Stable Parabolic Bundles over Elliptic Surfaces and over Orbifold Riemann Surfaces

For an elliptic surface $q:Y \to \Sigma$, with prescribed singular fibres, Stefan Bauer proved directly via algebraic geometry that the stable bundles over $Y$, whose chern classes are pull backs from $\Sigma$, correspond to the stable (V-)bundles over $\Sigma$. We show, via a short proof in differential geometry, a generalisation to stable parabolic bundles. This uses extensions of Donaldson's deep result, giving the existence of Hermitian-Yang-Mills (or anti-self-dual) connections on stable parabolic bundles. In our cases these connections are flat and hence, correspond to representations of certain fundamental groups, which in turn are isomorphic, by Ue's work. To generalize Bauer's equivalence of the corresponding moduli spaces of stable bundles, we combine his arguments with Kronheimer & Mrowka's construction of the moduli spaces of stable parabolic bundles. Finally, we consider the pulling back of smooth parabolic bundles via $q$.