A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface

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 $\beta \in H^0(X, {ad}(E_G))$, where ${ad}(E_G)$ is the adjoint vector bundle for $E_G$, such that the conjugacy class $\beta (x) \in {\mathfrak g}/G$, $x \in X$, is independent of $x$. We give a sufficient condition for the existence of a holomorphic connection on $E_G$ such that $\beta$ is flat with respect to the induced connection on ${ad}(E_G)$.

2010 Mathematics Subject Classification: 14H60, 14F05, 53C07

Keywords and Phrases: Holomorphic connection, adjoint bundle, flatness, canonical connection

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