The manifold de¬ned by Eq. (9.36) can be covered by ¬ve open sets for

which z a = 0, a = 1, . . . , 5. On the ¬rst open set, for example, one can

de¬ne local coordinates w a = z a /z 1 , a = 2, 3, 4, 5. These satisfy

5

(wa )4 dwa = 0. (9.38)

a=2

9.3 Examples of Calabi“Yau manifolds 371

In terms of these coordinates the holomorphic three-form is given by

dw2 § dw3 § dw4

„¦= . (9.39)

(w5 )4

Note that Eq. (9.39) seems to single out one of the coordinates. However,

taking Eq. (9.38) into account one sees that the four coordinates w a ,

a = 2, . . . , 5, are treated democratically.

Weighted complex projective space: W P n1 ···kn+1

k £

One generalization entails replacing P n by weighted complex projective

£

n

space W Pk1 ···kn+1 . This n-dimensional complex space is de¬ned by starting

£

with n+1 and making the identi¬cations8

£

(»k1 z 1 , »k2 z 2 , . . . , »kn+1 z n+1 ) ∼ »N (z 1 , z 2 , . . . , z n+1 ), (9.40)

where k1 , . . . , kn+1 are positive integers, and N is their least common multi-

ple. Further generalizations consist of products of such spaces with dimen-

sions ni . One can impose m polynomial constraint equations that respect

the scaling properties of the coordinates. Generically, this produces a space

ni ’ m complex dimensions. Then one has to compute the ¬rst

with

Chern class, which is not so easy in general. Still this procedure has been

automated, and several thousand inequivalent Calabi“Yau three-folds have

been obtained. Other powerful techniques, based on toric geometry, which is

not discussed in this book, have produced additional examples. Despite all

this e¬ort, the classi¬cation of Calabi“Yau three-folds is not yet complete.

EXERCISES

EXERCISE 9.1

Show that up to normalization J § J § J is the volume of a compact six-

dimensional K¨hler manifold. Consider ¬rst the case of two real dimensions.

a

SOLUTION

¯

Here J = iga¯dz a §d¯b is the K¨hler form, which is discussed in the appendix.

z a

b

This result has to be true because h3,3 = 1 for a compact K¨hler manifold

a

in three complex dimensions. J § J § J, which is a (3, 3)-form, must be

8 Note that the » s have exponents and the z s have superscripts.

372 String geometry

proportional to the volume form (up to an exact form), since it is closed but

not exact. Still, it is instructive to demonstrate this explicitly. So let us do

that now.

For one complex dimension (or two real dimensions) the K¨hler form is

a

J = igz z dz § d¯, where z = x + iy. The metric components then take the

z

¯

form

gxx = gyy = 2gz z , gxy = 0.

¯

The K¨hler form describes the volume, V =

a J, since

√

J = igz z dz § d¯ = 2gz z dx § dy = gdx § dy.

z

¯ ¯

¯

This argument generalizes to n complex dimensions, where J = iga¯dz a §d¯b .

z

b

√

Setting z a = xa + iy a and using g = 2n det ga¯, one obtains for n = 3

b

√

1

J § J § J = gdx1 § · · · § dy 3 ,

6

which is the volume form. Thus,

1

J § J § J.

V=

6

2

EXERCISE 9.2

Consider the orbifold T 2 —T 2 / 3 , where 3 acts on the coordinates of T 2 —T 2

by (z 1 , z 2 ) ’ (ωz 1 , ω ’1 z 2 ), where ω = exp(2πi/3) is a third root of unity,

and (z 1 , z 2 ) are the coordinates of the two tori. Compute the cohomology of

M , including the contribution coming from the ¬xed points. Compare the

result to the cohomology of K3.

SOLUTION

In order for the 3 transformation to be a symmetry, let us choose the

complex structure of the tori such that the periods are

z a ∼ z a + 1 ∼ z a + eπi/3 a = 1, 2.

The 3 action has nine ¬xed points where each of the z a takes one of the

following three values:

1 2

√ eπi/6 , √ eπi/6 .

0,

3 3

9.3 Examples of Calabi“Yau manifolds 373