Complex dimension

In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or a complex algebraic variety V.[1] If the complex dimension is d, the real dimension will be 2d.[2] That is, the smooth manifold M has dimension 2d; and away from any singular point V will also be a smooth manifold of dimension 2d.

However, for a real algebraic variety (that is a variety defined by equations with real coefficients), its dimension refers commonly to its complex dimension, and its real dimension refers to the maximum of the dimensions of the manifolds contained in the set of its real points. The real dimension is not greater than the dimension, and equals it if the variety is irreducible and has real points that are nonsingular. For example, the equation $x^{2}+y^{2}+z^{2}=0$ defines a variety of (complex) dimension 2 (a surface), but of real dimension 0 — it has only one real point, (0, 0, 0), which is singular.[3]

The same points apply to codimension. For example a smooth complex hypersurface in complex projective space of dimension n will be a manifold of dimension 2(n − 1). A complex hyperplane does not separate a complex projective space into two components, because it has real codimension 2.

References

Cavagnaro, Catherine; Haight, William T., II (2001), Dictionary of Classical and Theoretical Mathematics, CRC Press, p. 22, ISBN 9781584880509.
Marsden, Jerrold E.; Ratiu, Tudor S. (1999), Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17, Springer, p. 152, ISBN 9780387986432.
Bates, Daniel J.; Hauenstein, Jonathan D.; Sommese, Andrew J.; Wampler, Charles W. (2013), Numerically Solving Polynomial Systems with Bertini, Software, Environments, and Tools, 25, SIAM, p. 225, ISBN 9781611972702.

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[1] Cavagnaro, Catherine; Haight, William T., II (2001), Dictionary of Classical and Theoretical Mathematics, CRC Press, p. 22, ISBN 9781584880509.

[2] Marsden, Jerrold E.; Ratiu, Tudor S. (1999), Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17, Springer, p. 152, ISBN 9780387986432.

[3] Bates, Daniel J.; Hauenstein, Jonathan D.; Sommese, Andrew J.; Wampler, Charles W. (2013), Numerically Solving Polynomial Systems with Bertini, Software, Environments, and Tools, 25, SIAM, p. 225, ISBN 9781611972702.