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 is a closed map (i.e. maps closed sets onto closed sets).
[a] This can be seen as an analogue of compactness in algebraic geometry: a topological space X is compact if and only if the above projection map is closed with respect to topological products.
[3] The first examples of non-projective complete varieties were given by Masayoshi Nagata[3] and Heisuke Hironaka.
[4] An affine space of positive dimension is not complete.
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.