(Complete)
A variety \(X\) over \(k\) is complete, equivalently proper, if the structure morphism \(X \to \Spec k\) is proper: separated, of finite type, and universally closed.
(Consequences of completeness)
Let \(k\) be algebraically closed and \(X\) a complete variety over \(k\). Then \(\OO_X(X) = k\), the image of \(X\) under any morphism of varieties is closed, and any morphism from \(X\) to an affine variety is constant. Every projective variety is complete; over \(\CC\), a variety is complete if and only if it is compact in the Euclidean topology.
\(\AA^1\) is not complete, although \(\AA^1\to\Spec k\) is a closed map: after base change, the projection \(\AA^1 \times \AA^1 \to \AA^1\) sends the closed hyperbola \(V(xy-1)\) to \(\AA^1 \sm \ts{0}\), which is not closed.
Complete varieties need not be projective: Hironaka constructed a nonsingular complete threefold that is not projective. Every complete curve and every nonsingular complete surface is projective, while there are complete normal surfaces that are not projective.
Dependencies
D-8XX95Proper morphisms and the valuative criteriauses