About: Complex dimension

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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 (or of non-singular points in the case of a variety) 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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rdfs:label	Complex dimension (en)
rdfs:comment	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 (or of non-singular points in the case of a variety) 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 . (en)
dcterms:subject	Complex manifolds Dimension Algebraic geometry
Wikipage page ID	2783513 (xsd:integer)
Wikipage revision ID	1063216625 (xsd:integer)
Link from a Wikipage to another Wikipage	Complex manifolds Mathematics Complex manifold Complex projective space Dimension of an algebraic variety Dimension Hyperplane Algebraic geometry Codimension Smooth manifold Singular point of an algebraic variety Real algebraic variety Complex hypersurface
sameAs	Complex dimension Complex dimension Complex dimension Complex dimension Complex dimension
dbp:wikiPageUsesTemplate	dbt:Reflist dbt:Mathanalysis-stub
has abstract	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 (or of non-singular points in the case of a variety) 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 . 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 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. The same considerations 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. (en)
prov:wasDerivedFrom	wikipedia-en:Complex_dimension?oldid=1063216625&ns=0
page length (characters) of wiki page	3239 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Complex_dimension
is rdfs:seeAlso of	Dimension of an algebraic variety
is Link from a Wikipage to another Wikipage of	Andreotti–Frankel theorem Riemann surface E7 (mathematics) Pure spinor Geometric genus Calabi–Yau manifold Complex line Complex manifold Complex torus Fujita conjecture E6 (mathematics) E8 (mathematics) Fractal string Courant bracket Cohomology ring Real structure Yakov Eliashberg Dimension (disambiguation) Moishezon manifold Real dimension
is Wikipage redirect of	Real dimension
is Wikipage disambiguates of	Dimension (disambiguation)
is foaf:primaryTopic of	wikipedia-en:Complex_dimension

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