Complete algebraic variety

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.