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 is a closed map (i.e. maps closed sets onto closed sets). This can be seen as an analogue of compactness in algebraic geometry: a topological space X is compact if and only if the above projection map is closed with respect to topological products. The image of a complete variety is closed and is a complete variety. A closed subvariety of a complete variety is complete. A complex variety is complete if and only if it is compact as a complex-analytic variety. The most common example of a complete variety is a projective variety, but there do exist complete non-projective varieties in dimensions 2 and higher. While any complete nonsingular surface is projective, there exist nonsingular complete varieties in dimension 3 and higher which are not projective. 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. (Wikipedia).
From the album Margerine Eclipse
From playlist the absolute best of stereolab
From playlist the absolute best of stereolab
From playlist the absolute best of stereolab
Stereolab - With Friends Like These
Stereolab - With Friends Like These
From playlist the absolute best of stereolab
From playlist the absolute best of stereolab
Schemes 26: Abstract and projective varieties
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We discuss the relation between abstract, projective, and complete varieties, and given an example found by Hironaka of a complete variety that is not projecti
From playlist Algebraic geometry II: Schemes
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From playlist Mathematics
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Sam Payne March 13, 2015 Workshop on Chow groups, motives and derived categories More videos on http://video.ias.edu
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Arithmetic models for Shimura varieties – Georgios Pappas – ICM2018
Number Theory | Algebraic and Complex Geometry Invited Lecture 3.8 | 4.11 Arithmetic models for Shimura varieties Georgios Pappas Abstract: We describe recent work on the construction of well-behaved arithmetic models for large classes of Shimura varieties and report on progress in the s
From playlist Algebraic & Complex Geometry
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We discuss vanishing theorems for the cohomology of Shimura varieties with torsion coefficients, under a genericity condition at an auxiliary prime. We describe two complementary approaches to these results, due to Caraiani-Scholze and Koshikawa, both of which rely on the geometry of the H
From playlist 2022 Summer School on the Langlands program
The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties - Will Sawin
Joint IAS/Princeton University Number Theory Seminar Topic: The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties Speaker: Will Sawin Affiliation: Columbia University Date: March 18, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
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From playlist All About Whole Numbers
Low Algebraic Dimension Matrix Completion -Laura Balzano
Virtual Workshop on Missing Data Challenges in Computation Statistics and Applications Topic: Low Algebraic Dimension Matrix Completion Speaker: Laura Balzano Date: September 11, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics