Complex Differential Geometry

Roger Bielawski

July 27, 2009

Complex manifolds

A complex manifold of dimension m is a topological manifold (M,U),such that the transition functions φU φ

−1V are holomorphic maps

between open subsets of Cm for every intersecting U,V ∈U.

We have a holomorphic atlas (or “we have local complexcoordinates on M.”)

Remark: Obviously, a complex manifold of dimension m is a smooth(real) manifold of dimension 2m. We will denote the underlying realmanifold by MR.

Example

Complex projective space CPm - the set of (complex) lines in Cm+1,i.e. the set of equivalence classes of the relation

(z0, . . . ,zm)∼ (αz0, . . . ,αzm), ∀α ∈ C∗

on Cm+1−0.In other words CPm =

(Cm+1−0

)/∼.

The complex charts are defined as for RPm:

Ui = [z0, . . . ,zm];zi 6= 0 , i = 1, . . . ,m

φi : Ui → Cm, φi([z0, . . . ,zm]) =

(z0

zi, . . . ,

zi−1

zi,zi+1

zi, . . . ,

zm

zi

).

Example

1. Complex Grassmanian Grp(Cm) - the set of all p-dimensional vectorsubspaces 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 ofsubmersions f : Cm+1 → C. If f is holomorphic and df (theholomorphic differential) does not vanish at any point of f−1(c), thenf−1(c) is a holomorphic manifold.For example Fermat hypersurfaces:(

(z0, . . . ,zm);m

∑i=0

zdii = 1

), d0, . . . ,dm ∈ N.

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 realmanifold of (real) dimension 2m. Obviously, the converse is not true,but it turns out that there is a characterisation of complex manifoldsamong real ones, which is much simpler than the existence of aholomorphic atlas.

Identifying R2n with Cn is equivalent to giving a linear mapj : R2n → R2n satisfying j2 =−Id.Let x ∈M and let (U,φU) be a holomorphic chart around x .Define an endomorphism JU of TxMR by JU(X) = φ

−1U j φU(X).

JU does not depend on U, and so we have an endomorphismJx : TxMR → TxMR satisfying J2

x =−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 complexstructure on MR.

Definition

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

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

Definition

A smooth map f : (M1,J1)→ (M2,J2) between two complex manifoldsis called holomorphic if ψV f φ

−1U is a holomorphic map between

open subsets in Cn, for any charts (U,φU) in M1 and (V ,ψV ) inM2. This is equivalent to the differential of f commuting with thecomplex structures, i.e. f∗ J1 = J2 f∗.

The complexified tangent bundle

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

T CM = TM⊗R C,

and we extend all linear endomorphisms and linear differentialoperators 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 iseasy 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 atangent space of M at a point p: T R

p M = TM - the real tangent space,

T Cp M - the complexified tangent space, and T 1,0

p M (resp. T 0,1p M) - the

holomorphic tangent space (resp. antiholomorphic tangent space).Let us choose local complex coordinates z = (z1, . . . ,zn) near z. If wewrite zi = xi +

√−1yi , then:

TpM = R

∂

∂xi,

∂

∂yi

,

T Cp M = C

∂

∂xi,

∂

∂yi

= C

∂

∂zi,

∂

∂zi

,

where

∂

∂zi=

12

(∂

∂xi−√−1

∂

∂yi

),

∂

∂zi=

12

(∂

∂xi+√−1

∂

∂yi

).

Consequently:

T 1,0p M = C

∂

∂zi

, T 0,1

p M = C

∂

∂zi

,

and T 1,0p M (resp. T 0,1

p M) is the space of derivations which vanish onanti-holomorphic functions (resp. holomorphic functions).Now observe that:

V ,W ∈ Γ(T 1,0M

)=⇒ [V ,W ] ∈ Γ

(T 1,0M

),

andV ,W ∈ Γ

(T 0,1M

)=⇒ [V ,W ] ∈ Γ

(T 0,1M

).

The Newlander-Nirenberg Theorem

Theorem (Newlander-Nirenberg)

Let (M,J) be an almost complex manifold. The almost complexstructure J comes from a holomorphic atlas if and only if

V ,W ∈ Γ(T 0,1M

)=⇒ [V ,W ] ∈ Γ

(T 0,1M

).

An almost complex structure, which comes from complex coordinatesis called a complex structure.Remark: An equivalent condition is the vanishing of the Nijenhuistensor:

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 isvery hard. See Kobayashi & Nomizu for a proof under an additionalassumption that M and J are real-analytic, and Hormander’s”Introduction to Complex Analysis in Several Variables” for a proof infull generality.

Definition

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

In local coordinates Z = ∑ni=1 gi

∂

∂zi, where all gi are holomorphic.

Definition

A real vector field X ∈ Γ(TM) is called real holomorphic if its(1,0)-component X − iJX is a holomorphic vector field.

LemmaThe 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

Λ1,0M =

ω ∈ Λ1M⊗C; ω(Z ) = 0 ∀Z ∈ T 0,1M

andΛ0,1M =

ω ∈ Λ1M⊗C; ω(Z ) = 0 ∀Z ∈ T 1,0M

.

