Principal \(G\)-bundles over elliptic curves. (English) Zbl 0937.14019

From the introduction: Let \(E\) be an elliptic curve with origin \(p_0\), and \(G\) be a complex simple algebraic group. The goal of this note is to announce some results concerning the moduli of principal \(G\)-bundles over \(E\). Detailed proofs, as well as a more thorough discussion of the case where \(E\) is allowed to be singular or to vary in families and of the connection with del Pezzo surfaces, elliptic \(K3\) surfaces, and Calabi-Yau manifolds which are elliptic or \(K3\) fibrations, will appear elsewhere. In section 2, we describe the moduli space of semistable \(G\)-bundles over \(E\) via flat connections for the maximal compact subgroup \(K\) of \(G\), or equivalently via conjugacy classes of representations \(\rho:\pi_1(E)\to K\). Such bundles, which for a simply connected group \(G\) are exactly the bundles whose structure group reduces to a Cartan subgroup, have an automorphism group which is as large as possible in a certain sense within a fixed \(S\)-equivalence class. The main result here is a theorem due to Looijenga and Bernshtein-Shvartsman which describes this moduli space as a weighted projective space. At the end of the section, we connect this description, in the case where \(G=E_6\), \(E_7\), \(E_8\), with the moduli space of del Pezzo surfaces of degree \(3,2,1\) respectively and with the deformation theory of simple elliptic singularities. – In section 3, we describe regular \(G\)-bundles, which by contrast with flat bundles have automorphism groups whose dimensions are as small as possible within a fixed \(S\)-equivalence class. – In section 4, we show how special unstable bundles over certain maximal parabolic subgroups can be used to give another description of the moduli space in terms of regular bundles and obtain a new proof of the theorem of Looijenga and Bernshtein-Shvartsman. – Finally, in the last section we discuss the existence of universal bundles and give a brief description of how our construction can be twisted with the help of a certain spectral cover.


14H60 Vector bundles on curves and their moduli
14L30 Group actions on varieties or schemes (quotients)
14H52 Elliptic curves
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