canonical metrics on complex manifold

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Canonical Metrics

on Complex

Manifold

University of Michigan

Shing-Tung Yau

Harvard University

1



Complex manifolds are topological spaces that

are covered by coordinate charts where the

coordinate changes are given by holomorphic

transformations. Riemann surfaces are one

dimensional complex manifolds.

In order to understand complex manifolds, it

is useful to introduce metrics that are com-

patible with the complex structure. In gen-

eral, we should have a pair (M,ds2M ) where

ds2

M is the metric.

2



We say that the metric is canonical if any bi-

holomorphisms of the complex manifolds are

automatically isometries. Such metrics can

then be used to describe invariants of the

complex structures of the manifold.

The first important examples of such metrics

were constructed by Poincare for Riemann

surfaces with genus greater than one.

Note that the flat metric on the torus is not

quite canonical unless we require the biholo-

morphisms to preserve the area. (In higher

dimensions, this is the same as preserving the

Kahler class.)

3



Poincare’s metrics are metrics with constant

negative curvature. It requires a proof of the

existence theorem for conformal deformation

of metrics. It is a nonlinear differential equa-

tion and hence difficult higher dimensional

problem. Generalizations of Poincare’s work

took some time.

So far we have discussed Riemannian metrics.

But there are other types of metrics.

4



Caratheodory metric:

This is a metric on a Riemann surface X con-

structed by the following procedure:

Given a tangent vector v at a point p ∈ X ,

we define

v = supf ∗(v) : f is a holomorphic map from

X to the unit disk equipped

with the Poincare metric

5



In the 1930s, Ahlfors proved a powerful gen-

eralization of the Schwarz lemma in complex

analysis.

Theorem. Holomorphic maps from the Poincare

disk to a Riemann surface with curvature less

than -1 is distance decreasing.

Grauert-Reckziegel generalized Ahlfors-Schwarz

lemma in the 1960s for maps into higher di-

mensional complex manifold.

6



Based on these works, Kobayashi introduced

the concept of the Kobayashi metric.

Kobayashi metric:

For a complex tangent vector v at a point p

on a complex manifold, consider all holomor-

phic maps f : D → M such that f (0) = p and

f ∗(a ∂ ∂z ) = v. Define

v = inf f

|a|

It is easy to see that the Kobayashi metric

dominates the Caratheodory metric.

7



Both metrics satisfy a stronger property than

the canonical metric condition mentioned above.

If f : M → N is holomorphic, then f decreases

distance in the sense that

f ∗ds2N ≤ ds2

M .

These canonical metrics are not Riemannian

metrics, in general.

In 1970, Royden proved a remarkable theo-

rem that on the Teichmuller space, the Kobayashi

metric is the same as the Teichmuller met-

ric. He also deduced that the group of au-

tomorphisms of the Teichmuller space is the

mapping class group.

8



Bergman metric:

Stefan Bergman introduced another intrin-

sic metric. He looked at the space of holo-

morphic n-form f dz1 ∧ · · · ∧ dzn that are L2-

integrable. Note that the inner product

|f |2dz1 ∧ · · · ∧ dzndz1 ∧ · · · ∧ dzn

makes good sense.

With respect to such an inner product, we

can find an orthonormal basis f 1 dz1 ∧ · · · ∧dzn, f 2 dz1 ∧ · · · ∧ dzn, . . . , and Bergman de-

fined a metric (called the Bergman metric)

by

√ −1∂ ∂ log

|f i|2

This metric is canonically defined if there are

enough L2 holomorphic n-forms.

9



Kobayashi observed that the Bergman met-

ric is the same as the induced metric from an

embedding of the manifold into the complex

projective space. The embedding is given by

the holomorphic n-forms which are orthonor-

mal.

Many years ago, Lu in China proved that the

Bergman metric dominates the Caratheodory

metric. On the other hand, there is no clear

relationship between the Bergman metric and

the Kobayashi metric.

10



Definition. A complex manifold M n is called

holomorphic homogeneous regular if there are

positive constants r < R such that for each

point p ∈ M , there is a one to one holomor-

phic map f : M → Cn such that

1. f ( p) = 0.

2. Br ⊂ f (M ) ⊂ BR where Br and BR are

balls with radius r and R , respectively.

The Bers embedding theorem says that the

Teichmuller space is holomorphic homoge-

neous regular with (r, R) = (2, 6).

