- Published 2008

We introduce the notion of a multi-fan. It is a generalization of that of a fan in the theory of toric varieties in algebraic geometry. Roughly speaking a toric variety is an algebraic variety with an action of an algebraic torus of the same dimension as that of the variety, and a fan is a combinatorial object associated with the toric variety. Algebro-geometric properties of the toric variety can be described in terms of the associated fan. We develop a combinatorial theory of multi-fans and define “topological invariants” of a multi-fan. A smooth manifold with an action of a compact torus of half the dimension of the manifold and with some orientation data is called a torus manifold. We associate a unique multi-fan with a torus manifold, and apply the combinatorial theory to describe topological invariants of the torus manifold. A similar theory is also given for torus orbifolds. Though a multi-fan may correspond to more than one torus manifolds or orbifolds, it is shown that every complete simplicial multi-fan is assoiated with a certain torus orbifold. As a related subject a generalization of the Ehrhart polynomial concerning the number of lattice points in a convex polytope is discussed. The notion of multi-polytope generalizes that of convex polytope, and the number of lattice points in a multi-polytope is defined properly.

@inproceedings{Hattori2008TheoryOM,
title={Theory of Multi-fans},
author={Akio Hattori},
year={2008}
}