Orbifold principal bundles on an elliptic fibration and parabolic principal bundles on a Riemann surface. II. (Q816229)

Let \(X\) be a smooth projective curve over \(\mathbb C\), \(S=\{ p_1,\dots ,p_h\} \) a finite subset of \(X\), \(m_1,\dots ,m_h\) positive integers and \(C={\mathbb C}/({\mathbb Z}+{\mathbb Z}\tau )\) an elliptic curve. One can construct an elliptic fibration \(f : Z\rightarrow X\) such that : (1) \(C\) acts on \(Z\) with \(X\) as quotient, (2) the action of \(C\) on \(Z\setminus f^{-1}(S)\) is free, and (3) \(f^{-1}(p_i)_{\text{red}}\simeq {\mathbb C}/({\mathbb Z}/m_i{\mathbb Z})\) , \(i=1,\dots ,h\). Let \(G\) be a reductive linear algebraic group over \(\mathbb C\). An orbifold \(G\)-bundle over \(Z\) is a holomorphic principal \(G\)-bundle \(E_G\) over \(Z\) equipped with a holomorphic action of \(C\) on \(E_G\) such that the projection \(E_G\rightarrow Z\) is \(C\)-equivariant and the actions of \(G\) and \(C\) commute. Combining results of \textit{I. Biswas} [Collect. Math. 54, 293--308 (2003; Zbl 1049.14023)] and \textit{V. Balaji, I. Biswas} and \textit{D. S. Nagaraj} [J. Ramanujam Math. Soc. 18, 123--138 (2003; Zbl 1095.14032)] one sees that \(E^{\prime}_G := E_G/C\) is a ramified \(G\)-bundle over \(X\) with the property that \(\forall z\in E^{\prime}_G\) over \(p_i\in S\), the order of the isotropy subgroup \(G_z\subseteq G\) divides \(m_i\) and, conversely, every such ramified \(G\)-bundle over \(X\) is obtained from an orbifold \(G\)-bundle over \(Z\). An orbifold connection on \(E_G\) is an usual connection \(\nabla \) on the \(G\)-bundle \(E_G\), invariated by the action of \(C\) and such that, \(\forall z\in E_G\), \(\nabla \) vanishes on the tangent space in \(z\) at the \(C\)-orbit of \(z\). Assume, now, that \(G\) is simple. The author shows that, if \(E_G\) is a polystable orbifold \(G\)-bundle, the Hermite-Einstein connection on \(E_G\) is a flat holomorphic orbifold connection. Let \(P\subset G\) be a proper parabolic subgroup and \({\lambda}_0\) an antidominant character of \(P\) (such that the quotient \(G\times {\mathbb C}_{{\lambda}_0}/P\) by the action of \(P\) given by \((g,c)\cdot p := (gp,{\lambda}_0(p)^{-1}c)\) is an ample line bundle on \(G/P\)). Let \(m\) be the l.c.m. of \(m_1,\dots ,m_h\) and \(\lambda := {\lambda}_0^{\otimes m}\). The author shows that the line bundle \(E^{\prime}_G(\lambda ) := E^{\prime}_G\times {\mathbb C}_{\lambda}/P\) over \(E^{\prime}_G/P\) is numerically effective, but not ample. He also shows that if the genus of \(X\) is at least 2 and the polystable orbifold \(G\)-bundle \(E_G\) is ``general'' then the restriction of \(E^{\prime}_G(\lambda )\) to every proper closed subvariety of \(E^{\prime}_G/P\) is ample.