Integral cohomology rings of weighted Grassmann orbifolds and rigidity properties

TL;DR

This paper introduces weighted Grassmann orbifolds, classifies them via Pl"ucker weight vectors, and explicitly computes their integral cohomology rings, revealing torsion-free conditions and equivariant structure constants.

Contribution

It defines weighted Grassmann orbifolds using Pl"ucker weights, classifies them, and explicitly computes their integral cohomology rings, including structure constants.

Findings

Classification of weighted Grassmann orbifolds via Pl"ucker weights

Conditions for torsion-free integral cohomology

Explicit formulas for equivariant structure constants

Abstract

In this paper, we introduce `Pl\"{u}cker weight vector' and establish the definition of a weighted Grassmann orbifold ${\rm Gr}_{\bf b}(k,n)$, corresponding to a Pl\"{u}cker weight vector `${\bf b}$'. We achieve an explicit classification of weighted Grassmann orbifolds up to certain homeomorphism in terms of the Pl\"{u}cker weight vectors. We study the integral cohomology of ${\rm Gr}_{\bf b}(k,n)$ and provide some sufficient conditions such that the integral cohomology of ${\rm Gr}_{\bf b}(k,n)$ has no torsion. We explicitly describe the formula of the equivariant structure constants with respect to the equivariant Schubert basis in equivariant cohomology ring of divisive weighted Grassmann orbifolds with integer coefficients. Eminently, we compute the integral cohomology rings of divisive weighted Grassmann orbifolds explicitly.