11



Theorem. (Liu, Sun, Yau. S.K.Yeung) For

holomorphic homogeneous regular manifolds,

the Bergman metric, the Kobayashi metric,

and the Caratheodory metric are equivalent.

The main estimate behind this theorem and

the definition of Kobayashi metric and Caratheodor

metric is the Schwarz lemma, which in turn

depends on the negativity of the holomorphic

sectional curvature.

12



Perhaps a more convenient definition is the

following:

Let P (T ∗(M )) be the projectified cotangent

bundle of M , and let O(1) be the canonical

holomorphic line bundle over P (T ∗(M )). If

O(1) admits a metric with positive curvature,

then M would admit a Finsler metric with

negative holomorphic sectional curvature.

In general, we do not expect O(1) > 0, but

there can exist a line bundle L > 0 over

P (T ∗(M )) such that for some m > 0 , (m O(1))L−1

admits a non-trivial holomorphic section. This

condition gives rise to a Finsler metric which

may degenerate along some divisor on P (T ∗(M )).

But the holomorphic sectional curvature is

strongly negative.

13



Hence, I proposed the following definition to

replace the Caratheodory metric.

Definition. For any tangent vector v ∈ T p(M ),

define

v = supvD : where D is a Finsler metric

which may degenerate along a divisor

in the tangent bundle of M and

the holomorphic sectional curvature

of

D is less than

−1

14



Based on a theorem of Bogomolov, Lu and

I observed that for an algebraic surface M

such that c21 > c2, the metric exists and

that either

(1) there is a divisor D ⊆ M such that for

every holomorphic map f : C → M , the

image f (C) ⊆ D ; or

(2) there is an algebraic foliation with singu-

larity on M such that f maps C to a leaf

of such foliation.

15



If one assumes the stronger condition, thatof c2

1 > 2 c2, then an argument of Miyaoka

allows one to conclude that the above con-

dition (2) is unnecessary.

This says that Lang’s conjecture holds for

algebraic surface with c21 > 2 c2.

The metric defined above can be made to

be more algebraic geometric if one uses the

metric obtained by the projective embedding

for L where (m O(1))L−1 admits a section.

The Kobayashi metric always dominates the

metric I just defined. However, it is a much

more transcendental object.

16



However we can propose metric which is more

algebraic than the Kobayashi metric. Namely

in the definition of Kobayashi metric, we can

replace the Poincare disc by an algebraic curve

equipped with a metric whose curvature is

equal to

−1.

In this way, we obtain a more algebraic ge-

ometric definition of the Kobayashi metric

while part of the properties of the Kobayashi

metrics are kept.

On the other hand, the definition is good

only when the manifold is algebraic so that

we have plenty of algebraic curves.

Note that in all of the above definitions, there

is corresponding definitions for canonical vol-

ume forms. It is interesting to find relation-

17



ship between canonical volumes with canon-

ical metrics.



So far, the canonical metrics we reviewed

have no clear properties for the curvature.

Without such information, it is difficult to ap-

ply tools from Riemannian geometry to study

complex structures of the manifold.

The only resonable curvature constrain is the

Ricci tensor of the metric. The natural equa-

tion is the Einstein equation.

Namely, the Ricci tensor should be a multi-

ple of the metric. These metrics are called

Kahler-Einstein metrics.

18



The existence of such metrics was posed by

Calabi. For the case where the first Chern

class is negative, this was proved indepen-

dently by Aubin and Yau. For the zero first

Chern class case, the existence was proved

by me and such metrics has been especially

important for string theory.

These metrics are canonical. But we need to

generalize the concept of canonical metric inthe following way.

If f : M → N is biholomorphic and if ωM

and ωN are the Kahler forms of the Ricci flat

metric of M and N respectively such that

f ∗ωN is cohomologous to ωM , then f is an

isometry.

19



I have applied the existence of such Kahler-

Einstein metrics to questions in algebraic man-

ifolds. For examples:

(1) global rigidity of complex sturcture on

CPn;

(2) algebraic geometric criterions for a man-

ifold to be a Shimura variety.

20



For c1 > 0, the problem of the existence of a

Kahler-Einstein metric has not been solved.

The conjecture that I made twenty-five years

ago gives the guiding principle: for such man-

ifolds, the existence of a Kahler-Einstein met-

ric is equivalent to the stability of the mani-

fold.

