Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Klyachko vector bundles over topological toric manifolds

Yong Cui

TL;DR

The paper extends Klyachko's toric-vector-bundle classification to topological toric manifolds by introducing topological and smooth Klyachko vector bundles (TKVB/SKVB) and their graded filtrations (TKVS/SKVS) encoded by a poset of subspaces indexed by a topological fan. It proves categorical equivalences: TKVB_X ≃ TKVS_X and SKVB_X ≃ SKVS_X, bridging equivariant bundles with combinatorial poset data, and shows that holomorphic Klyachko bundles on a complex topological toric manifold are biholomorphic to toric bundles on a toric variety. The work includes a canonical exact sequence of equivariant bundles and a non-toric example to demonstrate applicability beyond classical toric varieties. Collectively, the results provide a unified algebraic-combinatorial framework for understanding equivariant vector bundles on topological toric spaces and their holomorphic specializations, with potential for computing cohomology and Chern data from poset filtrations.

Abstract

We give a Klyachko-type classification of topological/smooth/holomorphic $(\mathbb{C}^{*})^n$-equivariant vector bundles that are equivariantly trivial over invariant affine charts. This generalizes Klyachko's classification of toric vector bundles.

Klyachko vector bundles over topological toric manifolds

TL;DR

The paper extends Klyachko's toric-vector-bundle classification to topological toric manifolds by introducing topological and smooth Klyachko vector bundles (TKVB/SKVB) and their graded filtrations (TKVS/SKVS) encoded by a poset of subspaces indexed by a topological fan. It proves categorical equivalences: TKVB_X ≃ TKVS_X and SKVB_X ≃ SKVS_X, bridging equivariant bundles with combinatorial poset data, and shows that holomorphic Klyachko bundles on a complex topological toric manifold are biholomorphic to toric bundles on a toric variety. The work includes a canonical exact sequence of equivariant bundles and a non-toric example to demonstrate applicability beyond classical toric varieties. Collectively, the results provide a unified algebraic-combinatorial framework for understanding equivariant vector bundles on topological toric spaces and their holomorphic specializations, with potential for computing cohomology and Chern data from poset filtrations.

Abstract

We give a Klyachko-type classification of topological/smooth/holomorphic -equivariant vector bundles that are equivariantly trivial over invariant affine charts. This generalizes Klyachko's classification of toric vector bundles.

Paper Structure

This paper contains 9 sections, 30 theorems, 115 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background on topological toric manifolds
  3. Orbit-Cone correspondence
  4. Topological Klyachko vector bundles
  5. Smooth Klyachko vector bundles
  6. Applications and examples
  7. A canonical exact sequence
  8. Non-toric example
  9. Holomorphic Klyachko vector bundles

Key Result

Theorem 1

The category of toric vector bundles on $X(\Delta)$ is naturally equivalent to the category of finite-dimensional $k$-vector spaces $E$ with collections of decreasing filtrations $\{E^{\rho}(i)\}$ indexed by the rays of $\Delta$, satisfying Klyachko's Compability Condition.

Theorems & Definitions (71)

  • Definition 1: Klyachko's Compatibility Condition, cf. p2 of Pay07
  • Theorem 1: cf. p5 of Pay07
  • Definition 2: Def. \ref{['TKVS']}
  • Theorem 2: Theorems \ref{['main']}, \ref{['main2']}
  • Theorem 3: Theorem \ref{['main4']}
  • Definition 3: Def. \ref{['SKVB-real']}
  • Definition 4: Def. \ref{['SKVS-real']}
  • Theorem 4: Theorem \ref{['main3']}
  • Definition 1.1
  • Definition 1.2
  • ...and 61 more