Geodesic
A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface
Documenta mathematica, Tome 18 (2013), pp. 111-120.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 ${ad}(E_G)$.
Classification : 14H60, 14F05, 53C07
Keywords: holomorphic connection, adjoint bundle, flatness, canonical connection 
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     author = {Biswas, Indranil},
     title = {A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a {Riemann} surface},
     journal = {Documenta mathematica},
     pages = {111--120},
     publisher = {mathdoc},
     volume = {18},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a45/}
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Biswas, Indranil. A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface. Documenta mathematica, Tome 18 (2013), pp. 111-120. http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a45/