On a Morelli type expression of cohomology classes of torus orbifolds

Abstract

Let X be a complete toric variety of dimension n and \del the fan in a lattice N associated to X. For each cone \sigma of \del there corresponds an orbit closure V(\sigma) of the action of complex torus on X. The homology classes {[V(\sigma)]| \dim \sigma=k} form a set of specified generators of H_{n-k}(X,Q). Then any x\in H_{n-k}(X,Q) can be written in the form \[ x=\sum_{\sigma\in\del_X, \dim\sigma=k}\mu(x,\sigma)[V(\sigma)]. \] A question occurs whether there is some canonical way to express \mu(x,\sigma). Morelli gave an answer when X is non-singular and at least for x= \T_{n-k}(X) the Todd class of X. However his answer takes coefficients in the field of rational functions of degree 0 on the Grassmann manifold G_{n-k+1}(N_Q) of (n-k+1)-planes in N_Q. His proof uses Baum-Bott's residue formula for holomorphic foliations applied to the action of complex torus on X. On the other hand there appeared several attempts for generalizing non-singular toric varieties in topological contexts. Such generalized manifolds of dimension 2n acted on by a compact n dimensional torus T are called by the names quasi-toric manifolds, torus manifolds, toric manifolds, toric origami manifolds, topological toric manifolds and so on. Similarly torus orbifold can be considered. To a torus orbifold $X$ a simplicial set \del_X called multi-fan of X is associated. A question occurs whether a similar expression to Morelli's formula holds for torus orbifolds. It will be shown the answer is yes in this case too at least when the rational cohomology ring H^*(X)_Q is generated by H^2(X)_Q. Under this assumption the equivariant cohomology ring with rational coefficients H^*_T(X,Q) is isomorphic to H^*_T(\del_X,Q), the face ring of the multi-fan \del_X, and the proof is carried out on H^*_T(\del_X,Q) by using completely combinatorial terms.