Donaldson has made the most important con-

tribution. They are related to a proposalthat I made on how to approximate Kahler-

Einstein metrics by suitably defined Bergman

metrics. The stability of the manifold is re-

lated to the embedding of the manifold intoa complex projective space by high multiples

of a line bundle. In the present case, we use

the canonical line bundle.

21



The embedding gives rise to the Bergman

metric. The proof of the approximation was

done in Tian’s thesis, using ideas from Siu-

Yau.

When we embed the manifold, it is important

to be able to make the embedding balanced

and this was observed by my student Lo in his

thesis. The balanced condition here can be

interpreted to be related to the moment map

and was used by Donaldson to understand

the related concept of stability.

22



When we deform the complex structure of a

manifold, one can form a metric on the mod-

uli space using the Kahler-Einstein metric as

a background metric. This metric is called

the Weil-Petersson metric.

For a Riemann surface, the Weil-Petersson

metric is not complete and is very differentfrom the other metrics. It turns out that mi-

nus of its Ricci tensor defines a Kahler met-

ric which is equivalent to the Kahler-Einstein

metric and the Bergman metric (work of Liu,

Sun, and Yau).

23



Liu, Sun, and Yau have also proved that theKahler-Einstein metric is good in the sense of

Mumford on the Deligne-Mumford compact-

ification of the moduli space. In particular,

the moduli space is log stable.

Kahler-Einstein metrics can also be general-

ized to manifolds with singularities. However,

the manifold has to be Kahler. In dimension

greater than two, there are many complex

manifolds which are not Kahler. I conjecture:

Every almost complex manifold with dimen-

sion greater than two admit an integrable

complex structure.

24



When a manifold is not Kahler, it is difficult

to find canonical metric that are useful. Here,

physics can provide some help.

In 1986, Strominger proposed the following

system of equations for a bundle V

defined

on a complex manifold M which admits a

holomorphic three-form Ω.

(1) F h ∧ ω2

= 0

(2) F 2,0h = F

0,2h = 0

(3) ∂ ∂ω = √ −1tr(F h ∧ F h) − √ −1 trRg ∧ Rg

(4) d∗ω =√ −1(∂ − ∂ )log Ωω

25



Above, h is a Hermitian metric on V and ωis a Hermitian metric on M .

Equation (1) and (2) simply means V satis-

fies the Hermitian-Yang-Mills condition. Equa-

tion (4) means that the metric is conformally

balanced. And (3) is the anomaly equation

required for quantum consistency.

For a general complex manifold, the balanced

metric is a Hermitian metric given by the Her-

mitian form ω =√ −1

gi jdzi ∧ dz j such that

d(ωn

−1) = 0 .

This class of metrics was first studied by

Michelsohn in early 1980s.

26



It is difficult to solve Strominger’s four equa-

tions. But we have made progress in the pasttwo years.

Below I present two types of solutions to

Strominger’s system:

1. Li and Yau: On a Calabi-Yau manifold,

the equations can be solved by perturb-

ing the Calabi-Yau metric and vector bun-

dles.

2. Fu and Yau: On the non-Kahler mani-

fold, T

2

bundle over K 3, a metric ansatzcan be used and the delicate estimates

required for a smooth solution have been

obtained.

27



I. Li-Yau: Perturbation method

Let E s be a smooth family of holomorphic

vector bundles over a Calabi-Yau space X .

Let h0 be a Hermitian-Yang-Mills connection

on E 0.

We would like to extend h0 to a smooth fam-

ily of Hermitian-Yang-Mills connections.

28



The interesting case is when h0 is reducible.

Let (X, ω0) be Kahler.

Let (E 1, D1) and (E 2, D

2) be degree zero and

slope-stable vector bundles.

Let h1 and h2 be the Hermitian metrics on

E 1 and E 2 respectively.

Then h1⊕ et h2 is still a Hermitian metric that

corresponds to the connection D0 = D1⊕D2.

29



Suppose we are given a deformation of holo-

morphic structure Ds of D

0. Then Kodaira-

Spencer identifies the first order deformationof D

s at 0 to an element

k ∈ H 1(X, ε∗ ⊗ ε)

where ε is the sheaf of holomorphic section s

of (E, D0).

