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.^[1]

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^[2] 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.

Notes

^ Here the product variety X × Y does not carry the product topology, in general; the Zariski topology on it will have more closed sets (except in very simple cases).
^ Existence theorems for nonprojective complete algebraic varieties, Illinois J. Math. 2 (1958) 490–498.

References

Section II.4 of Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Chapter 7 of Milne, James S. (2009), Algebraic geometry, v. 5.20, http://jmilne.org/math/CourseNotes/ag.html, retrieved 2010-08-04
Section I.9 of Mumford, David (1999), The red book of varieties and schemes, Lecture notes in mathematics, 1358 (Second, expanded ed.), Springer-Verlag, doi:10.1007/b62130, ISBN 978-3-540-63293-1