1. Kahler geometry

Á. Let Ì be a complex manifold of complex dimension ç and (ae ) (ë = 1, 2, — , ç)

a system of complex local coordinates on an open subset U of M. Let

zk = ÷ë + ß / .

Then ( Ä 1 , y 1 , · · · , xn, yn) is a system of (real) local coordinates of the differentiable

manifold Ì on U. For each ÷ G U, we define a linear transformation / of the tangent

vector Space Ô (Ì) by the condition

/

ldxA

1

dy*

ldyÄ dxA

(ë = 1, 2, · · · , ç).

Then / satisfies the condition I2 = - id and the assignment I:x —* I defines a ten-

X X ° X

sor field of type (1, 1) on the differentiable manifold M. We call / the tensor of com-

plex structure of M. Let Ô {Ì) and Ô (Ì) be the complexifications of the tangent

X X

í

m

_m\

and the cotangent vector Spaces respectively. We define the elements d/dz , d/dz

and dzX, dz~X of Ô (M)C and T*(M)C by

dz"

1_

~2

+ é

dxA dyÄ

dzk = dxk-+ i dyX, dzk= dol·- i dy^.

The endomorphism / of the vector Space Ô (Ì) defines naturally the endomor-

phism of the complex vector Spaces Ô (Ì) and we have

(1) TxiMc=T*(B Ê,Ô^Ã,

where Ô (resp, T") consists of all u € Ô (Ì) such that / u - iu (resp. / u = - iu)

jC

X X

X.

. X

and ~~ denotes the conjugation of Ô (M)C. The elements \d/dzË1, \d/dzË|

(ë = 1, 2, · · · , ç) form bases of Ô and Ã" respectively at each point ÷ of the coorr

dinate neighbourhood V.

Á complex vector field X on Ì is a map which assigns to each point % of Ì an

element X of Ô (Ì) with an obvious differentiability condition. On a coordinate

neighbourhood U, we can express a complex vector field X uniquely in the form

1