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A Riemann-Roch Theorem For One-Dimensional Complex Groupoids

Denis Perrot
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Abstract

We consider a smooth groupoid of the form Σ\rtimesΓwhere Σis a Riemann surface and Γa discrete pseudogroup acting on Σby local conformal diffeomorphisms. After defining a K-cycle on the crossed product C_0(Σ)\rtimesΓgeneralising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L^{\infty}(Σ)\rtimesΓ.

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Denis Perrot. 2001-03-13. A Riemann-Roch Theorem For One-Dimensional Complex Groupoids. https://doi.org/10.1007/s002200100404

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