The most common example of a complete variety is a projective variety, but there do exist complete and non-projective varieties in dimensions 3 and higher. The first example of a non-projective complete variety was given by Heisuke Hironaka. An affine space of dimension > 0 is not complete.
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Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry.
Basically, a variety is a scheme whose structure sheaf is a sheaf of K-algebras with the property that the rings R that occur above are all domains and are all finitely generated K-algebras, i.e., quotients of polynomial algebras by prime ideals.
A completevariety is a variety such that any map from an open subset of a nonsingular curve into it can be extended uniquely to the whole curve.
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