −1 for any n. Let E and F be bundles of ranks m and n, respectively. Then: m (i) c1(Λ E) = c1(E),

(ii) c1(E ⊕ F) = c1(E) + c1(F),

(iii) c1(E ⊗ F) = nc1(E) + mc1(F), ∗ (iv) c1(E ) = −c1(E), ∗ ∗ (v) c1(f E) = f c1(E).

Let M be a complex manifold. The ﬁrst Chern class c1(M) of M is 1,0 ∗ c1 T M = c1(K ), i.e. the ﬁrst Chern class of the anti-canonical bundle of M. Example

n ∗ ∗ ⊗(n+1) ∗ c1(P ) = c1(K ) = c1 (J ) = (n + 1)c1(J ) = n + 1.

2 For n = 1, we get c1(P ) = 2.

This is just the Gauss-Bonnet theorem: for any hermitian metric on a compact surface S :

1 Z c1(S) = K ω = χ(S). 2π S

Examples of manifolds with c1(M) = 0: Observe that c1(M) = 0, if the canonical bundle KM is trivial, i.e. there exists a non-vanishing holomorphic n-form on M (n = dimC M). Cn. Other boring examples include any M with H2(M) = 0. Quotients of Cn by ﬁnite groups of holomorphic isometries, e.g by lattices: abelian varieties (complex tori). 3 2 2 2 The quadric Q = {(z1,z2,z3) ∈ C ; z1 + z2 + z3 = 1}. This is a complexiﬁcation of S2 and so H2(Q) 6= 0. The following

holomorphic 2-form is non-vanishing on Q and trivialises KQ:

z1dz2 ∧ dz3 + z2dz3 ∧ dz1 + z3dz1 ∧ dz2. The famous K3-surface (one of them, anyway): 3 4 4 4 4 S = {[z0,z1,z2,z3] ∈ P ; z0 + z1 + z2 + z3 = 0}.

We’ll show how to compute c1 of hypersurfaces. Let V ⊂ M be a complex submanifold of dimension 1. We have the exact sequence of holomorphic vector bundles over V : ∗ 1,0 1,0 0 → NV → Λ M|V → Λ V → 0. ∗ Taking the maximal exterior power gives: KM |V ' KV ⊗ NV , so

KV = KM |V ⊗ NV .

Now, observe that, if f = 0 is a local (say, on U ⊂ V ) deﬁning equation ∗ for V , then, by deﬁnition, df generates NV over U. If, on U0 ⊂ V , the equation of V is f 0 = 0, then on U ∩ U0:

f f f f df = d f 0 = d f 0 + df 0 = df 0, f 0 f 0 f 0 f 0 ∗ 0 so the transition functions for NV are f /f . ∗ We can now easily ﬁnish the computation for K3: N ' K 3 . S P |S

Higher Chern classes

The first Chern class of a complex vector bundle was defined using the trace of the curvature RD ∈ ΓΛ2M ⊗ Hom(E,E). We can use other invariant polynomials defined on matrices, e.g. the determinant or the sum of the squares of eigenvalues. For a k × k matrix M, let us write

k k−i det(t + M) = ∑ Pi (M)t . i=0

so that P1(M) = trM, Pk (M) = detM, etc. Each Pi is a homogeneous polynomial of degree k in the entries of A.

We can allow matrices over any commutative algebra - the Pi (M) are well deﬁned - in particular over the even part of the exterior algebra of a smooth manifold. Applying this to RD or its curvature matrix, we D 2i obtain closed forms Pi (R ) = Pi (Θ) ∈ Ω M. Deﬁnition The i-th Chern class of a complex vector bundle E over a smooth manifold M is the cohomology class √ −1 D 2i ci (E) = Pi R ∈ H (M), i = 1,...,k. 2π DR

Once again, it does not depend on the connection D (exercise). Also, for a complex manifold M, we deﬁne the Chern classes of M to be the Chern classes of T 1,0M. The Gauss-Bonnet theorem generalise to higher dimensions as follows: if M is a compact complex manifold of dimension n, then

cn(M) = χ(M).