We haveΛ1M⊗C = Λ1,0M⊕Λ0,1M.

and, consequently, we can decompose the k -th exterior power ofΛ1M⊗C as

Λk M⊕C = Λk (Λ1,0M⊕Λ0,1M)

=M

p+q=k

Λp (Λ1,0M)⊗Λq (Λ0,1M

).

We writeΛp,qM = Λp (Λ1,0M

)⊗Λq (Λ0,1M

),

so thatΛk M⊕C =

Mp+q=k

Λp,qM.

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

dzi1 ∧·· ·∧dzip ∧dzj1 ∧dzjq .

Remark: The above decomposition of Λk M⊗C is valid on any almostcomplex manifold.The difference between ”complex” and ”almost complex” lies in thebehaviour of the exterior derivative d .

Theorem

Let (M,J) be an almost complex manifold of dimension 2n. Thefollowing 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 elementaryformula for the exterior derivative of a 1-form:

2dω(Z ,W ) = Z(ω(W )

)−W

(ω(Z )

)−ω([Z ,W ]).

Using statement (iii), we decompose the exterior derivatived : 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 = d2 = (∂+ ∂)2 = ∂2 + ∂2 +(∂∂+ ∂∂) and the three operators in thelast 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. Ap-form ω of type (p,0) is called holomorphic if ∂ω = 0.

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

Z p,q∂

(M) = ω ∈ Ωp,q(M); ∂ω = 0 − ∂-closed forms.

Hp,q∂

(M) =Z p,q

∂(M)

∂Ωp,q−1(M).

A holomorphic map f : M → N between complex manifolds induces amap

f ∗ : Hp,q∂

(N)→ Hp,q∂

(M).

Warning: Dolbeault cohomology is not a topological invariant: itdepends on the complex structure.

The ordinary Poincare lemma says that every closed form on Rn isexact, and, hence, that the de Rham cohomology of Rn or a ballvanishes. Analogously:

Lemma (Dolbeault Lemma)

For B a ball in Cn, Hp,q∂

(B) = 0, if p +q > 0.

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

Example (Dolbeault cohomology of P1=CP1)

First of all Ω0,2(P1) = Ω2,0(P1) = 0, since dimC P1 = 1. HenceH0,2

∂(P1) = H2,0

∂(P1) = 0. Also H0,0

∂(P1) = C.

Secondly, ∂Ω1,0 = dΩ1, and, hence H1,1∂

(P1) = C.

Thirdly, if ω ∈ Z 1,0∂

(P1) = 0, then dω = 0 and using the vanishing of

H1(S2), we conclude that H1,0∂

(P1) = 0. This also shows that there

are no global holomorphic forms on P1, i.e. Z 1,0∂

(P1) = 0.

Finally, we compute H0,1∂

(P1) = 0:

Complex and holomorphic vector bundles

Let M be a smooth manifold. A (smooth) complex vector bundle (ofrank k ) on M consists of a family of Exx∈M of (k -dimensional)complex vector spaces parameterised by M, together with a C∞

manifold structure on E =S

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 thereexists a diffeomorphism

φU : π−1(U)−→ U×Ck

taking the vector space Ex isomorphically onto x×Ck for eachx ∈ 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 linebundle.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∞-mapgUV : U ∩V → GL(k ,C) given by

gUV (x) =(φU φ

−1V

)|x×Ck .

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∞-mapsgαβ : Uα∩Uβ → GL(k ,C) satisfying these identities, there is a uniquecomplex 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 , exteriorpowers 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 functionsgαβ and hαβ, respectively, then the transition functions of E ⊕Fand E ⊗F are(

gαβ(x) 00 hαβ(x)

)∈GL

(Ck⊕Cl), gαβ(x)⊗hαβ(x)∈GL

(Ck⊗Cl).

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 ⊂ Exx∈M

of subspaces of the fibres Ex such that F =S

x∈M Fx is a submanifoldof E . This means that there are trivialisations of E , relative to whichthe transition functions look like

gUV (x) =

(hUV (x) kUV (x)

0 jUV (x)

).

The bundle F has transition functions hUV , while jUV are transitionfunctions of the quotient bundle E/F , given by Ex/Fx .A homomorphism between vector bundles E and F on M is given by aC∞-map f : E → F , such that fx = f|Ex : Ex → Fx and fx is linear. Wehave the obvious notions of ker(f ) - a subbundle of E , and Im(f ) - asubbundle of F . Also, f is an isomorphism, if each fx is anisomorphism.A vector bundle E on M is trivial, if E is isomorphic to the productbundle M×Ck .

Given a C∞-map f : M → N and a vector bundle Eπ→ 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 → Esuch that s(x) ∈ Ex for all x ∈M (just like a vector field). The space ofsections is denoted by Γ(E).Observe that trivialising a rankk bundle E over U is equivalent togiving k sections s1, . . . ,sk , which are linearly independent at everypoint 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 bundleE

π→M is defined as a complex vector bundle, except that C∞ isreplaced by “holomorphic” everywhere.Examples: T 1,0M, Λp,0M (but not Λp,qM if q 6= 0). The line bundleΛn,0M, n = dimC M is called the canonical bundle of M, and is denotedby KM . Its sections are holomorphic n-forms. The dualK ∗

M = Λn(T 1,0M

)is the anti-canonical bundle.

