2013/08/12 by Akio Hattori, Hattori, Akio
Mathematics · #14M25 #52B29 #57R91 #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:14M25 #msc:52B29 #msc:57R91
paper · pdf · doi:10.48550/arxiv.1308.2439
This is an updated version of "On a Morelli type expression of cohomology classes of toric varieties" arXiv:1007.2046
arxiv created 2013/08/12 · arxiv updated 2013/08/13
Let X be a complete toric variety of dimension n and \del the fan in a lattice N associated to X. For each cone σof \del 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 Hn-k(X,Q). Then any x∈ Hn-k(X,Q) can be written in the form x=∑σ∈\delX, dimσ=kμ(x,σ)[V(σ)]. A question occurs whether there is some canonical way to express μ(x,σ). Morelli gave an answer when X is non-singular and at least for x= \Tn-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 Gn-k+1(NQ) of (n-k+1)-planes in NQ. 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 \delX 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 H2(X)Q. Under this assumption the equivariant cohomology ring with rational coefficients H^*T(X,Q) is isomorphic to H^*T(\delX,Q), the face ring of the multi-fan \delX, and the proof is carried out on H^*T(\delX,Q) by using completely combinatorial terms.