Unitary toric manifolds, multi-fans and equivariant index

We develop the theory of toric varieties from a topological point of view using equivariant cohomology. Indeed, we introduce a geometrical object called a unitary toric manifold and associate a combinatorial object called a multi-fan to it. This generalizes (in one direction) the well-known correspondence between a compact nonsingular toric variety and a (regular) fan. The multi-fan is a collection of cones which may overlap unlike a usual fan. It turns out that the degree of the overlap of cones is essentially the Todd genus of the unitary toric manifold. Since the Todd genus of a compact nonsingular toric variety is one, this explains why cones do not overlap in a usual fan. A moment map relates a unitary toric manifold equipped with an equivariant complex line bundle to a 'twisted polytope", and the equivariant Riemann-Roch index for the equivariant line bundle can be described in terms of the moment map. We apply this result to establish a generalization of Pick's formula.