Example (The tautological and canonical bundles on CPn)

The tautological line bundle J → CPn is defined by setting JA = A,where A is a line in Cn+1 defining a point of CPn.We have KCPn = Jn+1.

For every holomorphic bundle E →M we define the bundlesΛp,qE = Λp,qM⊗E of E-valued forms of type (p,q). The space ofsections is denoted by Ωp,qE . If we choose a trivialisation of E , i.e. alocal 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 wehave 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∂(fσ) = ∂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 = Λ0,0Eis a remarkable property of holomorphic vector bundles. On a generalcomplex vector bundle there is no canonical way of differentiatingsections. It has to be introduced ad hoc, via the concept of connection:

Connections and their curvature

DefinitionA connection (or covariant derivative) on a complex vector bundleE →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 = ∑j

θijej ,

where the matrix of 1-forms is called the connection matrix withrespect to e.The data e and θ determine D. The connection matrix depends on thechoice of e: if (e′1, . . . ,e

′k) is another frame with e′(z) = g(z)e(z),

g(z) ∈ GL(k ,C), then:

θe′ = gθeg−1 +dgg−1.

Curvature

One can extend any connection D to act on ΩpE , i.e. on E-valuedp-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(fσ) = D(df ⊗σ+ f (Dσ)

)=−df ∧Dσ+df ∧Dσ+ fD2

σ = fD2σ.

Consequently, D2 is induced by a bundle map E → Λ2M⊗E , i.e. by asection of Λ2M⊗Hom(E ,E). This Hom(E ,E)-valued 2-form RD iscalled the curvature of the connection D.The curvature matrix. In a local frame, we had Dei = ∑j θijej , and,hence:

D2ei = D(∑θijej

)= ∑

(dθij −∑θip ∧θpj

)⊗ej .

Therefore, we get the Cartan structure equations for the curvaturematrix with respect to a frame e:

Θe = dθe−θe∧θe.

Hermitian metrics

DefinitionLet E →M be a complex vector bundle. A hermitian metric on E is asmoothly varying hermitian inner product on each fibre Ex , i.e. ifσ = (s1, . . . ,sk) is a frame for E , then the functionshij(x) = 〈si(x),sj(x)〉 are C∞.

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

D is compatible with the metric, i.e.

d〈s1,s2〉= 〈Ds1,s2〉+ 〈s1,Ds2〉,i.e. the metric is parallel for D, andD is compatible with the complex structure, i.e. if we writeD = D1,0 +D0,1 with D1,0 : Γ(E)→ Ω1,0E andD0,1 : Γ(E)→ Ω0,1E , then D0,1 = ∂.

TheoremIf E →M is a hermitian vector bundle over a complex manifold, thenthere is unique connection D (called the Chern connection) compatiblewith both the metric and the complex structure.

Proof.

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

dhij = 〈Dei ,ej〉+ 〈ei ,Dej〉= ∑p

θiphpj +∑p

θpjhip.

The first term is of type (1,0) and the second one of type (0,1) and,hence:

∂hij = ∑p

θiphpj , ∂hij = ∑p

θpjhip.

Therefore ∂h = θh and ∂h = hθT and θ = ∂hh−1 is the unique solutionto both equations.

Remark on the curvature of the Chern connection. If D iscompatible with the complex structure, then the (0,2)-component ofthe 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

0 = d2〈ei ,ej〉= 〈D2ei ,ej〉+2〈Dei ,Dej〉+ 〈ei ,D2ej〉

and so the (2,0)-component of D2 is the negative hermitian transposeof 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 anyframe 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,and e is a holomorphic frame, then we computed: θe = ∂hh−1, wherehij = 〈ei ,ej〉. We compute dθe:

(∂+ ∂)θe = ∂θe +∂(∂hh−1) = ∂θe−∂h∧∂(h−1) = ∂θe +∂hh−1∧∂hh−1.

Therefore, the curvature matrix of the Chern connection w.r.t. aholomorphic frame is given by

Θ = ∂θ.

In the case of a line bundle, a local non-vanishing holomorphic sections and h = 〈s,s〉:

θ = ∂ logh, Θ = ∂∂ logh.

Example (Curvature of the tautological bundle of CPn)

J is a subbundle of the trivial bundle Pn×Cn+1:

J = l× z ∈ Pn×Cn+1; z ∈ l.

We have a hermitian metric on Pn×Cn+1 given by the standardhermitian inner product on Cn+1. It induces a hermitian metric on theline bundle J. We compute the curvature of the associated Chernconnection in the patch U0 (z0 6= 0), in which J is trivialised viaφ : Cn×C→ J: (

(z1, . . . ,zn),u)7−→ u(1,z1, . . . ,zn).

We have a non-vaninshing section s(z1, . . . ,zn) = (1,z1, . . . ,zn), and,consequently, h = 〈s,s〉= 1+∑

ni=1 |zi |2. Thus, the curvature is

∂∂ logh. =n

∑i=1

1+∑r 6=i |zr |2

(1+∑nr=1 |zr |2)2 dzi ∧dzi .

