Orbifold principal bundles on an elliptic fibration and parabolic principal bundles on a Riemann surface (Q1429240)

Let \(C\) be an elliptic curve, \(X\) a compact Riemann surface, \(S = \{ p_1, \dots ,p_h \} \) a finite set of distinct points of \(X\) and \(m_1, \dots ,m_h\) positive integers. The author considers an elliptic fibration \(f : Z\rightarrow X\), constructed by logarithmic transformations from \(X\times C\), such that \(f\) is trivial over \(X\setminus S\), \(f^{-1}(p_i)_{\text{red}} = C/(\mathbb Z/m_i\mathbb Z)\) and \(f^{-1}(p_i) = m_if^{-1}(p_i)_{\text{red}}\), \(i = 1, \dots ,h\). An orbifold vector bundle \(V\) over \(Z\) is a vector bundle over \(Z\) equipped with a lift of the action of \(C\). The author remarks that such a lift is unique, if it exists. The author constructs a bijective correspondence between orbifold vector bundles over \(Z\) and parabolic vector bundles over \(X\) with parabolic structure over \(S\) such that the parabolic weight at any \(p_i\) is an integral multiple of \(1/m_i\). The idea is that if \(V\) is an orbifold vector bundle over \(Z\) then \(f_{\ast}V\) is a vector bundle over \(X\) and \(V\) can be obtained from \(f^{\ast}({\mathcal O}_X(S)\otimes f_{\ast}V)\) by elementary transformations. Under this correspondence, semistable (resp., polystable) orbifold vector bundles over \(Z\) (with respect to a canonical polarization) correspond to semistable (resp., polystable) parabolic bundles over \(X\). The author further extends this correspondence to the context of principal \(G\)-bundles, where \(G\) is a connected semisimple algebraic group.