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**Holomorphic principal bundles over elliptic curves. II: The parabolic construction.**
*(English)*
Zbl 1033.14016

For part I of this paper see: http://front.math.ucdavis.edu/math.AG/981130.

From the paper: Let \(E\) be a smooth elliptic curve and let \(G\) be a simple complex algebraic group of rank \(r\). We shall always assume that \(\pi_1(G)\) is cyclic and \(c\) is a generator. The goal of this paper is to continue the study, begun in [part I], of the moduli space \({\mathcal M}(G,c)\) of semistable holomorphic \(G\)-bundles \(\xi\) with \(c_1(\xi)=c\). In part I this space was studied from the transcendental viewpoint of \((0,1)\)-connections using the results of Narasimhan-Seshadri and Ramanathan that in every \(S\)-equivalence class there is a unique representative whose holomorphic structure is given by a flat connection. This viewpoint, however, is not suitable for many questions, such as finding universal bundles, studying singular elliptic curves, or generalizing to families of elliptic curves. In this paper, which is largely independent of [part I], we describe \({\mathcal M}(G,c)\) from an algebraic point of view.

The moduli space is constructed by considering deformations of a minimally unstable \(G\)-bundle. The set of all such deformations can be described as the \(\mathbb{C}^*\)-quotient of the cohomology group of a sheaf of unipotent groups, and we show that this quotient has the structure of a weighted projective space. We identify this weighted projective space with the moduli space of semistable \(G\)-bundles, giving a new proof of a theorem of E. Looijenga [Invent. Math. 38, 17–32 (1976; Zbl 0358.17016)].

From the paper: Let \(E\) be a smooth elliptic curve and let \(G\) be a simple complex algebraic group of rank \(r\). We shall always assume that \(\pi_1(G)\) is cyclic and \(c\) is a generator. The goal of this paper is to continue the study, begun in [part I], of the moduli space \({\mathcal M}(G,c)\) of semistable holomorphic \(G\)-bundles \(\xi\) with \(c_1(\xi)=c\). In part I this space was studied from the transcendental viewpoint of \((0,1)\)-connections using the results of Narasimhan-Seshadri and Ramanathan that in every \(S\)-equivalence class there is a unique representative whose holomorphic structure is given by a flat connection. This viewpoint, however, is not suitable for many questions, such as finding universal bundles, studying singular elliptic curves, or generalizing to families of elliptic curves. In this paper, which is largely independent of [part I], we describe \({\mathcal M}(G,c)\) from an algebraic point of view.

The moduli space is constructed by considering deformations of a minimally unstable \(G\)-bundle. The set of all such deformations can be described as the \(\mathbb{C}^*\)-quotient of the cohomology group of a sheaf of unipotent groups, and we show that this quotient has the structure of a weighted projective space. We identify this weighted projective space with the moduli space of semistable \(G\)-bundles, giving a new proof of a theorem of E. Looijenga [Invent. Math. 38, 17–32 (1976; Zbl 0358.17016)].

### MSC:

14H10 | Families, moduli of curves (algebraic) |

14D20 | Algebraic moduli problems, moduli of vector bundles |

14H60 | Vector bundles on curves and their moduli |