Hermitian metrics on complex manifolds

We now look at the particular case E = T 1,0M (M - a complexmanifold). A choice of holomorphic frame is given by a choice of localholomorphic coordinates z1, . . . ,zn (ei = ∂

∂zi), and a hermitian metric

on T 1,0M can be written as

g = ∑i,j

hijdzi ⊗dzj .

where hij is hermitian matrix, h∗ = h. Notice that this also defines aHermitian metric on T CM, and by restriction a C-valued hermitianmetric on TM.Its real part is a real-valued symmetric bilinear form, which we alsodenote by g, and refer to as hermitian metric on M. Observe thatg(JX ,JY ) = g(X ,Y ) ∀X ,Y ∈ TxM.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 ), andin local coordinates:

g =12

Re∑i,j

hijdzidzj ,

ω =

√−12 ∑

i,jhijdzi ∧dzj .

Observe that ω ∈ Ω1,1M.

Example (Connection and curvature in dimension one)

z = x + iy - a local coordinate, ∂

∂z - a local holomorphic frame, and ahermitian metric on T 1,0M is written as hdz⊗dz, for a local functionh > 0. The connection matrix of the Chern matrix is ∂hh−1 = ∂ logh

∂z dz,and the curvature matrix is

Θ = ∂∂ logh =∂2 logh∂z∂z

dz ∧dz =(−1

4∆ logh

)dz ∧dz.

Now, the fundamental form on M is√−12 hdz ∧dz and hence,

Θ =−√−1K ω,

where K = (−∆ logh)/2h is the usual Gaussian curvature of asurface.

Recall the different behaviour of positively and negatively curvedsurfaces. Similarly, we say that the curvatureRE ∈ Γ

(Λ1,1M⊗Hom(E ,E)

)of a Chern connection on a hermitian

vector bundle is positive at x ∈M (notation RE(x) > 0), if thehermitian matrix RE(x)(v , v) ∈ Hom(Ex ,Ex) is positive definite. Inlocal 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. ifthe curvature is positive everywhere. Observe that the curvature of thetautological bundle JPn (with induced metric) is negative.Let us compute the curvature RF of the Chern connection of aholomorphic subbundle F ⊂ E (with the induced hermitian metric). LetN = F⊥. This is a C∞ complex subbundle of E .If s ∈ Γ(F) and t ∈ Γ(N), then:

0 = d〈s, t〉= 〈Ds, t〉+ 〈s,Dt〉,

so that in a frame for E consisting of frames for F and N, theconnection matrix of D = DE is

θE =

(θF A−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

ΘE = dθE −θE ∧θE =

(dθF −θF ∧θF +A∧A∗ ∗

∗ ∗

),

and so ΘF = ΘE |F −A∧A∗. Therefore

RF ≤ RE |F ,

so that the curvature decreases in holomorphic subbundles. Inparticular, if E is a trivial bundle with Euclidean metric (so that RD ≡ 0)and F ⊂ E is a holomorphic subbundle with the induced metric, thenRD

F ≤ 0.Applying this to a submanifold M of Cn and F = T 1,0M ⊂ T 1,0Cn

∣∣M

with the induced hermitian metric, we see that the curvature of T 1,0Mis always non-positive. In particular, if M is a Riemann surface, then itsGaussian curvature K ≤ 0.

Observe that the same calculation for the quotient bundle Q = E/Fand 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 sectionss1, . . . ,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 ,(x ,λ) 7→ ∑

ki=1 λisi(x).

Thus, E is a quotient bundle of a trivial bundle and, if we give E themetric induced from the Euclidean metric on M×Ck , then RQ ≥ 0.

The first Chern class

We go back to a non-holomorphic setting and consider complex vectorbundles over smooth manifolds. We equip such a bundle E →M witha connection D. Recall that the curvature RD of D is a section ofΛ2M⊗Hom(E ,E), so basically a matrix of 2-forms. We can speak ofthe 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 isa closed 2-form. It is called the Ricci form of D.

Lemma

The cohomology class[trRD

]∈ H2

DR(M) does not depend on D.

Proof.

A = D−D′ is a section of Λ1M⊗Hom(E ,E), so trRD − trRD′= d trA,

which is an exact form.

The cohomology class c1(E) =√−1

2π

[trRD

]∈ H2

DR(M) is called the1st Chern class of E . It is a topological invariant.

Example

We compute c1(JP1). Recall that for the Chern connection inducedfrom P1×C2, we computed the curvature matrix in the chart U0 withholomorphic coordinate z as

1

(1+ |z|2)2 dz ∧dz.

Now, H2DR(P1) is identified with C via integration: ω 7→

RP1 ω. Thus

c1(JP1) =

√−1

2π

ZC

1

(1+ |z|2)2 dz∧dz =1

2π

Z[0,2π]×[0,∞)

r

(1+ r2)2 dθ∧dr ,

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

c1(JP1) =−12π

Z +∞

0

Z 2π

0

r

(1+ r2)2 dθdr =−1.

It follows from this that c1(JPn) =−1 for any n.

Let E and F be bundles of ranks m and n, respectively. Then:

(i) c1(ΛmE) = 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 isc1(T 1,0M

)= c1(K ∗), i.e. the first Chern class of the anti-canonical

bundle of M.

Example

c1(Pn) = c1(K∗) = c1

((J∗)⊗(n+1)

)= (n +1)c1(J

∗) = n +1.

For n = 1, we get c1(P2) = 2.

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

c1(S) =1

2π

ZS

K ω = χ(S).

