ON A MORELLI TYPE EXPRESSION OF COHOMOLOGY CLASSES OF TORUS ORBIFOLDS

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Let X be a complete toric variety of dimension n and Δ the fan in a lattice N associated to X. For each cone σ of Δ there corresponds an orbit closure V(σ) of the action of complex torus on X. The homology classes {[V(σ)] | dim σ = k} form a set of specified generators of H_<n-k>(X,Q). Then any x ∈ H_<n-k>(X,Q) can be written in the form x = Σ__<σ∈Δ_<x>, dim σ = k> μ(x, σ)[V(σ)]. A question occurs whether there is some canonical way to express (x, ). Morelli [12] 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 [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 Δ_<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> (Δ_<X>, Q), the face ring of the multi-fan Δ_<X> , and the proof is carried out on H^<*>_<T> (Δ_<X> , Q) by using completely combinatorial terms.

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  • Osaka Journal of Mathematics

    Osaka Journal of Mathematics 51 (4), 1113-1132, 2014-10

    Osaka University and Osaka City University, Departments of Mathematics

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