There are of course purely topological ways of deﬁning Chern classes!

Chern classes of a holomorphic bundles

Now E is hermitian holomorphic vector bundle over a complex manifold M. Recall that the curvature of a Chern connection√D is of −1 D type (1,1) and that its matrix is skew-hermitian, so that Pi 2π R is a real (i,i)-form, and consequently, for any holomorphic bundle

i,i 2i ci (E) ∈ H (M) ∩ HDR(M;R). Recall also that we deﬁned notions of the curvature being > 0,< 0,≥ 0,≤ 0. For example, nonnegativity meant that RD(v,v¯) has all eigenvalues nonnegative, for any v ∈ T 1,0M. This means that

D 1,0 Pi (R )(v1,v¯1,...,vi ,v¯i )) ≥ 0 ∀v1,...,vi ∈ T M, and consequently that Z ci (E) ≥ 0 Z for any i-dimensional complex submanifold Z of M. In particular, this holds for any holomorphic vector bundle generated by its sections.

Note that these conditions depend only on ci (E), not on the connection or its curvature. Prescribing the Ricci curvature of Chern connections

Thus, we say that c1(E) ≥ 0 (resp. > 0,< 0,≤ 0), if the cohomology 1,1 class c1(E) ∈ H (M) can be locally represented by a closed real 2-form φ, such that in local complex coordinates √ −1 φ = φ ¯dzα ∧ dz¯ , 2π ∑ αβ β

and the hermitian matrix [φαβ¯] has all eigenvalues nonnegative (resp. positive, negative, nonpositive). E.g., any holomorphic vector bundle generated by its sections satisﬁes

c1(E) ≥ 0. Let now a closed real (1,1)-form φ represent c1(E). We ask √ Is there a connection D on E such that trRD = −2π −1φ?

For example, suppose that c1(E) = 0. It is clearly represented by φ ≡ 0 and we would like to know whether there is a hermitian metric, such that the associated Chern connection is Ricci-ﬂat (i.e. trRD ≡ 0).

Let h,i be a hermitian metric on E. Recall that in a local holomorphic

frame (e1,...,en) with the associated matrix hij = hei ,ej i, we had the following formula for RD:

Θ = ¯∂∂hh−1.

Therefore, the Ricci form trRD is represented in this local frame by

¯∂∂(logdeth).

Now we modify h,i by taking ef /k h,i, where k = rankE and f is a 0 f /k 0 f smooth real function on M. Let hij = e hei ,ej i. Then deth = e deth and the Ricci forms of the two Chern connections are related by

0 trRD − trRD = ¯∂∂f .

0 Therefore we ﬁnd a hermitian metric with trRD = φ, providing we can solve the equation ¯∂∂f = φ − trRD.

The right-hand side is√ a closed imaginary (1,1)-form cohomologous to D 0: φ − trR = −2π −1(c1(E) − c1(E)). Therefore the answer to our question: can we ﬁnd a hermitian metric on E, the Ricci form of which is a given form φ, is yes on manifolds M for which the following condition (called the global ∂¯∂-lemma) holds: √ Any exact real (1,1)-form β on M is of the form −1¯∂∂f , for a smooth function f : M → R. Lemma Let M be a complex manifold with H0,1(M) = 0. Then the global ∂¯∂-lemma holds on M.

Proof. Since β is exact, there exists a real 1-form α such that dα = β. We decompose α as τ + τ0, where τ is (1,0) and τ0 is (0,1). We must have τ0 = ¯τ. Now, β = (∂ + ¯∂)(τ +¯τ) = ∂τ + (¯∂τ + ∂¯τ) + ¯∂¯τ. Comparing the types, we get: β = ¯∂τ + ∂¯τ, ∂τ = 0, ¯∂¯τ = 0.