Examples of manifolds with c1(M) = 0:Observe that c1(M) = 0, if the canonical bundle KM is trivial, i.e. thereexists 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.gby lattices: abelian varieties (complex tori).

The quadric Q = (z1,z2,z3) ∈ C3; z21 + z2

2 + z23 = 1. This is a

complexification of S2 and so H2(Q) 6= 0. The followingholomorphic 2-form is non-vanishing on Q and trivialises KQ :

z1dz2∧dz3 + z2dz3∧dz1 + z3dz1∧dz2.

The famous K3-surface (one of them, anyway):S = [z0,z1,z2,z3] ∈ P3; z4

0 + z41 + z4

2 + z43 = 0.

We’ll show how to compute c1 of hypersurfaces. Let V ⊂M be acomplex submanifold of dimension 1. We have the exact sequence ofholomorphic vector bundles over V :0→ N∗

V → Λ1,0M|V → Λ1,0V → 0.Taking the maximal exterior power gives: KM |V ' KV ⊗N∗

V , so

KV = KM |V ⊗NV .

Now, observe that, if f = 0 is a local (say, on U ⊂ V ) defining equationfor V , then, by definition, df generates N∗

V over U.If, on U ′ ⊂ V , the equation of V is f ′ = 0, then on U ∩U ′:

df = d( f

f ′f ′)

= d( f

f ′)f ′+

ff ′

df ′ =ff ′

df ′,

so the transition functions for N∗V are f/f ′.

We can now easily finish the computation for K3: N∗S ' KP3 |S .

Higher Chern classes

The first Chern class of a complex vector bundle was defined using thetrace of the curvature RD ∈ Γ

(Λ2M⊗Hom(E ,E). We can use other

invariant polynomials defined on matrices, e.g. the determinant or thesum of the squares of eigenvalues.For a k × k matrix M, let us write

det(t +M) =k

∑i=0

Pi(M)tk−i .

so that P1(M) = trM, Pk(M) = detM, etc. Each Pi is a homogeneouspolynomial of degree k in the entries of A.We can allow matrices over any commutative algebra - the Pi(M) arewell defined - in particular over the even part of the exterior algebra ofa smooth manifold. Applying this to RD or its curvature matrix, weobtain closed forms Pi(RD) = Pi(Θ) ∈ Ω2iM.

Definition

The i-th Chern class of a complex vector bundle E over a smoothmanifold M is the cohomology class

ci(E) =

[Pi

(√−1

2πRD)]

∈ H2iDR(M), i = 1, . . . ,k .

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 theChern classes of T 1,0M.The Gauss-Bonnet theorem generalise to higher dimensions asfollows: 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 complexmanifold M. Recall that the curvature of a Chern connection D is oftype (1,1) and that its matrix is skew-hermitian, so that Pi

(√−12π

RD)

isa real (i, i)-form, and consequently, for any holomorphic bundle

ci(E) ∈ H i,i(M)∩H2iDR(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

Pi(RD)(v1, v1, . . . ,vi , vi))≥ 0 ∀v1, . . . ,vi ∈ T 1,0M,

and consequently that ZZ

ci(E)≥ 0

for any i-dimensional complex submanifold Z of M. In particular, thisholds for any holomorphic vector bundle generated by its sections.Note that these conditions depend only on ci(E), not on theconnection or its curvature.

Prescribing the Ricci curvature of Chern connections

Thus, we say that c1(E)≥ 0 (resp. > 0,< 0,≤ 0), if the cohomologyclass c1(E) ∈ H1,1(M) can be locally represented by a closed real2-form φ, such that in local complex coordinates

φ =

√−1

2π∑φ

αβdzα∧dzβ,

and the hermitian matrix [φαβ

] has all eigenvalues nonnegative (resp.positive, negative, nonpositive).E.g., any holomorphic vector bundle generated by its sections satisfiesc1(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 byφ≡ 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 〈,〉 be a hermitian metric on E . Recall that in a local holomorphicframe (e1, . . . ,en) with the associated matrix hij = 〈ei ,ej〉, we had thefollowing formula for RD:

Θ = ∂(∂hh−1).

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

∂∂(logdeth).

Now we modify 〈,〉 by taking ef/k〈,〉, where k = rankE and f is asmooth real function on M. Let h′ij = ef/k〈ei ,ej〉. Then deth′ = ef dethand the Ricci forms of the two Chern connections are related by

trRD′− trRD = ∂∂f .

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 to0:[φ− trRD

]=−2π

√−1(c1(E)− c1(E)).

Therefore the answer to our question: can we find a hermitian metricon E , the Ricci form of which is a given form φ, is yes on manifolds Mfor 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+ τ′, where τ is (1,0) and τ′ is (0,1). We musthave τ′ = τ.Now, β = (∂+ ∂)(τ+ τ) = ∂τ+(∂τ+∂τ)+ ∂τ. Comparing the types,we get:

β = ∂τ+∂τ, ∂τ = 0, ∂τ = 0.

Since H0,1(M) = 0, there exists a function u : M → C such thatτ = ∂u. Therefore τ = ∂u, andβ = ∂τ+∂τ = ∂∂u +∂∂u = ∂∂(u− u) = 2i∂∂(Imu).

