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

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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MSC: Primary 14L30, 53D20; Secondary 52A39
Received: January 17, 2011; in revised form January 14, 2012
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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
2. Perrin N., “On the Geometry of Spherical Varieties”, Transform. Groups, 19:1 (2014), 171–223
3. Boucksom S., “Body of Okounkov [According to Okounkov, Lazarsfeld-Mustata and Kaveh-Khovanskii]”, Asterisque, 2014, no. 361, 1–41
4. Harada M., Kaveh K., “Integrable Systems, Toric Degenerations and Okounkov Bodies”, Invent. Math., 202:3 (2015), 927–985
5. Kaveh K., “Crystal Bases and Newton-Okounkov Bodies”, Duke Math. J., 164:13 (2015), 2461–2506
6. Fulger M., Zhou X., “Schur Asymptotics of Veronese Syzygies”, Math. Ann., 362:1-2 (2015), 529–540
7. K. Kaveh, A. G. Khovanskii, “Complete intersections in spherical varieties”, Sel. Math.-New Ser., 22:4, SI (2016), 2099–2141
8. Ya. Deng, “Transcendental Morse inequality and generalized Okounkov bodies”, Algebraic Geom., 4:2 (2017), 177–202
9. D. Schmitz, H. Seppaenen, “Global Okounkov bodies for Bott–Samelson varieties”, J. Algebra, 490 (2017), 518–554
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
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