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.
The most common example of a complete variety is a projective variety, but there do exist complete and non-projective varieties in dimensions 2 and higher. 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.
Famous quotes containing the words complete and/or variety:
“It is easier to live through someone else than to become complete yourself.”
—Betty Friedan (b. 1921)
“Life is a series of diminishments. Each cessation of an activity either from choice or some other variety of infirmity is a death, a putting to final rest. Each loss, of friend or precious enemy, can be equated with the closing off of a room containing blocks of nerves ... and soon after the closing off the nerves atrophy and that part of oneself, in essence, drops away. The self is lightened, is held on earth by a gram less of mass and will.”
—Coleman Dowell (1925–1985)