The result follows with f = 2Imu.

Kahler metricsWe now consider the case E = TM, where M is a complex manifoldwith a complex structure J. TM is a complex vector bundle - J allowsus to identify TxM with Cn, 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 productg on each TxM, such that g(JX ,JY ) = g(X ,Y ) for all X ,Y ∈ TxM.We will now consider connections on TM. First, of all observe, that forsuch a connection D : Γ(TM)→ Ω1(M)⊗Γ(TM), the (covariant)directional derivative DZ (X) := D(X)(Z ) of a vector field is again avector field.We now reinterpret the two properties defining the Chern connectionD of g:1. D being metric, i.e. dg(X ,Y ) = g(DX ,Y )+g(X ,DY ), is equivalentto

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 = ∂, isequivalent 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 matrixin the holomorphic frame has type (1,0)).On the other hand, since g is a Riemannian metric, i.e. a smoothlyvarying inner product, there exists a unique connection (theLevi-Civita connection) ∇ on TM, which is again metric and it has zerotorsion, i.e.

∇X Y −∇Y X = [X ,Y ], ∀X ,Y ∈ Γ(TM).

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

Theorem

Let g be a hermitian metric on a complex manifold (M,J). Thefollowing 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 thatω(X ,Y ) = g(JX ,Y )).

(v) For each point p ∈M, there exists a smooth real function in aneighbourhood of p, such that ω = i∂∂f .

(vi) For each point p ∈M, there exist holomorphic coordinates wcentred at p, such that g(w) = 1+O(|w |2).

Proof. (i), (ii), and (iii) are clearly equivalent from the uniquenessproperties. We show (i) =⇒ (iv) =⇒ (v) =⇒ (vi) =⇒ (i).

(i) =⇒ (iv). Since ∇g = 0 and ∇J = 0, ∇ω = 0. For any p-form andany torsion-free connection ∇ one has (exercise):

dω(X0, . . . ,Xp) =p

∑i=0

(−1)i(∇Xi ω

)(X0, . . . , Xi , . . . ,Xp),

which implies that parallel forms are closed.

(iv) =⇒ (v). This is the ∂∂-lemma applied to a ball B in Cn

(Dolbeault Lemma says that H0,1(B) = 0).

(v) =⇒ (vi). We can choose local coordinates around p0, so thatω = i ∑l,m ωlmdzl ∧dzm, where

ωlm =12

δlm +∑j(ajlmzj +bjlmzj)+O(|z|2).

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

ajlm =∂3f

∂zj∂zl∂zm,

so ajlm = aljm. We now put

wm = zm +∑j,l

ajlmzjzl , =⇒ dwm = dzm +2∑j,l

ajlmzjdzl ,

and compute:

12

i ∑m

dwm∧dwm =12

i ∑m

dzm∧dzm + i ∑j,l,m

ajlmzjdzl ∧dzm

+ i ∑j,l,m

ajlmzjdzm∧dzl +O(|z|2)

= i ∑l,m

ωlmdzl ∧dzm +O(|z|2) = ω+O(|z|2) = ω+O(|w |2).

(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, ∇Jvanishes at p0, and, since p0 is arbitrary, it vanishes everywhere.

A hermitian metric on a complex manifold satisfying the equivalentconditions (i)− (vi) is called a Kahler metric.

The fundamental form ω of a Kahler metric is called a Kahler form andthe local function f in (v) is called a local Kahler potential.

Examples:1. The standard Euclidean metric on Cn.

g =12

Ren

∑s=1

dzs⊗dzs, ω =i2

Ren

∑s=1

dzs ∧dzs =i2

∂∂|z|2.

Thus f (z) = 12 |z|

2 is a global Kahler potential on Cn. The pair (g,J) isU(n)-invariant.

2. The Fubini-Study metric on CPn.For a z = (z0, . . . ,zn) ∈ Cn+1−0, 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 correspondingsymmetric form g(X ,Y ) = ω(X ,JY ) is positive-definite. We computefirst ω in the chart U0 = z0 6= 0, where the local coordinates arewi = zs

z0, s = 1, . . . ,n:

ω = i∂∂ log(1+ |w |2) = ∂

(i

1+ |w |2n

∑s=1

wsdws

)=

i1+ |w |2

n

∑s=1

dws∧dws−i

(1+ |w |2)2

(n

∑s=1

wsdws

)∧

(n

∑s=1

wsdws

)

Observe that ω is U(n +1)- invariant, and so it is enough to checkpositivity 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 onCPn ' Sn+1/S1 obtained from the standard round metric on Sn+1.

These two basic examples (Cn and CPn) yield plenty of others,because:

LemmaA complex submanifold of a Kahler manifold, equipped with theinduced metric, is Kahler.

Proof. Let (N,J,gN)⊂ (M,J,gM) as in the statement. Thefundamental form ωN of gN is just the pullback (restriction) of thefundamental 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 Kahlermetric, because:

Lemma

If M is a compact Kahler manifold, then H2qDR(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:

vol(M) =Z

Mω

n =Z

Md(ψ∧ω

n−q) = 0,

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 ΩqMinduces an injective map

Hq,0∂

(M) → HqDR(M,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 unitaryframe φi as η = ∑ fIφI (I runs over subsets of cardinality q and φI isthe wedge product of φj with j ∈ I). Then η∧ η = ∑ fI fJφI ∧ φJ .