Therefore

k

∈ ⊕2i,j=1H 1(ε∗

i

⊗ε j).

30



Theorem. Suppose k12 and k21 are nonzero.

Then there is a unique t so that for s suffi-

ciently small h0(t) = h1 ⊕ eth2 extends to a

smooth family of Hermitian-Yang-Mills met-rics on (E, D

s ).

31



The fourth equation of Strominger system is

equivalent to

dΩωω2

= 0 .

Let H(X ) be the cone of positive definite

Hermitian form on X .

Let H(E )0 be the space of determinant one

Hermitian metric on the bundle E (i.e. the

induced metric on ∧rE CX is the constant

one metric).

32



We define

L = L1

⊕L2

⊕L3 :

H(E )0 × H(X ) −→Ω3,3(End0E ) ⊕ Im

√ −1∂ ∂ ⊕ Im d∗0

where

L1(H, ω) = √ −1F H ∧ ω2

L2(H, ω) =√ −1∂ ∂ω + trE (F H ∧ F H )

− trT (Rg ∧ Rg)

L3(H, ω) = ∗0d Ωωω2

.

33



We shall apply the implicit function theorem

to L.

Fix a determinant one Hermitian metric

,

on E . We can write other determinant one

Hermitian metric on E by a unique positive

definite , Hermitian symmetric endomor-

phism H of Z satisfying det H = 1.

Such spaces H will be denoted by Γ(End+h E ),

identity I ∈ Γ(End+h E ).

The tangent space at I is Γ(End0hE ) traceless

symmetric endomorphisms of E .

34



δL1(I, ω0)(δh,δω)

= DDH δh + 2F H ∧ ω0 ∧ δω


=√

−1∂ ∂ (δω) + 2(trE δF I (δh)

∧ F I )

− trT δRg0(δg) ∧ Rg0


= 2d∗0

(δω)−

d∗0

((δω,ω0

)ω0

).

35



We can construct irreducible solutions to Stro-

minger’s system perturbatively.

Start with a Calabi–Yau manifold,

(E, D0) = C

⊕(r−3)X

⊕T X ,

the metric is identified with I : E −→ E .

For all c > 0, (I , c ω0) is a solution to L = 0.

36



Let

W 1 = Ω3,3R

(End0hE )L

pk−2

W 2 = (Im√ −1 ∂ ∂ )L p

k−2⊕ (Im d∗

0)L pk−1

V 0 =

A ⊕ aI T X | A ∈ EndC

⊕(r−3)X

are constant matrices such that

A = A−t, tr A + 3a = 0

V 1 = ω3

0 ⊗ V 0.

Then ∃ C > 0 such that for all c > C ,

δL1(I,cω0) ⊕ δL2(I,cω0) ⊕ δL3(I,cω0)

: Γ(End0

hE )

L pk ⊕

Ω1,1(X

)−→ W 1/V 1 ⊕ W 2

is surjective.

37



Theorem. Let X be a Calabi-Yau three-fold

with ω a Ricci-flat Kahler form. Let Ds be

a smooth deformation of holomorphic struc-

ture D0 on E = CX ⊕T X . Suppose the associ-

ated cohomology classes [C 12] and [C 21] are

non-zero. Then for sufficiently large c , there

is a family of pairs of Hermitian metrics and

Hermitian forms (H s, ωs) for 0 ≤ s < ε such

that

1. ω0 = c ω and the harmonic part of ωs is

equal to c ω.

2. The pair (H s, ωs) is a solution to Stro-

minger’s system for the holomorphic vec-

tor bundle (E, Ds ).

38



Let

Ds =D

0 + A3, As ∈ Ω0,1(End E )

A0 =

C 11 C 12C 21 C 22

∈ Ω0,1(End E ).

We can assume C ij are D0 harmonic. Since

H 1(X, OX ) = 0, C 11 = 0.

39



In general, we consider the r +3 holomorphic

vector bundle C⊕rX ⊕ T X . We also have

D0

= 0 C 12

C 21 C 22

where

C 12 =(α1, . . . , αr)t ∈ Ω0,1(T X )⊕ j

C 21 =(β 1, . . . , β r)

∈Ω0,1(T rX )

⊕ j

C 22 ∈ Ω0,1(End T X ).

Suppose [α1], . . . , [αr] ∈ H 1(X, T ∗X ) are lin-

early independent and [β 1], . . . , [β r] ∈ H 1(X, T ∗X )

are linearly independent. Then the above

theorem holds.

