Geodesic
On existence of double coset varieties
Artem Anisimov1
1 Department of Higher Algebra Faculty of Mechanics and Mathematics Lomonosov Moscow State University Leninskie Gory 1 GSP-1, Moscow 119991, Russia
Colloquium Mathematicum, Tome 126 (2012) no. 2, pp. 177-185 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Let ${\rm G}$ be a complex affine algebraic group and ${\rm H}, {\rm F} \subset {\rm G}$ be closed subgroups. The homogeneous space ${\rm G}/ {\rm H}$ can be equipped with the structure of a smooth quasiprojective variety. The situation is different for double coset varieties $\rm F \hspace{0.5pt} \backslash\hspace{-3pt}\backslash\hspace{-.5pt}{G}% \hspace{.5pt} /\hspace{-1pt}/\hspace{.5pt} H \hspace{1pt}$. We give examples showing that the variety $\rm F \hspace{0.5pt} \backslash\hspace{-3pt}\backslash\hspace{-.5pt}{G}% \hspace{.5pt} /\hspace{-1pt}/\hspace{.5pt} H \hspace{1pt}$ does not necessarily exist. We also address the question of existence of $\rm F \hspace{0.5pt} \backslash\hspace{-3pt}\backslash\hspace{-.5pt}{G}% \hspace{.5pt} /\hspace{-1pt}/\hspace{.5pt} H \hspace{1pt}$ in the category of constructible spaces and show that under sufficiently general assumptions $\rm F \hspace{0.5pt} \backslash\hspace{-3pt}\backslash\hspace{-.5pt}{G}% \hspace{.5pt} /\hspace{-1pt}/\hspace{.5pt} H \hspace{1pt}$ does exist as a constructible space.
DOI : 10.4064/cm126-2-3
Keywords: complex affine algebraic group subset closed subgroups homogeneous space equipped structure smooth quasiprojective variety situation different double coset varieties hspace backslash hspace backslash hspace hspace hspace hspace hspace examples showing variety hspace backslash hspace backslash hspace hspace hspace hspace hspace does necessarily exist address question existence hspace backslash hspace backslash hspace hspace hspace hspace hspace category constructible spaces under sufficiently general assumptions hspace backslash hspace backslash hspace hspace hspace hspace hspace does exist constructible space

Artem Anisimov 1

1 Department of Higher Algebra Faculty of Mechanics and Mathematics Lomonosov Moscow State University Leninskie Gory 1 GSP-1, Moscow 119991, Russia 
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Artem Anisimov. On existence of double coset varieties. Colloquium Mathematicum, Tome 126 (2012) no. 2, pp. 177-185. doi: 10.4064/cm126-2-3

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