0,1 Since H (M) = 0, there exists a function u : M → C such that ¯τ = ¯∂u. Therefore τ = ∂u¯, and β = ¯∂τ + ∂¯τ = ¯∂∂u¯ + ∂¯∂u = ∂¯∂(u − u¯) = 2i∂¯∂(Imu).

The result follows with f = 2Imu. Kahler¨ metrics We now consider the case E = TM, where M is a complex manifold with a complex structure J. TM is a complex vector bundle - J allows n us to identify Tx M with C , for every n. In other words we identify TM ' T 1,0M, X 7−→ X − iJX. Recall that a hermitian metric on M is a smoothly varying inner product g on each Tx M, such that g(JX,JY ) = g(X,Y ) for all X,Y ∈ Tx M. We will now consider connections on TM. First, of all observe, that for such a connection D : Γ(TM) → Ω1(M) ⊗ Γ(TM), the (covariant) directional derivative DZ (X) := D(X)(Z) of a vector field is again a vector field. We now reinterpret the two properties defining the Chern connection D of g: 1. D being metric, i.e. dg(X,Y ) = g(DX,Y ) + g(X,DY ), is equivalent to

Z.g(X,Y ) = g(DZ X,Y ) + g(X,DZ Y ) ∀X,Y ,Z ∈ Γ(TM). 2. D being compatible with the complex structure, i.e. D0,1 = ¯∂, is equivalent to

DJ = 0, i.e. DZ (JX) = JDZ (X) ∀X,Z ∈ Γ(TM). i.e. J is parallel for D (follows from the fact that the connection matrix in the holomorphic frame has type (1,0)). On the other hand, since g is a Riemannian metric, i.e. a smoothly varying inner product, there exists a unique connection (the Levi-Civita connection) ∇ on TM, which is again metric and it has zero torsion, i.e.

∇X Y − ∇Y X = [X,Y ], ∀X,Y ∈ Γ(TM).

Thus, on a complex manifold with a hermitian metric, we have two natural connections: Chern and Levi-Civita. We ask for which hermitians metrics the two coincide.

Theorem Let g be a hermitian metric on a complex manifold (M,J). The following conditions are equivalent: (i) J is parallel for the Levi-Civita connection ∇. (ii) The Chern connection D has zero torsion. (iii) The Levi-Civita and the Chern connections coincide. (iv) The fundamental form ω of g is closed, dω = 0, (recall that ω(X,Y ) = g(JX,Y )). (v) For each point p ∈ M, there exists a smooth real function in a neighbourhood of p, such that ω = i∂¯∂f . (vi) For each point p ∈ M, there exist holomorphic coordinates w centred at p, such that g(w) = 1 + O(|w|2).

Proof. (i), (ii), and (iii) are clearly equivalent from the uniqueness properties. We show (i) =⇒ (iv) =⇒ (v) =⇒ (vi) =⇒ (i). (i) =⇒ (iv). Since ∇g = 0 and ∇J = 0, ∇ω = 0. For any p-form and any torsion-free connection ∇ one has (exercise): p d (X ,...,X ) = (−1)i (X ,...,X ,...,X ), ω 0 p ∑ ∇Xi ω 0 bi p i=0 which implies that parallel forms are closed. ¯ n (iv) =⇒ (v). This is the ∂∂-lemma applied to a ball B in C (Dolbeault Lemma says that H0,1(B) = 0).

