Complete algebraic variety

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In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism

X × Y → Y

is a closed map, i.e. maps closed sets onto closed sets. (NB Here the cartesian product variety does not carry the product topology, in general; the Zariski topology on it will, except in very simple cases, have more closed sets.)

The most common example of a complete variety is a projective variety, but there do exist complete and non-projective varieties in dimensions 3 and higher. The first example of a non-projective complete variety was given by Heisuke Hironaka. An affine space of dimension > 0 is not complete.

The morphism taking a complete variety to a point is a proper morphism, in the sense of scheme theory. An intuitive justification of 'complete', in the sense of 'no missing points', can be given on the basis of the valuative criterion of properness, which goes back to Claude Chevalley.