Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009...

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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 φ U φ -1 V are holomorphic maps between open subsets of C m 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 M R .

Transcript of Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009...

Page 1: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 2: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 3: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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∗.

Page 4: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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

).

Page 5: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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 ].

Page 6: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 7: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 8: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 9: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 10: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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 .

Page 11: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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:

Page 12: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 13: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 14: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 15: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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 ,

Page 16: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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

Page 17: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 18: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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

[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

ZC

1

(1+ |z|2)2 dz∧dz =1

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.

Page 19: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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

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.

Page 20: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 21: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 22: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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)).

Page 23: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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).

Page 24: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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):

Page 25: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 26: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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!

Page 27: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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

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.

Page 28: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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:

Page 29: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 30: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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 .

Page 31: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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

).

Page 32: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 33: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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!

Page 34: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.

Page 35: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such

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.