On a Morelli type expression of cohomology classes of torus orbifolds

TL;DR

This paper extends Morelli's formula for expressing cohomology classes of toric varieties to torus orbifolds, using combinatorial methods under certain conditions, and explores the structure of their equivariant cohomology rings.

Contribution

It generalizes Morelli's cohomology class expression from smooth toric varieties to torus orbifolds with rational cohomology generated by H^2, using combinatorial face rings.

Findings

The formula applies to torus orbifolds with cohomology generated by H^2.

The equivariant cohomology ring is isomorphic to the face ring of the multi-fan.

The proof utilizes combinatorial descriptions of the face ring.

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.