Stable principal bundles and reduction of structure group

Abstract

Let $E_G$ be a stable principal $G$--bundle over a compact connected Kaehler manifold, where $G$ is a connected reductive linear algebraic group defined over the complex numbers. Let $H\subset G$ be a complex reductive subgroup which is not necessarily connected, and let $E_H\subset E_G$ be a holomorphic reduction of structure group. We prove that $E_H$ is preserved by the Einstein-Hermitian connection on $E_G$. Using this we show that if $E_H$ is a minimal reductive reduction in the sense that there is no complex reductive proper subgroup of $H$ to which $E_H$ admits a holomorphic reduction of structure group, then $E_H$ is unique in the following sense: For any other minimal reductive reduction $(H^{\prime },E_{H^{\prime }})$ of $E_G$, there is some element $g$ of $G$ such that $H^{\prime }=g^{-1}Hg$ and $E_{H^{\prime }}=E_{H}g$. As an application, we give an affirmative answer to a question of Balaji and Kollár.

10 pages

Cited by in corpus (1)

• Katz's middle convolution algorithm