Equivariant vector bundles over topological toric manifolds
Yong Cui, Amin Gholampour
TL;DR
The paper extends Klyachko's equivariant classification from algebraic toric varieties to topological toric manifolds, proving that every $\mathbb T = (\mathbb C^*)^n$-equivariant vector bundle on a $2n$-dimensional topological toric manifold is a topological/smooth Klyachko vector bundle. It accomplishes this by constructing $G=(S^1)^n$-eigenframes near fixed points and extending them to full $\mathbb T$-eigenframes on affine charts $U_I$, thereby obtaining equivariant trivializations on each chart and encoding the data in a Klyachko-type filtration framework. The results establish an affirmative answer to whether all equivariant bundles arise from Klyachko-type data in the topological setting, and they extend Cui25's framework to the smooth category as well. This provides a robust combinatorial/topological mechanism for classifying equivariant vector bundles over topological toric manifolds, with potential implications for downstream geometric and topological applications.
Abstract
We prove that every topological/smooth $\T=(\C^{*})^{n}$-equivariant vector bundle over a topological toric manifold of dimension $2n$ is a topological/smooth Klyachko vector bundle in the sense of arXiv:2504.02205.