Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$G$-Connections on principal bundles over complete $G$-varieties

Bivas Khan, Mainak Poddar

Abstract

Let $X$ be a complete variety over an algebraically closed field $k$ of characteristic zero, equipped with an action of an algebraic group $G$. Let $H$ be a reductive group. We study the notion of $G$-connection on a principal $H$-bundle. We give necessary and sufficient criteria for the existence of $G$-connections extending the Atiyah-Weil type criterion for holomorphic connections obtained by Azad and Biswas. We also establish a relationship between the existence of $G$-connection and equivariant structure on a principal $H$-bundle, under the assumption that $G$ is semisimple and simply connected. These results have been obtained by Biswas et al. when the underlying variety is smooth.

$G$-Connections on principal bundles over complete $G$-varieties

Abstract

Let be a complete variety over an algebraically closed field of characteristic zero, equipped with an action of an algebraic group . Let be a reductive group. We study the notion of -connection on a principal -bundle. We give necessary and sufficient criteria for the existence of -connections extending the Atiyah-Weil type criterion for holomorphic connections obtained by Azad and Biswas. We also establish a relationship between the existence of -connection and equivariant structure on a principal -bundle, under the assumption that is semisimple and simply connected. These results have been obtained by Biswas et al. when the underlying variety is smooth.
Paper Structure (13 sections, 17 theorems, 126 equations)

This paper contains 13 sections, 17 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Atiyah sequence for G-connection
  4. Atiyah sequence for group action
  5. $G$-connection on vector bundle
  6. Homomorphisms and induced $G$-connections
  7. Infinitesimal deformation map and connection
  8. Equivariant structure and G-connection
  9. Examples
  10. Toric varieties
  11. Line bundle
  12. Trivial Action
  13. Tautological $G$-Connection

Key Result

Theorem 1.1

Let $H$ be a reductive linear algebraic group. Then a principal $H$-bundle $\mathcal{P}$ admits a $G$-connection if and only if the following conditions hold:

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • Proposition 3.7
  • ...and 29 more