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

Abstract

In this paper we study the GKM theory of torus-equivariant vector bundles and their projectivizations. We introduce the combinatorial notion of a leg bundle. This notion generalizes the notion of a GKM-graph by allowing noncompact edges and dropping the pairwise linear independence condition on the labels. We also introduce the projectivization of a leg bundle and study its canonical connection. We prove the Leray-Hirsh type theorem and the Borel-Hirzebruch type formula for the graph equivariant cohomology of the projectivization for a leg bundle if it is again a GKM graph by using the GKM theoretical arguments.