(v) =⇒ (vi). We can choose local coordinates around p0, so that ω = i ∑l,m ωlmdzl ∧ dz¯m, where

1 2 ωlm = δlm + ∑(ajlmzj + bjlmz¯j ) + O(|z| ). 2 j

Reality implies that ajlm = bjml . On the other hand, using (v), we have

∂3f ajlm = , ∂zj ∂zl ∂z¯m so ajlm = aljm. We now put

wm = zm + ∑ajlmzj zl , =⇒ dwm = dzm + 2∑ajlmzj dzl , j,l j,l and compute:

1 1 i ∑dwm ∧ dw¯m = i ∑dzm ∧ dz¯m + i ∑ ajlmzj dzl ∧ dz¯m 2 m 2 m j,l,m 2 + i ∑ ajlmz¯j dzm ∧ dz¯l + O(|z| ) j,l,m 2 2 2 = i ∑ωlmdzl ∧ dz¯m + O(|z| ) = ω + O(|z| ) = ω + O(|w| ). l,m

√ (vi) =⇒ (i) If we write zi = xi + −1yi for the coordinates obtained in (v), then the connection matrix of the Levi-Civita connection

(Christoffel symbols) is equal to zero at p0. Consequently, ∇J vanishes at p0, and, since p0 is arbitrary, it vanishes everywhere. A hermitian metric on a complex manifold satisfying the equivalent conditions (i) − (vi) is called a Kahler¨ metric. The fundamental form ω of a Kahler¨ metric is called a Kahler¨ form and the local function f in (v) is called a local Kahler¨ potential. Examples: 1. The standard Euclidean metric on Cn. n n 1 i i ¯ 2 g = Re ∑ dzs ⊗ dz¯s, ω = Re ∑ dzs ∧ dz¯s = ∂∂|z| . 2 s=1 2 s=1 2

1 2 n Thus f (z) = 2 |z| is a global Kahler¨ potential on C . The pair (g,J) is U(n)-invariant.

2. The Fubini-Study metric on CPn. n+1 For a z = (z0,...,zn) ∈ C − {0}, we put

ω = i∂¯∂log(|z|2).

∗ This is invariant under rescaling by C and, so, it induced a real closed (1,1)-form on Pn. We need to check that the corresponding symmetric form g(X,Y ) = ω(X,JY ) is positive-deﬁnite. We compute ﬁrst ω in the chart U0 = {z0 6= 0}, where the local coordinates are zs wi = , s = 1,...,n: z0 ! i n = i ¯log(1 + |w|2) = w dw¯ = ω ∂∂ ∂ 2 ∑ s s 1 + |w| s=1 ! ! i n i n n dw ∧ dw¯ − w¯ dw ∧ w dw¯ 2 ∑ s s 2 2 ∑ s s ∑ s s 1 + |w| s=1 (1 + |w| ) s=1 s=1

Observe that ω is U(n + 1)- invariant, and so it is enough to check positivity at any point p ∈ Pn, e.g p = [1,0,...,0], i.e. w = 0. OK! Remark. The Fubini-Study metric is the quotient metric on n n+1 1 n+1 CP ' S /S obtained from the standard round metric on S . These two basic examples (Cn and CPn) yield plenty of others, because: Lemma A complex submanifold of a Kahler¨ manifold, equipped with the induced metric, is Kahler.¨

Proof. Let (N,J,gN ) ⊂ (M,J,gM ) as in the statement. The fundamental form ωN of gN is just the pullback (restriction) of the fundamental form ωM of gM , hence it is closed. Observe also that a product of two Kahler¨ manifolds is Kahler.¨ On the other hand, many complex manifolds do not admit any Kahler¨ metric, because: Lemma 2q If M is a compact Kahler¨ manifold, then HDR(M) 6= 0, q ≤ n = dimC M.

Proof.

Let ω be the Kahler¨ form. Then ωq is a closed form, which is not exact. Indeed, had we ωq = dψ, then ωn = d(ψ ∧ ωn−q), and, so: Z Z vol(M) = ωn = d(ψ ∧ ωn−q) = 0, M M

which is impossible. Thus, for example, there is no Kahler¨ metric on S1 × S2n−1 for n ≥ 2 (this is a complex manifold - see Examples Set 1). Another easily provable topological restriction is

