Criteria for Hattori-Masuda multi-polytopes via Duistermaat-Heckman functions and winding numbers
The theory of multi-fans was first developed by \textit{A. Hattori} and \textit{M. Masuda} [Osaka J. Math. 40, No. 1, 1--68 (2003; Zbl 1034.57031]. A multi-fan is a combinatorial object that generalizes an ordinary fan in algebraic geometry. To every ordinary fan one can associate a toric variety, and this correspondence is one-to-one. Indeed, there is a rich interplay between the combinatorics of fans and the geometry and topology of toric varieties. Torus manifolds are typical geometric realizations of multi-fans, and in the theory of multi-fans, the analogue of a toric variety is a torus manifold. However, this correspondence is not one-to-one; different torus manifolds may correspond to the same multi-fan, and it is not known whether every regular complete multi-fan can be realized as a multi-fan associated with a torus manifold. While one can associate a convex polytope to every ordinary fan, a multi-polytope can be associated to a multi-fan. In [loc. cit.], Hattori and Masuda defined the Duistermaat-Heckman function, the winding number and the equivariant index in combinatorial terms for multi-fans and multi-polytopes. In this paper, the authors give some criteria for a multi-polytope to be an ordinary polytope in terms of the Duistermaat-Heckman functions and winding numbers. They also prove a generalized Pick formula for simple lattice multi-polytopes and study their Ehrhart polynomials.
scientific article; zbMATH DE number 6914287
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