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