Lemma Let M be a compact Kahler¨ manifold. Then the identity map on ΩqM induces an injective map

q,0 H (M) ,→ Hq (M, ), ¯∂ DR C

i.e. every holomorphic (q,0)-form is closed and never exact. Proof. Let η be a holomorphic (q,0)-form. Write η in a local unitary frame {φi } as η = ∑fI φI (I runs over subsets of cardinality q and φI is the wedge product of with j ∈ I). Then ∧ ¯ = f ¯f ∧ ¯ . φj √ η η ∑ I J φI φJ −1 ¯ On the other hand, ω = 2 ∑φi ∧ φi , and (n = dimC M): n−q ¯ ω = cq ∑ φK ∧ φK . #K =n−q Hence ¯ n−q 0 2 n η ∧ η ∧ ω = cq ∑ |fI | ω , #I=q since the only non-zero wedge-products arise for K disjoint from I and J (so for I = J). In particular, if η 6= 0, then the integral over M of the LHS is non-zero. If, however, η = dψ, then :

η ∧ η¯ ∧ ωn−q = d ψ ∧ η¯ ∧ ωn−q, since dη¯ = 0 and dω = 0, and we obtain a contradiction. Thus, a non-zero holomorphic (q,0)-form is never exact. To show that η is closed, note: dη = (∂ + ¯∂)η = ∂η, and, so, dη is a holomorphic (q + 1,0)-form, hence dη = 0.

The last result is a particular case of a much stronger fact, which goes under the name of Hodge relations or Hodge decomposition:

Theorem (Hodge) The complex cohomology of a compact Kahler¨ manifold satisﬁes:

r M p,q p,q H (M,C) = H (M), H (M) = Hq,p(M). p+q=r

In particular: for r odd, dimHr (M) is even. To give even an idea of a proof of this theorem of Hodge requires a substantial detour. Before, let us see how badly this theorem fails for non-compact Kahler¨ manifolds: Example

2 1 M = C − {0}, so that HDR(M) = 0. We shall show that 0,1 dimH (M) = +∞. Let U = {z 6= 0} = ∗ × , ¯∂ 1 1 C C ∗ ∗ ∗ U2 = {z2 6= 0} = C × C , so that U1 ∪ U2 = M, U1 ∩ U2 = C × C . Let λ1,λ2 : M → R be a partition of unity subordinated to U1,U2 and let f be a holomorphic function on U1 ∩ U2. Then g1 = λ2f defines a smooth function on U1 and g2 = −λ1f defines a smooth function on ¯ ¯ U2. Observe that on U1 ∩ U2, ∂(g1 − g2) = ∂f = 0, and so we can define a (0,1)-form ω on M, with ¯∂ω = 0, by (¯ ¯ ∂g1 = f ∂λ2 on U1 ω = ¯ ¯ −∂g2 = −f ∂λ1 on U2.

¯ ∞ ¯ Suppose that ω = ∂h for some h ∈ C (M). Then ∂(g1 − h) = 0 on U1 ¯ and ∂(g2 − h) = 0, and, so g1 − h is holomorphic on U1, g2 − h is holomorphic on U2. Then f = g1 − g2 = (g1 − h) − (g2 − h), and, hence, f = u1 + u2, where ui is holomorphic on Ui , i = 1,2. m n But any convergent Laurent series f on U1 ∩ U2, which contains z1 z2 , m,n < 0, is not u1 + u2, so ω deﬁned by such an f is not exact.

An idea of a proof of the Hodge theorem

Let V be a vector space, equipped with an inner product h,i. Then there is an induced inner product on the tensor powers V ⊗k . By restriction, there is an inner product on the exterior powers Λk V . It

can be described by choosing an orthonormal basis (e1,...,en) and decreeing the following basis of Λk V :

