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Stable parabolic fibrations and flat singular connections. (Fibrés paraboliques stables et connexions singulières plates.) (French) Zbl 0769.53013

The theorem of Narasimhan and Seshadri on the existence of flat unitary connections on stable bundles over a Riemannian surface was given a different derivation by S. K. Donaldson [J. Differ. Geom. 18, 269- 277 (1983; Zbl 0504.49027)] using moment map ideas and Uhlenbeck’s weak compactness theorem for gauge potentials. The setting for this result has now broadened considerably in scope, thanks to the new 3-manifold invariants which require a more detailed study of the moduli spaces of flat connections on surfaces. In particular, the study of connections on surfaces with punctures is fundamental. The analogue of Narasimhan and Seshadri’s theorem in this case is given in [V. B. Mehta and C. S. Seshadri, Math. Ann. 248, 205-239 (1980; Zbl 0454.14006)]. In the paper under review, the author provides a new proof of this based on Donaldson’s original work. What is required is to develop an appropriate analysis for connections with a certain type of singularity at the marked points. The author uses weighted Sobolev spaces and derives Sobolev inequalities and compactness theorems in this context, culminating in a proof of the theorem of Mehta and Seshadri. There are other approaches to this problem, some using orbifold methods which deal effectively with rational weights, but this is probably the first head-on analytical attack on the question.

MSC:

53C05 Connections (general theory)
58D27 Moduli problems for differential geometric structures
32Q20 Kähler-Einstein manifolds
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References:

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