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The Non-abelian Hodge Correspondence for Non-Compact Curves

In this talk I will describe the non-abelian Hodge theory of a non-compact curve. This was worked out by Simpson in the paper Harmonic Bundles on Non-Compact Curves, and as such almost everything I say here can be found in that paper in more detail. Let X be a smooth non-compact curve, with smooth completion j : X ↪→ X, and S = X \X a finite set of punctures. For simplicity, I’ll assume we have only a single puncture S = {s}. To produce a non-abelian Hodge correspondence in this setting, we’ll need to consider filtered analogues of flat and Higgs bundles, and impose tameness conditions on the behaviour of the Higgs field or connection with respect to this filtration.