Complete algebraic variety

A complete algebraic variety in algebraic geometry is the analogue of a compact manifold in differential geometry . So an algebraic variety is complete if it has certain "compact" properties.

definition

Let be an algebraic variety, so that for all varieties the projection with respect to the Zariski topology is a closed mapping , that is, for a Zariski-closed subset is also closed. Then means complete. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \pi _{Y}\colon X\times Y\to Y}$ ${\displaystyle A\times B\subseteq X\times Y}$ ${\displaystyle \pi _{Y}(A\times B)\subseteq Y}$ ${\displaystyle X}$

Examples

The most important example of complete varieties are projective varieties . Affine varieties , however, are only complete if they are finite. Examples of non-projective complete varieties can also be constructed with greater effort. Examples are, for example, singular non-projective complete surfaces or smooth complete non-projective three-dimensional varieties.

Inheritance of completeness

The property of completeness is retained under certain constructions. For example:

• Completed sub-varieties of complete varieties are complete.
• Complete sub-varieties of varieties are complete.
• Complete variety products are complete.
• Pictures of complete varieties under morphisms are complete and complete.

Characteristics of full varieties

Regular functions of complete varieties

The regular functions of connected complete varieties are precisely the constant functions.

Completeness sometimes enforces projectivity

Complete quasi-projective varieties, complete curves, and smooth complete surfaces are projective varieties.

Nagata theorem

The following embedding result goes back to Masayoshi Nagata :

Each variety can be densely embedded as an open subset in a complete variety.

Borelian Fixed Point Theorem

The following fixed point theorem is relevant for the theory of algebraic groups :

If a connected solvable algebraic group operates on a complete non-empty variety over an algebraically closed field , a fixed point exists.

Similar terms

Relation to compactness

With the following characterization of the compactness of a Hausdorff space , the connection to the completeness of an algebraic variety becomes clear:

A Hausdorff space is compact if and only if, for all topological spaces, the projection with regard to the product topology is on a closed image. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \pi _{Y}\colon X\times Y\to Y}$ ${\displaystyle X\times Y}$

Connection with actual morphisms

The morphisms corresponding to the complete varieties are the actual morphisms. Therefore, complete varieties are sometimes also referred to as actual varieties . Thus every morphism that is defined on a complete variety is a proper morphism and a variety is complete just when the constant morphism from the variety to a point is a proper morphism.

literature

• James E. Humphreys : Linear Algebraic Groups. Springer, New York 1975, ISBN 978-1-4684-9445-7 , 6. Complete Varieties.
• Robin Hartshorne : Algebraic Geometry. Springer, New York 1977, ISBN 978-1-4419-2807-8 , II.4 Separated and Proper Morphisms.
• Karl-Heinz Fieseler, Ludger Kaup: Algebraic Geometry. Heldermann Verlag, Lemgo 2005, ISBN 3-88538-113-3 , 5th projective algebraic varieties.

Individual evidence

1. ^ Fieseler, Kaup: Algebraic Geometry. 2005, 5.14 definition
2. ^ Humphreys: Linear Algebraic Groups. 1975, 6.1
3. ^ Fieseler, Kaup: Algebraic Geometry. 2005, 5.25 Corollary
4. ^ Humphreys: Linear Algebraic Groups. 1975, 6.2 theorem
5. ^ Fieseler, Kaup: Algebraic Geometry. 2005, 5.19 Corollary
6. ^ Humphreys: Linear Algebraic Groups. 1975, 6.1 Proposition, (e)
7. ↑ Masayoshi Nagata: On the imbeddings of abstract surfaces in projective varieties. Mem. College Sci. Univ. Kyoto Ser. A Math. 30 (1957), no.3, 231-235.
8. ^ Hartshorne: Algebraic Geometry. 1977, Ex. II.7.13, Ex. III.5.9
9. ↑ Heisuke Hironaka: On the theory of birational blowing-up. Harvard 1960.
10. ↑ Masayoshi Nagata: Existence theorems for nonprojective complete algebraic varieties. Illinois J. Math. 2 (1958) 490-498.
11. ^ Hartshorne: Algebraic Geometry. 1977, Appendix B, Example 3.4.1
12. ^ Fieseler, Kaup: Algebraic Geometry. 2005, 5.17 Lemma
13. ^ Humphreys: Linear Algebraic Groups. 1975, 6.1 Proposition
14. ^ Fieseler, Kaup: Algebraic Geometry. 2005, 5.19 Corollary, 1)
15. ^ Humphreys: Linear Algebraic Groups. 1975, 6.1 Proposition (f)
16. ^ Hartshorne: Algebraic Geometry. 1977, Ex. III.5.8
17. ^ Oscar Zariski : Introduction to the Problem of Minimal Models in the Theory of Algebraic Surfaces. American Journal of Mathematics Vol. 80, No. 1 (Jan. 1958), 146-184
18. ↑ Masayoshi Nagata: Imbedding of an abstract variety in a complete variety. Journal of Mathematics of Kyoto University (2) 1962, 1-10.
19. ↑ Masayoshi Nagata: A generalization of the imbedding problem of an abstract variety in a complete variety. Journal of Mathematics of Kyoto University (3) 1963, 89-102.
20. ^ Humphreys: Linear Algebraic Groups. 1975, 21.2 Fixed Point Theorem
21. ↑ Armand Borel : Groupes Line Aires Algebriques. Annals of Mathematics, Second Series, Vol. 64, No. 1 (Jul. 1956), 20-82
22. ↑ Nicolas Bourbaki : General Topology I. 10.2, Corollary 1 to Theorem 1
23. ^ Fieseler, Kaup: Algebraic Geometry. 2005, 13 Introduction
24. ^ Hartshorne: Algebraic Geometry. 1977, p. 105 definition
25. ^ Fieseler, Kaup: Algebraic Geometry. 2005, 13.4 remark