# A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface

### Indranil Biswas

School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400005 India

## Abstract

Let $E_{G}$ be a holomorphic principal $G$-bundle over a compact connected Riemann surface, where $G$ is a connected reductive affine algebraic group defined over $C$, such that $E_{G}$ admits a holomorphic connection. Take any $β∈H_{0}(X,ad(E_{G}))$, where $ad(E_{G})$ is the adjoint vector bundle for $E_{G}$, such that the conjugacy class $β(x)∈g/G,x∈X$, is independent of $x$. We give a sufficient condition for the existence of a holomorphic connection on $E_{G}$ such that $β$ is flat with respect to the induced connection on $ad(E_{G})$.

## Cite this article

Indranil Biswas, A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface. Doc. Math. 18 (2013), pp. 111–120

DOI 10.4171/DM/393