Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

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.