A Riemann-roch Theorem for One-dimensional Complex Groupoids

We consider a smooth groupoid of the form Σ ⋊ Γ where Σ is a Riemann surface and Γ a discrete pseudogroup acting on Σ by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C0(Σ) ⋊ Γ generalising 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∞(Σ)⋊ Γ.