Equivariant structure constants for ordinary and weighted projective space

TL;DR

This paper computes the equivariant cohomology rings of weighted and ordinary projective spaces under different torus actions, providing explicit formulas and basis relations, advancing understanding of their algebraic structures.

Contribution

It introduces new computations of equivariant cohomology rings for weighted projective spaces under two torus actions and relates basis classes to those of ordinary projective space.

Findings

Explicit integral torus-equivariant cohomology rings for weighted projective spaces.

Identification of basis classes as multiples of ordinary projective space classes.

Simple formula for structure constants using divided difference operators.

Abstract

We compute the integral torus-equivariant cohomology ring for weighted projective space for two different torus actions by embedding the cohomology in a sum of polynomial rings $\oplus_{i=0}^n \Z[t_1, t_2,..., t_n]$. One torus action gives a result complementing that of Bahri, Franz, and Ray. For the other torus action, each basis class for weighted projective space is a multiple of the basis class for ordinary projective space; we identify each multiple explicitly. We also give a simple formula for the structure constants of the equivariant cohomology ring of ordinary projective space in terms of the basis of Schubert classes, as a sequence of divided difference operators applied to a specific polynomial.