Introduction and Examples
A complex manifold is a paracompact Hausdorff space which has a covering by neighborhoods each homeomorphic to an open set in the m-dimensional complex number space such that where two neighborhoods overlap the local coordinates transform by a complex analytic transformation. That is, if z1,..., zm are local coordinates in one such neighborhood and if w1,..., wm are local coordinates in another neighborhood, then where they are both defined, we have wi = wi (z1,..., zm), where each wi is a holomorphic (or analytic) function of the z’s and the functional determinant ∂(w1,..., wm)/∂(z1,..., zm) ≠ 0.
KeywordsLine Bundle Holomorphic Function Complex Manifold Functional Determinant Gaussian Integer
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