On a Morelli type expression of cohomology classes of torus orbifolds

Abstract

Let $X$ be a complete toric variety of dimension $n$ and $\Delta$ the fan in a lattice $N$ associated to $X$. For each cone $\sigma$ of $\Delta$ there corresponds an orbit closure $V(\sigma)$ of the action of complex torus on $X$. The homology classes $\{[V(\sigma)] \mid \dim \sigma = k\}$ form a set of specified generators of $H_{n-k}(X, \mathbb{Q})$. Then any $x \in H_{n-k}(X, \mathbb{Q})$ can be written in the form \begin{equation*} x = \sum_{\sigma \in \Delta_{X}{,} \dim\sigma = k}\mu(x, \sigma)[V(\sigma)]. \end{equation*} A question occurs whether there is some canonical way to express $\mu(x, \sigma)$. Morelli [12] gave an answer when $X$ is non-singular and at least for $x = \mathcal{T}_{n-k}(X)$ the Todd class of $X$. However his answer takes coefficients in the field of rational functions of degree $0$ on the Grassmann manifold $G_{n-k+1}(N_{\mathbb{Q}})$ of $(n-k+1)$-planes in $N_{\mathbb{Q}}$. 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 [4, 10, 7, 11, 9, 2]. 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 $\Delta_{X}$ 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 $H^{*}(X)_{\mathbb{Q}}$ is generated by $H^{2}(X)_{\mathbb{Q}}$. Under this assumption the equivariant cohomology ring with rational coefficients $H^{*}_{T}(X, \mathbb{Q})$ is isomorphic to $H^{*}_{T}(\Delta_{X}, \mathbb{Q})$, the face ring of the multi-fan $\Delta_{X}$, and the proof is carried out on $H^{*}_{T}(\Delta_{X}, \mathbb{Q})$ by using completely combinatorial terms.