1. Kahler geometry
Á. Let Ì be a complex manifold of complex dimension ç and (ae ) = 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
= 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|
= 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
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