Complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism
is a closed map. This can be seen as an analogue of compactness in algebraic geometry: a topological space is compact if and only if the above projection map is closed with respect to topological products.
The image of a complete variety is closed and is a complete variety. A closed subvariety of a complete variety is complete.
A complex variety is complete if and only if it is compact as a complex-analytic variety.
The most common example of a complete variety is a projective variety, but there do exist complete non-projective varieties in dimensions 2 and higher. While any complete nonsingular surface is projective, there exist nonsingular complete varieties in dimension 3 and higher which are not projective. The first examples of non-projective complete varieties were given by Masayoshi Nagata and Heisuke Hironaka. An affine space of positive dimension 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.