One of the most useful tools in the study of convex polytope is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on the combinatorics of the polytopes. This construction requires that the polytope is rational which is a real restriction when the polytope is general (neither simple nor simplicial). Often we would like to consider general polytopes and even polyhedral spheres (and more general objects) where the toric variety construction does not work.

I am aware of very general constructions by M. Davis, and T. Januszkiewicz, (one relevant paper might be Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) and several subsequent papers). Perhaps these constructions allow you to start with arbitrary polyhedral spheres and perhaps even in greater generality.

I ask about an explanation of the scope of these constructions, and, in simple terms as possible, how does the construction go?

One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on the combinatorics of the polytopes. This construction requires that the polytope is rational which is a real restriction when the polytope is general (neither simple nor simplicial). Often we would like to consider general polytopes and even polyhedral spheres (and more general objects) where the toric variety construction does not work.

I am aware of very general constructions by M. Davis, and T. Januszkiewicz, (one relevant paper might be Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) and several subsequent papers). Perhaps these constructions allow you to start with arbitrary polyhedral spheres and perhaps even in greater generality.

I ask about an explanation of the scope of these constructions, and, in simple terms as possible, how does the construction go?