Borel-Hirzebruch type formula for the graph equivariant cohomology of a projective bundle over a GKM-graph

TL;DR

This paper develops a GKM-theoretic framework for equivariant vector bundles and their projectivizations, establishing a Borel-Hirzebruch type formula for graph equivariant cohomology, and explores their realization as GKM fiber bundles.

Contribution

It introduces the concept of leg bundles in GKM theory, defines their projectivization, and proves a Borel-Hirzebruch type formula for their cohomology, advancing the understanding of GKM graph structures.

Findings

Established a Borel-Hirzebruch type formula for projectivized leg bundles.

Connected projectivizations with GKM fiber bundles as per Guillemin-Sabatini-Zara.

Provided a framework for realizing projective GKM fiber bundles from leg bundles.

Abstract

In this paper, we introduce the GKM theoretical counterpart of the equivariant complex vector bundles as the "leg bundle". We also provide a definition for the projectivization of a leg bundle and prove the Borel-Hirzebruch type formula for its graph equivariant cohomology, assuming that the projectivization is again a GKM graph. Furthermore, we study the realization of the projective GKM fiber bundle, in the sense of Guillemin-Sabatini-Zara, can be obtained from the projectivization of a leg bundle.