Log-concavity of complexity one Hamiltonian torus actions

Yunhyung Cho; Min Kyu Kim

doi:10.1016/j.crma.2012.07.004

Differential Geometry

Log-concavity of complexity one Hamiltonian torus actions
[Log-concavité des actions toriques hamiltoniennes de complexité un]

Présenté par : the Editorial Board

Yunhyung Cho ¹ ; Min Kyu Kim ²

¹ School of Mathematics, Korea Institute for Advanced Study, 87 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea
² Department of Mathematics Education, Gyeongin National University of Education, San 59-12, Gyesan-dong, Gyeyang-gu, Incheon, 407-753, Republic of Korea

Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 845-848.

Résumé
Abstract

Soit $(M, ω)$ une variété symplectique de dimension 2n munie dʼune action hamiltonienne du tore $T^{n - 1}$ . Le théorème de convexité dʼAtiyah–Guillemin–Sternberg implique que lʼimage de lʼapplication moment est un polytope convexe de dimension $(n - 1)$ . Dans cette Note, nous montrons que la fonction de densité de la mesure de Duistermaat–Heckman est log-concave sur lʼimage de lʼapplication moment.

Let $(M, ω)$ be a closed 2n-dimensional symplectic manifold equipped with a Hamiltonian $T^{n - 1}$ -action. Then Atiyah–Guillemin–Sternberg convexity theorem implies that the image of the moment map is an $(n - 1)$ -dimensional convex polytope. In this Note, we show that the density function of the Duistermaat–Heckman measure is log-concave on the image of the moment map.

Reçu le : 2012-05-31
Accepté le : 2012-07-11
Publié le : 2012-09-01

DOI : 10.1016/j.crma.2012.07.004

Affiliations des auteurs :

Yunhyung Cho ¹ ; Min Kyu Kim ²

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     title = {Log-concavity of complexity one {Hamiltonian} torus actions},
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     doi = {10.1016/j.crma.2012.07.004},
     language = {en},
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Yunhyung Cho; Min Kyu Kim. Log-concavity of complexity one Hamiltonian torus actions. Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 845-848. doi : 10.1016/j.crma.2012.07.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.07.004/

Bibliographie
Cité par

[1] Y. Cho The log-concavity conjecture on semifree symplectic $S^{1}$ -manifolds with isolated fixed points | arXiv

[2] J.J. Duistermaat; G.J. Heckman On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., Volume 69 (1982), pp. 259-268

[3] W. Graham Logarithmic convexity of push-forward measures, Invent. Math., Volume 123 (1996), pp. 315-322

[4] V. Guillemin; S. Sternberg Convexity property of the moment mapping, Invent. Math., Volume 67 (1982), pp. 491-513

[5] Y. Karshon Example of a non-log-concave Duistermaat–Heckman measure, Math. Res. Lett., Volume 3 (1996), pp. 537-540

[6] Y. Karshon Periodic Hamiltonian flows on four dimensional manifolds, Mem. Amer. Math. Soc., Volume 141 (1999) no. 672

[7] Y. Lin The log-concavity conjecture for the Duistermaat–Heckman measure revisited, Int. Math. Res. Not. (2008) no. 10 (Art. ID rnn027, 19 pp)

[8] E. Meinrenken Symplectic surgery and the ${Spin}^{c}$ -Dirac operators, Adv. Math., Volume 134 (1998), pp. 240-277

[9] A. Okounkov Why would multiplicities be log-concave?, Marseille, 2000 (Progress in Mathematics), Volume vol. 213, Birkhäuser Boston, Boston, MA (2003), pp. 329-347

[10] A. Okounkov Log-concavity of multiplicities with application to characters of $U (\infty)$ , Adv. Math., Volume 127 (1997), pp. 258-282

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