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Mosc. Math. J., 2012, Volume 12, Number 2, Pages 369–396 (Mi mmj471)  

This article is cited in 10 scientific papers (total in 10 papers)

Convex bodies associated to actions of reductive groups

Kiumars Kaveha, Askold G. Khovanskiibcd

a Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260
b Mathematics, University of Toronto, Toronto, Canada
c Moscow Independent University
d Institute for Systems Analysis, Russian Academy of Sciences

Abstract: We associate convex bodies to a wide class of graded $G$-algebras where $G$ is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in the graded algebra. We extend the notion of Duistermaat–Heckman measure to graded $G$-algebras and prove a Fujita type approximation theorem and a Brunn–Minkowski inequality for this measure. This in particular applies to arbitrary $G$-line bundles giving an equivariant version of the theory of volumes of line bundles. We generalize the Brion–Kazarnowskii formula for the degree of a spherical variety to arbitrary $G$-varieties. Our approach follows some of the previous works of A. Okounkov. We use the asymptotic theory of semigroups of integral points and Newton–Okounkov bodies developed in [15].

Key words and phrases: Reductive group action, multiplicity of a representation, Duistermaat–Heckman measure, moment map, graded $G$-algebra, $G$-line bundle, volume of a line bundle, semigroup of integral points, convex body, mixed volume, Brunn–Minkowski inequality.

DOI: https://doi.org/10.17323/1609-4514-2012-12-2-369-396

Full text: http://www.ams.org/.../abst12-2-2012.html
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Bibliographic databases:

MSC: Primary 14L30, 53D20; Secondary 52A39
Received: January 17, 2011; in revised form January 14, 2012
Language:

Citation: Kiumars Kaveh, Askold G. Khovanskii, “Convex bodies associated to actions of reductive groups”, Mosc. Math. J., 12:2 (2012), 369–396

Citation in format AMSBIB
\Bibitem{KavKho12}
\by Kiumars~Kaveh, Askold~G.~Khovanskii
\paper Convex bodies associated to actions of reductive groups
\jour Mosc. Math.~J.
\yr 2012
\vol 12
\issue 2
\pages 369--396
\mathnet{http://mi.mathnet.ru/mmj471}
\crossref{https://doi.org/10.17323/1609-4514-2012-12-2-369-396}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2978761}
\zmath{https://zbmath.org/?q=an:06126178}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309365900009}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Kaveh K. Khovanskii A.G., “Newton-Okounkov Bodies, Semigroups of Integral Points, Graded Algebras and Intersection Theory”, Ann. Math., 176:2 (2012), 925–978  crossref  zmath  isi  elib  scopus
    2. Perrin N., “On the Geometry of Spherical Varieties”, Transform. Groups, 19:1 (2014), 171–223  crossref  mathscinet  zmath  isi  elib  scopus
    3. Boucksom S., “Body of Okounkov [According to Okounkov, Lazarsfeld-Mustata and Kaveh-Khovanskii]”, Asterisque, 2014, no. 361, 1–41  zmath  isi
    4. Harada M., Kaveh K., “Integrable Systems, Toric Degenerations and Okounkov Bodies”, Invent. Math., 202:3 (2015), 927–985  crossref  zmath  isi  elib  scopus
    5. Kaveh K., “Crystal Bases and Newton-Okounkov Bodies”, Duke Math. J., 164:13 (2015), 2461–2506  crossref  zmath  isi  elib  scopus
    6. Fulger M., Zhou X., “Schur Asymptotics of Veronese Syzygies”, Math. Ann., 362:1-2 (2015), 529–540  crossref  zmath  isi  elib  scopus
    7. K. Kaveh, A. G. Khovanskii, “Complete intersections in spherical varieties”, Sel. Math.-New Ser., 22:4, SI (2016), 2099–2141  crossref  mathscinet  zmath  isi  scopus
    8. Ya. Deng, “Transcendental Morse inequality and generalized Okounkov bodies”, Algebraic Geom., 4:2 (2017), 177–202  crossref  mathscinet  zmath  isi
    9. D. Schmitz, H. Seppaenen, “Global Okounkov bodies for Bott–Samelson varieties”, J. Algebra, 490 (2017), 518–554  crossref  mathscinet  zmath  isi  scopus
    10. S. Boucksom, T. Hisamoto, M. Jonsson, “Uniform k-stability, Duistermaat–Heckman measures and singularities of pairs”, Ann. Inst. Fourier, 67:2 (2017), 743–841  crossref  mathscinet  zmath  isi
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