1

0 0

(9.29)

1 20 1

0 0

1

Thus, the nonzero Betti numbers of K3 are b0 = b4 = 1, b2 = 22, and the

Euler characteristic is χ = 24. The 22 nontrivial harmonic two-forms consist

of three self-dual forms (b+ = 3) and 19 anti-self-dual forms (b’ = 19).

2 2

9.3 Examples of Calabi“Yau manifolds 369

Calabi“Yau n-folds

The complete classi¬cation of Calabi“Yau n-folds for n > 2 is an unsolved

problem, and it is not even clear that the number of compact Calabi“Yau

three-folds is ¬nite. Many examples have been constructed. Here we mention

a few of them.

Submanifolds of complex projective spaces

Examples of a Calabi“Yau n-folds can be constructed as a submanifold of

P n+1 for all n > 1. Complex projective space, P n , sometimes just de-

£ £

noted P n , is a compact manifold with n complex dimensions. It can be

constructed by taking n+1 /{0}, that is the set of (z 1 , z 2 , . . . , z n+1 ) where

£

the z i are not all zero and making the identi¬cations

(z 1 , z 2 , . . . , z n+1 ) ∼ (»z 1 , »z 2 , . . . , »z n+1 ), (9.30)

for any nonzero complex » = 0. Thus, lines7 in n+1 correspond to points £

in P n . £

P n is a K¨hler manifold, but it is not a Calabi“Yau manifold. The sim-

a

£

plest example is P 1 , which is topologically the two-sphere S 2 . Obviously,

£

it does not admit a Ricci-¬‚at metric. The standard metric of P n , called the £

Fubini“Study metric, is constructed as follows. First one covers the manifold

by n + 1 open sets given by z a = 0. Then on each open set one introduces

local coordinates. For example, on the open set with z n+1 = 0, one de¬nes

wa = z a /z n+1 , with a = 1, . . . , n. Then one introduces the K¨hler potential

a

(for this open set)

n

|wa |2 .

K = log 1 + (9.31)

a=1

This determines the metric by formulas given in the appendix. A crucial

requirement is that the analogous formulas for the K¨hler potential on the

a

other open sets di¬er from this one by K¨hler transformations. You are

a

asked to verify this in a homework problem.

Examples of Calabi“Yau manifolds can be obtained as subspaces of com-

plex projective spaces. Speci¬cally, let G be a homogenous polynomial of

degree k in the coordinates z a of n+2 , that is, £

G(»z 1 , . . . , »z n+2 ) = »k G(z 1 , . . . , z n+2 ). (9.32)

P n+1 de¬ned by

The submanifold of £

G(z 1 , . . . , z n+2 ) = 0 (9.33)

7 A line in a complex manifold has one complex dimension.

370 String geometry

is a compact K¨hler manifold with n complex dimensions. This submanifold

a

has vanishing ¬rst Chern class for k = n+2. One way of obtaining this result

is to explicitly compute c1 (X). To do so note that c1 (X) can be expressed

through the volume form since X is K¨hler. As a volume form on X one can

a

use the pullback of the (n ’ 1)-power of the K¨hler form of CP n+1 . Another

a

way of obtaining this result is to use the adjunction formula of algebraic

geometry, which implies

c1 (X) ∼ [k ’ (n + 2)] c1 (£ P n+1 ). (9.34)

This vanishes for k = n + 2.

P 3 ) one obtains K3 mani-

• In the case of n = 2 (quartic polynomials in £

folds. As an example consider

4

(z a )4 = 0, (9.35)

a=1

as a quartic equation representing K3. Di¬erent choices of quartic poly-

nomials give K3 manifolds that are di¬eomorphic to each other but have

di¬erent complex structures. Deformations of Calabi“Yau manifolds, in

particular deformations of the complex structure, are discussed in Sec-

tion 9.5.

• In the case of n = 3 this construction describes the quintic hypersurface

in P 4 . This manifold can be described by the polynomial

£

5

(z a )5 = 0, (9.36)

a=1

or a more general polynomial of degree ¬ve in ¬ve variables. This manifold

has the Hodge numbers

h1,1 = 1 h2,1 = 101,

and (9.37)

which gives an Euler number of χ = ’200. Varying the coe¬cients of the