On existence of double coset varieties

Volume 126 / 2012

Artem Anisimov Colloquium Mathematicum 126 (2012), 177-185 MSC: Primary 14L30; Secondary 14M17, 14R20. DOI: 10.4064/cm126-2-3

Abstract

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.

Authors

  • Artem AnisimovDepartment of Higher Algebra
    Faculty of Mechanics and Mathematics
    Lomonosov Moscow State University
    Leninskie Gory 1
    GSP-1, Moscow 119991, Russia
    e-mail

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