40



Example

Consider

X = z50 + · · · + z5

x = 0 ∈ P4

0 0

0 −→ T X −→ T X P4 −→ OX (5) −→ 0

0 −→ F −→ OX (1)⊕5 −→ OX (5) −→ 0 OX OX

0 0

Here F is the cokernel of OX

(1)⊕

5

−→ OX (5)

and fill in

0 −→ OX −→ F −→ T X −→ 0.

41



The above sequence is a non-split extension.

Making use of this element in Ext1(T X , OX )

we can perform a deformation of the holo-

morphic structure Dt with C 12 = 0 and C 21 = 0.

Hence we have proved:

Let X be a smooth quintic three-fold and ω

be any Kahler form on X . Then for large c >

0, there is a smooth deformation of CX ⊕ T X

such that for small s, there are pairs (H s, ωs)

of Hermitian metrics on E and Hermitian forms

ωs on X that solves Strominger’s system.

42



For the Calabi–Yau manifold with three gen-

erations that I constructed:

X ⊂ P3 × P3

given by

x3

i = 0

y

3

i = 0

xiyi = 0

quotient by Z3. One can also construct ir-

reducible solution to Strominger’s system on

T X ⊕ C⊕2X .

43



II. Fu-Yau: Non-Kahler manifolds

Let (S, ωS , ΩS ) be the K3 surface. Let ω12π ,ω2

2π

∈H 2(S,Z) and let ω1 and ω2 be anti-self-dual

(1,1)-forms. Then there is a non-Kahler man-

ifold X such that π : X → S is a holomorphic

T 2 bundle over S .

If we write locally ω1 = dα1 and ω2 = dα2,

then there are coordinates of the T 2 fiber, x

and y, such that dx + π∗α1 and dy + π∗α2 are

globally defined 1-forms on X .

44



Let

θ = dx + π∗α1 +√ −1(dy + π∗α2) .

Then the Hermitian form on X is

ω0 = π∗ωS + √ −12

θ ∧ θ

and the holomorphic 3-form is

Ω = π∗ΩS ∧ θ .

ω0 satisfies the fourth equation d( Ω ω0 ω20) = 0.

Let u be any smooth function on S and let

ωu = π∗(euωS ) +

√ −1

2 θ ∧ θ .

Then ωu is a Hermitian metric on X and

(ωu, Ω) also satisfies d( Ω ωu ω2u) = 0.

45



If we let Ru be the curvature of the Hermitian

connection of the metric ωu on the holomor-

phic tangent bundle, then

tr Ru ∧ Ru =tr RS ∧ RS + 2 ∂ ∂u ∧ ∂ ∂u

+2√ −1 ∂ ∂ (e−uf ωS ),

where for ω2 = n ω1, n ∈ Z, f = 1+n2

4 ω12ωS

.

So the third equation in Strominger’s system

can be reduced to√ −1∂ ∂eu ∧ ωS − 2 ∂ ∂u ∧ ∂ ∂u − 2

√ −1 ∂ ∂ (e−uf ωS )

=tr RS ∧

RS −

tr F H ∧

F H −

(

ω1

2 +

ω2

2)ω2

S

2!.

(1)

46



Let (E, H ) be the Hermitian-Yang-Mills vec-

tor bundle over S with the gauge group SU (r).

Then (V = π∗E, H ) is also the Hermitian-

Yang-Mills vector bundle over X . We can

consider equation (1) as the equation on the

K 3 surface S . Integrating equation (1) over

S , S trRS ∧ RS − trF H ∧ F H

= S

(

ω1

2ωS

+

ω2

2ωS

)ω2

S

2!

.

We use Q( ωi2π) to denote the intersection num-

ber of anti-self-dual (1,1)-form ωi2π. As

18π2 S trRS ∧ RS = 24, the above condition

can be written as

2(24 − c2(E )) = −

Q

ω1

2π

+ Q

ω2

2π

.

(2)

47



Certainly we can choose ω1 and ω2 and SU (r)

vector bundle E such that they satisfy the

condition (2). Then there is a smooth func-

tion µ such that

trRS ∧ RS − trF H ∧ F H

−ω12 + ω22

ω2S

2! = −µ

ω2S

2!.

