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Hodge Theory and Complex Algebraic Geometry I by Claire Voisin, Leila Schneps

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1

Holomorphic Functions of Many Variables

In this chapter, we recall the main properties of holomorphic functions of several complex variables. These results will be used freely in the remainder of the text, and will enable us to introduce the notions of a complex manifold, and a holomorphic function defined locally on a complex manifold.

The -valued holomorphic functions of the complex variables z1,..., zn are those whose differential is -linear, or equivalently, those which are annihilated by the operators . It follows immediately from this definition ...

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