Real structure

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In mathematics, several different mathematical objects have a notion of what a real structure on them is.

[edit] Real structure of a Vector Space

A real structure on a complex vector space V is an antilinear involution $\sigma: V \to V$ . A real structure defines a real subspace $V_{\mathbb{R}} \subset V$ , its fix locus, and the natural map

$\mathbb{C} \otimes_{\mathbb{R}} V_{\mathbb R} \to V$

is an isomorphism. Conversely any vector space that is the complexification of a real vector space has a natural real structure.

[edit] Real structure of an Algebraic Variety

For an algebraic variety defined over a subfield of the real numbers, the real structure is the complex conjugation acting on the points of the variey in complex projective or affine space. Its fixed locus are the space of real points of the variety (which may be empty).

[edit] Real structure of a Scheme

For a scheme defined over a subfield of the real numbers, complex conjugation is in a natural way a member of the Galois group of the algebraic closure of the basefield. The real structure is the Galois action of this conjugation on the extension of the scheme over the algebraic closure of the base field. The real points are the points whose residue field is fixed (which may be empty).