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Mathematics of the USSR-Sbornik, 1989, Volume 63, Issue 2, Pages 375–392
DOI: https://doi.org/10.1070/SM1989v063n02ABEH003280
(Mi sm1711)
 

This article is cited in 8 scientific papers (total in 8 papers)

Closed orbits of Borel subgroups

V. L. Popov
References:
Abstract: The author considers an algebraic action of a connected reductive algebraic group $G$ defined over an algebraically closed field $k$ on an affine irreducible algebraic variety $X$, and studies the question of when the action of a Borel subgroup $B$ of $G$ on $X$ is stable, i.e., the $B$-orbit of any point belonging to some nonempty open subset of $X$ is closed in $X$. A criterion for stability is obtained: Suppose that $\operatorname{char}k=0$. In order that the action of $B$ on $X$ be stable it is necessary, and, if $G$ is semisimple and the group of divisor classes $\mathrm{Cl}X$ is periodic, also sufficient that $X$ contain a point with a finite $G$-stabilizer. For an action $G:V$ defined by a linear representation $G\to GL(V)$ the cases when $B:V$ is not stable and either $G$ is simple or $G$ is semisimple and the action $G:V$ is irreducible are listed. A general criterion for an orbit of a connected solvable group acting on an affine variety to be closed is also obtained, and it is used to obtain a simple sufficient condition for an orbit of such a group, acting linearly, to be closed.
Bibliography: 30 titles.
Received: 18.02.1987
Russian version:
Matematicheskii Sbornik. Novaya Seriya, 1988, Volume 135(177), Number 3, Pages 385–402
Bibliographic databases:
Document Type: Article
UDC: 512
MSC: Primary 14L30, 20G05; Secondary 22E45, 14D25
Language: English
Original paper language: Russian
Citation: V. L. Popov, “Closed orbits of Borel subgroups”, Mat. Sb. (N.S.), 135(177):3 (1988), 385–402; Math. USSR-Sb., 63:2 (1989), 375–392
Citation in format AMSBIB
\Bibitem{Pop88}
\by V.~L.~Popov
\paper Closed orbits of Borel subgroups
\jour Mat. Sb. (N.S.)
\yr 1988
\vol 135(177)
\issue 3
\pages 385--402
\mathnet{http://mi.mathnet.ru/sm1711}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=937648}
\zmath{https://zbmath.org/?q=an:0713.20036}
\transl
\jour Math. USSR-Sb.
\yr 1989
\vol 63
\issue 2
\pages 375--392
\crossref{https://doi.org/10.1070/SM1989v063n02ABEH003280}
Linking options:
  • https://www.mathnet.ru/eng/sm1711
  • https://doi.org/10.1070/SM1989v063n02ABEH003280
  • https://www.mathnet.ru/eng/sm/v177/i3/p385
  • This publication is cited in the following 8 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
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    Abstract page:553
    Russian version PDF:169
    English version PDF:22
    References:77
    First page:2
     
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