{ei1 ∧ ··· ∧ eik ; 1 ≤ i1 < ··· < ik ≤ n} to be orthonormal. Now, if (M,g) is an oriented Riemannian manifold,

then we can do the above on each tangent space Tx M. Since a dual of a vector space with an inner product also has an inner product, we k ∗ get an inner product on Λ Tx M, and, if (M,g) is oriented, i.e. we have a nonvanishing volume form dV , and compact, then we can define an inner product on differential forms in Ωk M: Z hα,βi = hα|x ,β|x idV . M k The idea is that in every cohomology class in HDR(M) we seek a representative α with the smallest norm. We also show that such a representative must be unique (an element with the smallest norm in a closed affine subspace of a Hilbert space). p,q We do the same for the Dolbeault cohomology classes H (M). ¯∂ Finally, we show that, if M is Kahler,¨ then, if we decompose the k α ∈ HDR(M,C) with minimal norm into forms of type (p,q), p + q = k, p,q then each component has the minimal norm in H (M). ¯∂ How to find the element of smallest norm in a cohomology class k [ψ] ∈ HDR(M)? We are looking for the element of smallest norm in an affine subspace P = {ψ + dη; η ∈ Ωk−1M}. For M compact, Ωk (M) with the inner product h,i is a pre-Hilbert space (essentially an L2-space), and were P a closed subspace, we could find an element of the smallest norm by the orthogonal projection (Ωk (M) = dΩk−1M ⊕ (dΩk−1M)⊥). The orthogonal projection, on the other hand, can be expressed via the adjoint operator to d:

kψ + dηk2 = kψk2 + kdηk2 + 2hψ,dηi = kψk2 + kdηk2 + 2hd∗ψ,ηi.

Thus, if d∗ψ = 0, then ψ has the smallest norm in P.

Hodge dual

So cohomology classes should be represented by forms ψ such that dψ = 0 and d∗ψ = 0, once we define d∗ :Ωk+1 → Ωk . To define d∗, we go back to the situation described earlier: V is a vector space, dimV = n, equipped with an inner product h,i and an n orientation, i.e. a preferred element of Λ V , e.g. e1 ∧ ··· ∧ en. Any α ∈ Λn−k V defines a linear functional φ on Λk V by:

ω ∧ α = φ(ω)e1 ∧ ··· ∧ en. Such a functional must be represented by ω 7→ hω,τi for some τ ∈ Λk V . This way, we get a 1-1 correspondence between Λk V and Λn−k V : ∗ :Λk V → Λn−k V , τ 7→ α,

ω ∧ ∗τ = hω,τie1 ∧ ··· ∧ en. The operator ∗ is called the Hodge dual. Notice, in particular: 2 k(n−k) k ∗1 = e1 ∧ ··· ∧ en, h∗τ1,∗τ2i = hτ1,τ2i, ∗ = (−1) on Λ V . Again, we obtain such a ∗ on differential forms of an oriented Riemannian manifold (M,g). 3 3 Example: Take M = R with the Euclidean metric g = ∑i=1 dxi ⊗ dxi and the standard orientation dx1 ∧ dx2 ∧ dx3. We get:

∗dx1 = dx2 ∧ dx3, ∗dx2 = dx3 ∧ dx1, ∗dx3 = dx1 ∧ dx2.

Now, observe that ∗d∗ :Ωk+1 → Ωk and we have (dV denotes the volume form determining orientation):

hdα,βidV = dα ∧ ∗β = d(α ∧ ∗β) − (−1)k α ∧ d ∗ β, so hdα,βidV − d(α ∧ ∗β) = −(−1)k α ∧ d ∗ β = −(−1)k (−1)k(n−k)α ∧ ∗2d ∗ β = −(−1)knhα,∗d ∗ βidV . Therefore, the operator d∗ = (−1)kn+1 ∗ d∗ :Ωk+1 → Ωk is the adjoint of d, and is called the codifferential. A form ω, such that d∗ω = 0 is called co-closed. We need to show that on a compact oriented manifold (M,g), any cohomology class has a representative ψ with d∗ψ = 0 (and dψ = 0).

