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

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

@article{CRMATH_2012__350_17-18_845_0,
     author = {Yunhyung Cho and Min Kyu Kim},
     title = {Log-concavity of complexity one {Hamiltonian} torus actions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {845--848},
     publisher = {Elsevier},
     volume = {350},
     number = {17-18},
     year = {2012},
     doi = {10.1016/j.crma.2012.07.004},
     language = {en},
}

TY  - JOUR
AU  - Yunhyung Cho
AU  - Min Kyu Kim
TI  - Log-concavity of complexity one Hamiltonian torus actions
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 845
EP  - 848
VL  - 350
IS  - 17-18
PB  - Elsevier
DO  - 10.1016/j.crma.2012.07.004
LA  - en
ID  - CRMATH_2012__350_17-18_845_0
ER  -

%0 Journal Article
%A Yunhyung Cho
%A Min Kyu Kim
%T Log-concavity of complexity one Hamiltonian torus actions
%J Comptes Rendus. Mathématique
%D 2012
%P 845-848
%V 350
%N 17-18
%I Elsevier
%R 10.1016/j.crma.2012.07.004
%G en
%F CRMATH_2012__350_17-18_845_0

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

S. Sabatini; D. Sepe On topological properties of positive complexity one spaces, Transformation Groups, Volume 27 (2022) no. 2, pp. 723-735 | DOI:10.1007/s00031-020-09588-y | Zbl:1496.53082
Yunhyung Cho Log-concavity of the Duistermaat-Heckman measure for semifree Hamiltonian $S^{1}$ -actions, Proceedings of the American Mathematical Society, Volume 142 (2014) no. 7, pp. 2417-2428 | DOI:10.1090/s0002-9939-2014-12014-4 | Zbl:1309.37050

Cité par 2 documents. Sources : zbMATH

Commentaires - Politique