Difference between revisions of "Complex Dimensions"

Revision as of 14:30, 8 May 2024

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In mathematics, the concept of **complex dimension** usually refers to the dimension of a **complex manifold** or a **complex algebraic variety**.

1. Complex Manifold**:

  - A complex manifold is a space where local neighborhoods of points (or non-singular points in the case of a variety) are modeled on a Cartesian product of the form $$\mathbb{C}^d$$ for some positive integer $$d$$.
  - The **complex dimension** of a complex manifold is the exponent in this product.
  - Interestingly, because $$\mathbb{C}$$ (the set of complex numbers) can be modeled by $$\mathbb{R}^2$$ (the set of real numbers), a space with complex dimension will have a corresponding real dimension.
  - Specifically, if a smooth manifold has complex dimension $$d$$, its real dimension will be $$2d$$.

1. Complex Algebraic Variety**:

  - A complex algebraic variety is a geometric object defined by polynomial equations with complex coefficients.
  - Similar to complex manifolds, the complex dimension of a complex algebraic variety corresponds to half of its real dimension.
  - Away from any singular points, a complex algebraic variety will also be a smooth manifold of the same dimension.

1. Real vs. Complex Dimensions**:

  - For real algebraic varieties (varieties defined by equations with real coefficients), the term "dimension" commonly refers to its complex dimension.
  - The real dimension, in this context, refers to the maximum of the dimensions of the manifolds contained in the set of its real points.
  - For example, consider the equation $$x^2 + y^2 = 1$$, which defines a circle in the complex plane. This variety has a complex dimension of 1 (since it's a curve), but its real dimension is also 1 (since it's a smooth manifold).

In summary, the complex dimension provides insight into the geometric properties of spaces defined by complex numbers, and it's related to the real dimension through a factor of 2.

References

1. [Complex dimension - Wikipedia](https://en.wikipedia.org/wiki/Complex_dimension)
2. [Dimension - Wikipedia](https://en.wikipedia.org/wiki/Dimension) ²
3. [Complex dimension - Scientific Lib](https://www.scientificlib.com/en/Mathematics/LX/ComplexDimension.html) ³

Source: Conversation with Bing, 5/8/2024

(1) Dimension - Wikipedia. https://en.wikipedia.org/wiki/Dimension.
(2) Complex dimension - Scientific Lib. https://www.scientificlib.com/en/Mathematics/LX/ComplexDimension.html.
(3) Complex dimension - Wikipedia. https://en.wikipedia.org/wiki/Complex_dimension.