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

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2

Complex Manifolds

In this chapter, we introduce and study the notion of a complex structure on a differentiable or complex manifold. A complex manifold X of (complex) dimension n is a differentiable manifold locally equipped with complex-valued coordinates (called holomorphic coordinates) z1,. .., zn, such that the diffeomorphisms from an open set of n to an open set of n given by coordinate changes are holomorphic. By the definition of a holomorphic transformation, we then see that the structure of a complex vector space on the tangent space ...

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