The Riemannian Laplacian

Observe that such a form also satisﬁes (dd∗ + d∗d)ψ = 0. The operator ∆ = dd∗ + d∗d :Ωk M → Ωk M is called the Riemannian Laplacian or the Laplace-Beltrami operator. Let’s check that we really get the usual Laplacian on functions on Rn: ∗ ∗ ∗ ∂f ∂f (dd + d d)f = d df = −∗ d ∗ ∑ dxj = −∗ d ∑ ∗ dxj ∂xj ∂xj ∂2f ∂2f = − ∗ ∑ dxi ∧ ∗dxj = − ∗ ∑ 2 dV = ∆f . ∂xi ∂xj ∂xi

In general, let a Riemannian metric be given in local coordinates by ij g = ∑ij dxi ⊗ dxj . Let [g ] be the inverse matrix of [gij ] (this deﬁnes ∗ the metric on T M), and |g| = det[gij ]. Then 1 ∂ p ij ∂f ∆gf = −p ∑ |g|g . |g| i,j ∂xi ∂xj A form ψ such that ∆ψ = 0 is called harmonic. Thus, a form satisfying dψ = d∗ψ = 0 is harmonic. On a compact manifold, we also have the converse: Lemma If M is compact, then any harmonic form ψ satisﬁes dψ = d∗ψ = 0.

Proof. “Integration by parts”: Z Z Z 0 = h∆ψ,ψidV = hdd∗ψ+d∗dψ,ψidV = |d∗ψ|2 + |dψ|2dV . M M M

Corollary A harmonic function on a compact Riemannian manifold is constant.

Let H k (M) be the vector space of harmonic k-forms, i.e. H k (M) = {ψ ∈ Ωk M; ∆ψ = 0}.

For M compact and oriented, let h,i denote the global inner product on Ωk M (given by integration).

Theorem (Hodge-de Rham) On a compact oriented manifold (M,g):

Ωk M = H k (M) ⊕ dΩk−1M ⊕ d∗Ωk+1M, where the summands are orthogonal with respect to the global inner product h,i.

Before discussing the proof, let’s see some applications: Corollary (Hodge isomorphism) k k The natural map f : H (M) → HDR(M), given by ψ 7→ [ψ] is an isomorphism.

Proof. We known that dψ = 0, so the map is well deﬁned. Since H is orthogonal to exact forms, the kernel of f is zero. Finally, let k H ∗ H [ω] ∈ HDR(M) and decompose ω = ω + dλ + d φ with ω harmonic. Then 0 = hdω,φi = hdd∗φ,φi = hd∗φ,d∗φi. ∗ H H Therefore d φ = 0 and [ω] = [ω + dλ] = ω , so f is surjective. Corollary (Poincare´ duality) On a compact oriented n-dimensional manifold M, k n−k HDR(M) ' HDR (M).

Proof. Put any metric on M. The Hodge dual ∗ gives an isomorphism k n−k H (M) ' H (M). On the proof of the Hodge-de Rham Theorem It is clear that the three summands H k (M), dΩk−1M, d∗Ωk+1M are mutually orthogonal: if ω is harmonic, then hω,dφi = hd∗ω,φi = 0. Similarly hω,d∗ψi = 0, and, ﬁnally: hdφ,d∗ψi = hddφ,ψi = 0. The hard part is to show that their sum is all of Ωk M. First of all, the space of smooth differential forms with the inner product h,i is not a Hilbert space, i.e. it is not complete. This is just as for C∞ functions on an interval - they do not form a complete linear topological space. In both cases, the completed space (equivalence classes of Cauchy sequences) consists of L2-integrable objects. But then, we cannot differentiate them.

