Equivariant principal bundles and logarithmic connections on toric varieties

Indranil Biswas, Arijit Dey and Mainak Poddar

Vol. 280 (2016), No. 2, 315–325

DOI: 10.2140/pjm.2016.280.315

Abstract

Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$-orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$-bundle $E_G\to M$ admits a $T$-equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$-bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection singular over $D$.

Keywords

smooth toric variety, logarithmic connection, equivariant principal bundle

Mathematical Subject Classification 2010

Primary: 14L30, 14M27

Secondary: 14M17

Milestones

Received: 11 March 2015

Revised: 8 June 2015

Accepted: 8 July 2015

Published: 28 January 2016