Equivariant Chow cohomology of toric varieties

We show that the equivariant Chow cohomology ring of a toric variety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which localization holds in equivariant Chow cohomology with integer coefficients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions. If X = X(∆) is a smooth, complete complex toric variety then the following rings are canonically isomorphic: the equivariant singular cohomology ring H T (X), the equivariant Chow cohomology ring A ∗ T (X), the Stanley-Riesner ring SR(∆), and the ring of integral piecewise polynomial functions PP (∆). If X is simplicial but not smooth then H T (X) may have torsion and the natural map from SR(∆) takes monomial generators to piecewise linear functions with rational, but not necessarily integral, coefficients. In such cases, these rings are not isomorphic, but they become isomorphic after tensoring with Q. When X is not simplicial, there are still natural maps between these rings, for instance from A∗T (X)Q to H ∗ T (X)Q and from H ∗ T (X) to PP (∆), but these maps are far from being isomorphisms in general. The main purpose of this note is to construct a natural isomorphism from A∗T (X) to PP (∆) for an arbitrary toric variety; the map is obtained by restricting a Chow cohomology class to each of the T -orbits Oσ ⊂ X for cones σ ∈ ∆. The equivariant Chow cohomology of Oσ is naturally isomorphic to the ring SymMσ of integral polynomial functions on σ, where Mσ = M/(σ ⊥ ∩M) (for u ∈ M , the image of u in Mσ is identified with the first equivariant Chern class of the equivariant line bundle OX(divχ )|Oσ in A 1 T (Oσ)). The ring of integral piecewise polynomial functions on ∆ is defined by PP (∆) = {f : |∆| → R : f |σ ∈ SymMσ for each σ ∈ ∆}. The map f 7→ (f |σ)σ∈∆ identifies PP (∆) with a subring of ⊕ σ∈∆ SymMσ: PP (∆) ∼= {(fσ)σ∈∆ : fτ = fσ|τ for τ ≺ σ}. We write ισ for the inclusion of Oσ in X . Theorem 1 Let X = X(∆) be a toric variety. Then ⊕ σ∈∆ ι ∗ σ maps A ∗ T (X) isomorphically onto PP (∆).