Put in another way: after completing Ωk M, we can find element of smallest norm in the closure of every cohomology class. But why should such an element be a smooth differentiable form? The solution is to complete Ωk M not with respect to the norm kφk = hφ,φi1/2, but one that involves also integration of (covariant) derivatives up to high order s. Such a completion is called a Sobolev k space. Let us denote it by Ws M. It is a Hilbert space, and the k k Laplacian extends to a Fredholm operator ∆s : Ws (M) → Ws−2(M) (Fredholm = dimKer∆ < +∞, dimCoker∆ < +∞). Moreover Ker∆s = Ker∆, so that every “Sobolev class” harmonic form is actually smooth. ⊥ We have now a well-defined Y = (Ker∆s) and what we need is to show that Y ∩ Ωk M = dΩk−1M ⊕ d∗Ωk+1M. We have ∗ k k ∗ Y = ℑ∆s : Ws−2(M) → Ws (M), but on smooth forms, ∆s = ∆ (∆ is self-adjoint) so any smooth form ψ orthogonal to Ker∆ is in the image of ∆, i.e. ψ = ∆u = (dd∗ + d∗d)u = d(d∗u) + d∗(du). I hope that this rough outline gives some idea why the Hodge-de Rham theorem is true! We now wish to an analogous decomposition on a compact Hermitian manifold (M,h,J) using the operator ¯∂. Thus, we define the formal adjoint ¯∂∗ :Ωp,q+1M → Ωp,qM of ¯∂ and the complex Laplacian ¯∗¯ ¯¯∗ ∆¯∂ = ∂ ∂ + ∂∂ . The main points are: We have a hermitian inner product on Ωp,qM. There is a natural orientation on a complex manifold. The Hodge star maps Ωp,qM to Ωn−q,n−pM. As n is even, ∗2 = (−1)p+q. ¯∂∗ = − ∗ ∂∗.

The formula for ∆¯∂ in local coordinates is similar to the one given for the Riemannian Laplacian: if h = Re∑hij dzi ⊗ dzj , then (on functions) 1 ∂ p ij ∂f ∆¯∂f = −p ∑ |h|h . |h| i,j ∂zi ∂z¯j But note the crucial difference! ¯ A differential form φ, such that ∆¯∂φ = 0 is called ∂-harmonic. ¯ ¯ Just like for ∆g: if M is compact, then φ is ∂-harmonic iff ∂φ = 0 and ¯∂∗φ = 0.

We denote by H p,qM the space of ¯∂-harmonic forms of type (p,q). Theorem (Dolbeaut decomposition theorem) On a compact Hermitian manifold (M,h,J):

Ωp,qM = H p,q(M) ⊕ ¯∂Ωp,q−1M ⊕ ¯∂∗Ωp,q+1M, where the summands are orthogonal with respect to the global hermitian product h,i.

The proof is similar to that of the Hodge-de Rham theorem.; Again: Corollary (Dolbeaut isomorphism) The natural map H p,q(M) → Hp,q(M), given by ψ 7→ [ψ] is an isomorphism.

Corollary (Serre duality) On a compact complex n-dimensional manifold M, Hp,q(M) ' Hn−p,n−q(M).

p,q n−p,n−q Proof. ∗¯H (M) → H (M) is an isomorphism. Thus, on a Hermitian manifold (M,g,I) we have two Laplacians: the

Riemannian ∆h and the complex one ∆¯∂. In general, there is no relation between them; hence no relation between harmonic and ¯∂-haarmonic forms; hence no relation between De Rham and Dolbeault cohomology. A miracle of Kahler¨ geometry:

Proposition

If (M,h,J) is a Kahler¨ manifold, then ∆h = 2∆¯∂.

Proof. Both Laplacians, written in local coordinates, involve only ﬁrst derivatives of the metric. Thus, in normal Kahler¨ coordinates (complex coordinates in which the metric is Euclidean + O(|z|2) at a point p, the two Laplacians are ∆EUCL + O(|z|) and ∆¯∂−EUCL + O(|z|). Since the n Proposition holds in C , we have ∆h|p = 2∆¯∂|p and, hence, everywhere. On a compact Kahler¨ manifold we obtain now Hodge relations from this Proposition, the Hodge-De Rham and the Dolbeault decomposition theorems.