On a Morelli type expression of cohomology classes of torus orbifolds
Akio Hattori

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…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Geometric and Algebraic Topology · Algebraic Geometry and Number Theory