So we obtain the following equation:

√ −1∂ ∂eu ∧ ωS − ∂ ∂u ∧ ∂ ∂u

− √ −1∂ ∂ (e−uf ωS ) + µω2

S

2! = 0 .

(3)

which can be rewritten as the standard com-

plex Monge-Ampere equation:

∆(e−u − f e−u) + 8det ui j

det gi j

+ µ = 0 .

48



We solve equation (3) by the continuity method

and get

Theorem. Equation (3) has a smooth solu-

tion u such that

ω = (eu + f e−u)ωS + 2√

−1∂ ∂u

is a Hermitian metric on S .

The requirement that ω is hermitian is an

elliptic condition that we need to impose.

49



Theorem. Let S be a K 3 surface with Calabi-

Yau metric ωS . Let ω1 and ω2 be anti-self-

dual (1, 1)-forms on S such that ω12π ∈ H 2(S,Z)

and ω22π ∈ H 2(S,Z). Let X be a T 2-bundle

over S constructed by ω1 and ω2. Let E bea stable bundle over S with the gauge group

SU (r). Suppose ω1, ω2 and c2(E ) satisfy the

topological constraint (2). Then there exists

a smooth function u and a Hermitian-Yang-Mills metric H on E such that (V = π∗E, π∗F H , X , ωu

is a solution of Strominger’s system.

We will use the normalization condition

S

e−4u ω2S

2!

14

= A ,

S 1

ω2S

2! = 1 .

50



Zeroth order estimate

Let P = 2gi j ∂ 2

∂zi∂ z j. We have two methods of

calculating

S P (eku)

det gi j

det gi j

ω2S

2! .

Then using the Sobolev inequality, Moser it-

eration and Poincare inequality, we obtain

Proposition. If A < 1, then there is a con-

stant C 1 which depends on f , µ and the

Sobolev constant of ωS such that

inf S

u ≥ − ln(C 1A).

Moreover, if A is small enough such thatA < (C 1)−1, then there is an upper bound

of supS u which depends on f , µ, Sobolev

constant of ωS and A.

51



An estimate of the determinant

We need to estimate the lower bound of the

determinant

F =det g

i j

det gi j

.

We apply the maximum principle to the func-tion

G = 1 − e−u | u |2 +2e−u − 2e− inf u

and obtain

52



Proposition. Given any constant κ ∈ (0, 1),

we fix some positive constant satisfying

< min1, 2−1κ.

Suppose that A satisfies

A < min1, C −11 , 2(1 + sup f )−1

2C −11 ,

1

−κ

2C 3

1

C −1

1 ,

3

−6

C 4 C −1

1 , C 5,

where C 3 and C 4 depend on f and µ, C 4 also

depends on the curvature bound of ωS , and

C 5 depends on κ, and C 3. Then F > κe2u ≥

κ(C 1A)−2.

53



Second order estimate

Since

eu + f e−u + u ≥ F 12 > κ

12(C 1A)−1 > 0,

it is sufficient to have an upper estimate of

eu + f e−u + u. Applying the maximum prin-

ciple to the function

e−λ1u+λ2|u|2 · (eu + f e−u + u),

where λ1 and λ2 are positive constants that

can be determined, we can obtain the second

order estimate.

54



Third order estimate

Let

Γ = gi jgklu,iku, jl

Θ = girgs jgktu,i jku,rst

Ξ = gi jgklg pqu,ikpu, jlq

Φ = gi j

gkl

g pq

grs

u,ilpru, jkqs

Ψ = gi jgklg pqgrsu,ilpsu, jkqr,

where indices preceded by a comma indicate

covariant differentiation with respective to

the metric ωS .

55



Again, we apply the maximum principle to

the function

(κ1+u)Θ+κ2(m+u)Γ+κ3 | u |2 Γ+κ4Γ,

where all κi are positive constants that can

be determined and m is a fixed constant such

that m + u > 0. We can then obtain the

third order estimate.

56



Topology of the total space:

(1) h0,1(X ) = h0,1(S ) + 1

(2) h1,0(X ) = h1,0(S )

(3) b1(X ) = b1(S ) + 1

(4a) b2(X ) = b2(S ) − 1

if ω1 is a multiple of ω2

(4b) b2(X ) = b2(S ) − 2

if ω1 is not a multiple of ω2 .

canonical metrics on complex manifold

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