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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 18, Issue 3

Issues

Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers

Mi Ju Cho
  • Department of Mathematics Education, Chosun University, 309 Pilmun-daero, Dong-gu, Gwangju 61452, Republic of Korea
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/ Jin Hong Kim
  • Corresponding author
  • Department of Mathematics Education, Chosun University, 309 Pilmun-daero, Dong-gu, Gwangju 61452, Republic of Korea
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/ Hwa Lee
  • Department of Mathematics Education, Chosun University, 309 Pilmun-daero, Dong-gu, Gwangju 61452, Republic of Korea
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Published Online: 2018-07-18 | DOI: https://doi.org/10.1515/advgeom-2017-0023

Abstract

A multi-fan (respectively multi-polytope), introduced first by Hattori and Masuda, is a purely combinatorial object generalizing an ordinary fan (respectively polytope) in algebraic geometry. It is well known that an ordinary fan or polytope is associated with a toric variety. On the other hand, we can geometrically realize multi-fans in terms of torus manifolds. However, it is unfortunate that two different torus manifolds may correspond to the same multi-fan. The goal of this paper is to give some criteria for a multi-polytope to be an ordinary polytope in terms of the Duistermaat–Heckman functions and winding numbers. Moreover, we also prove a generalized Pick formula and its consequences for simple lattice multi-polytopes by studying their Ehrhart polynomials.

Keywords: Polytopes; multi-polytopes; Duistermaat–Heckman functions; winding numbers; Ehrhart polynomials; generalized Pick formula; generalized twelve-point formula

MSC 2010: 57R20; 57S15; 14M25

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About the article


Received: 2015-07-15

Revised: 2016-03-13

Revised: 2016-06-20

Published Online: 2018-07-18

Published in Print: 2018-07-26


Funding: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2014R1A1A2054683).


Citation Information: Advances in Geometry, Volume 18, Issue 3, Pages 355–372, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0023.

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