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In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points are modeled on a Cartesian product of the form for some , and the complex dimension is the exponent in this product. Because can in turn be modeled by , a space with complex dimension will have real dimension . That is, a smooth manifold of complex dimension has real dimension ; and a complex algebraic variety of complex dimension , away from any singular point, will also be a smooth manifold of real dimension .
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