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

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.

20 pages, 5 figures; 2nd version, rewrite the heavy notations and add the new results; 3rd version, reduce some figures and proofs; 4th version, revised following the referee's comments