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

Abstract

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.