On the other hand, ω =√−12 ∑φi ∧ φi , and (n = dimC M):

ωn−q = cq ∑

#K=n−q

φK ∧ φK .

Henceη∧ η∧ω

n−q = c′q ∑#I=q

|fI |2ωn,

since the only non-zero wedge-products arise for K disjoint from I andJ (so for I = J). In particular, if η 6= 0, then the integral over M of theLHS is non-zero. If, however, η = dψ, then :

η∧ η∧ωn−q = d

(ψ∧ η∧ω

n−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 thatη 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 goesunder the name of Hodge relations or Hodge decomposition:

Theorem (Hodge)

The complex cohomology of a compact Kahler manifold satisfies:

H r (M,C) =M

p+q=r

Hp,q(M), Hp,q(M) = Hq,p(M).

In particular: for r odd, dimH r (M) is even.

To give even an idea of a proof of this theorem of Hodge requires asubstantial detour.

Before, let us see how badly this theorem fails for non-compact Kahlermanifolds:

Example

M = C2−0, so that H1DR(M) = 0. We shall show that

dimH0,1∂

(M) = +∞. Let U1 = z1 6= 0= 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 andlet f be a holomorphic function on U1∩U2. Then g1 = λ2f defines asmooth function on U1 and g2 =−λ1f defines a smooth function onU2. Observe that on U1∩U2, ∂(g1−g2) = ∂f = 0, and so we candefine a (0,1)-form ω on M, with ∂ω = 0, by

ω =

∂g1 = f ∂λ2 on U1

−∂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 isholomorphic on U2. Then f = g1−g2 = (g1−h)− (g2−h), and,hence, f = u1 +u2, where ui is holomorphic on Ui , i = 1,2.But any convergent Laurent series f on U1∩U2, which contains zm

1 zn2 ,

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 〈,〉. Thenthere is an induced inner product on the tensor powers V⊗k . Byrestriction, there is an inner product on the exterior powers Λk V . Itcan be described by choosing an orthonormal basis (e1, . . . ,en) anddecreeing 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 TxM. Since a dualof a vector space with an inner product also has an inner product, weget an inner product on Λk T ∗

x M, and, if (M,g) is oriented, i.e. we havea nonvanishing volume form dV , and compact, then we can define aninner product on differential forms in Ωk M:

〈α,β〉=Z

M〈α|x ,β|x〉dV .

The idea is that in every cohomology class in HkDR(M) we seek a

representative α with the smallest norm.

We also show that such a representative must be unique (an elementwith the smallest norm in a closed affine subspace of a Hilbert space).We do the same for the Dolbeault cohomology classes Hp,q

∂(M).

Finally, we show that, if M is Kahler, then, if we decompose theα ∈ Hk

DR(M,C) with minimal norm into forms of type (p,q), p +q = k ,then each component has the minimal norm in Hp,q

∂(M).

How to find the element of smallest norm in a cohomology class[ψ] ∈ Hk

DR(M)?We are looking for the element of smallest norm in an affine subspaceP = ψ+dη; η ∈ Ωk−1M. For M compact, Ωk(M) with the innerproduct 〈,〉 is a pre-Hilbert space (essentially an L2-space), and wereP a closed subspace, we could find an element of the smallest normby the orthogonal projection (Ωk(M) = dΩk−1M⊕ (dΩk−1M)⊥).The orthogonal projection, on the other hand, can be expressed viathe adjoint operator to d :

‖ψ+dη‖2 = ‖ψ‖2 +‖dη‖2 +2〈ψ,dη〉= ‖ψ‖2 +‖dη‖2 +2〈d∗ψ,η〉.

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 thatdψ = 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 avector space, dimV = n, equipped with an inner product 〈,〉 and anorientation, i.e. a preferred element of ΛnV , 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→ 〈ω,τ〉 for someτ ∈ Λk V . This way, we get a 1-1 correspondence between Λk V andΛn−k V :

∗ : Λk V → Λn−k V , τ 7→ α,

ω∧∗τ = 〈ω,τ〉e1∧·· ·∧en.

The operator ∗ is called the Hodge dual. Notice, in particular:

∗1 = e1∧·· ·∧en, 〈∗τ1,∗τ2〉= 〈τ1,τ2〉, ∗2 = (−1)k(n−k) on Λk V .

Again, we obtain such a ∗ on differential forms of an orientedRiemannian manifold (M,g).Example: Take M = R3 with the Euclidean metric g = ∑

3i=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 thevolume form determining orientation):

〈dα,β〉dV = dα∧∗β = d(α∧∗β)− (−1)kα∧d ∗β, so

〈dα,β〉dV −d(α∧∗β) =−(−1)kα∧d ∗β

=−(−1)k(−1)k(n−k)α∧∗2d ∗β =−(−1)kn〈α,∗d ∗β〉dV .

Therefore, the operator d∗ = (−1)kn+1 ∗d∗ : Ωk+1 → Ωk is the adjointof d , and is called the codifferential. A form ω, such that d∗ω = 0 iscalled co-closed.

We need to show that on a compact oriented manifold (M,g), anycohomology 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. Theoperator

∆ = 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:

(dd∗+d∗d)f = d∗df =−∗d ∗(

∑∂f∂xj

dxj

)=−∗d

(∑

∂f∂xj

∗dxj

)=−∗

(∑

∂2f∂xi∂xj

dxi ∧∗dxj

)=−∗

(∑

∂2f∂x2

i

)dV = ∆f .

In general, let a Riemannian metric be given in local coordinates byg = ∑ij dxi ⊗dxj . Let [g ij ] be the inverse matrix of [gij ] (this definesthe metric on T ∗M), and |g|= det[gij ]. Then

∆g f =− 1√|g|∑i,j

∂

∂xi

(√|g|g ij ∂f

∂xj

).

A form ψ such that ∆ψ = 0 is called harmonic. Thus, a formsatisfying dψ = d∗ψ = 0 is harmonic. On a compact manifold, wealso have the converse:

Lemma

If M is compact, then any harmonic form ψ satisfies dψ = d∗ψ = 0.

Proof. “Integration by parts”:

0 =Z

M〈∆ψ,ψ〉dV =

ZM〈dd∗ψ+d∗dψ,ψ〉dV =

ZM

(|d∗ψ|2 + |dψ|2

)dV .

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 〈,〉 denote the global inner product onΩk 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 innerproduct 〈,〉.

Before discussing the proof, let’s see some applications:

Corollary (Hodge isomorphism)

The natural map f : H k(M)→ HkDR(M), given by ψ 7→ [ψ] is an

isomorphism.

Proof. We known that dψ = 0, so the map is well defined. Since H isorthogonal to exact forms, the kernel of f is zero. Finally, let[ω] ∈ Hk

DR(M) and decompose ω = ωH +dλ+d∗φ with ωH harmonic.Then

0 = 〈dω,φ〉= 〈dd∗φ,φ〉= 〈d∗φ,d∗φ〉.Therefore d∗φ = 0 and [ω] = [ωH +dλ] = ωH , so f is surjective.

Corollary (Poincare duality)

On a compact oriented n-dimensional manifold M,Hk

DR(M)' Hn−kDR (M).

Proof. Put any metric on M. The Hodge dual ∗ gives an isomorphismH k(M)'H n−k(M).

On the proof of the Hodge-de Rham TheoremIt is clear that the three summands H k(M), dΩk−1M, d∗Ωk+1M aremutually orthogonal: if ω is harmonic, then 〈ω,dφ〉= 〈d∗ω,φ〉= 0.Similarly 〈ω,d∗ψ〉= 0, and, finally:

〈dφ,d∗ψ〉= 〈ddφ,ψ〉= 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 innerproduct 〈,〉 is not a Hilbert space, i.e. it is not complete. This is just asfor C∞ functions on an interval - they do not form a complete lineartopological space. In both cases, the completed space (equivalenceclasses of Cauchy sequences) consists of L2-integrable objects. Butthen, we cannot differentiate them.

Put in another way: after completing Ωk M, we can find element ofsmallest norm in the closure of every cohomology class. But whyshould such an element be a smooth differentiable form?

The solution is to complete Ωk M not with respect to the norm‖φ‖= 〈φ,φ〉1/2, but one that involves also integration of (covariant)derivatives up to high order s. Such a completion is called a Sobolevspace. Let us denote it by W k

s M. It is a Hilbert space, and theLaplacian extends to a Fredholm operator ∆s : W k

s (M)→W ks−2(M)

(Fredholm = dimKer∆ < +∞, dimCoker∆ < +∞). MoreoverKer∆s = Ker∆, so that every “Sobolev class” harmonic form isactually smooth.We have now a well-defined Y = (Ker∆s)

⊥ and what we need is toshow that Y ∩Ωk M = dΩk−1M⊕d∗Ωk+1M. We haveY = ℑ∆∗

s : W ks−2(M)→W k

s (M), but on smooth forms, ∆∗s = ∆ (∆ is

self-adjoint) so any smooth form ψ orthogonal to Ker∆ is in the imageof ∆, 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-deRham theorem is true!

We now wish to an analogous decomposition on a compact Hermitianmanifold (M,h,J) using the operator ∂. Thus, we define the formaladjoint ∂∗ : 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 forthe Riemannian Laplacian: if h = Re∑hijdzi ⊗dzj , then (on functions)

∆∂f =− 1√

|h|∑i,j∂

∂zi

(√|h|hij ∂f

∂zj

).

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 globalhermitian product 〈,〉.

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 anisomorphism.

Corollary (Serre duality)

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

Proof. ∗H p,q(M)→H n−p,n−q(M) is an isomorphism.

Thus, on a Hermitian manifold (M,g, I) we have two Laplacians: theRiemannian ∆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 andDolbeault 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 firstderivatives of the metric. Thus, in normal Kahler coordinates (complexcoordinates in which the metric is Euclidean + O(|z|2) at a point p, thetwo Laplacians are ∆EUCL +O(|z|) and ∆

∂−EUCL +O(|z|). Since theProposition holds in Cn, we have ∆h|p = 2∆

∂|p and, hence,everywhere.

On a compact Kahler manifold we obtain now Hodge relations fromthis Proposition, the Hodge-De Rham and the Dolbeaultdecomposition theorems.