A Riemann-Roch Theorem For One-Dimensional Complex Groupoids

TL;DR

This paper extends the Riemann-Roch theorem to one-dimensional complex groupoids by defining a K-cycle on a crossed product algebra and computing its Chern character using cyclic cohomology and index theory.

Contribution

It introduces a generalized Riemann-Roch theorem for complex groupoids, including a new Euler class from modular automorphisms, linking index theory and cyclic cohomology.

Findings

Defined a K-cycle generalizing the Dolbeault complex

Computed the Chern character in cyclic cohomology

Connected the Euler class with the modular automorphism group

Abstract

We consider a smooth groupoid of the form \Sigma\rtimes\Gamma where \Sigma is a Riemann surface and \Gamma a discrete pseudogroup acting on \Sigma by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C_0(\Sigma)\rtimes\Gamma 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^{\infty}(\Sigma)\rtimes\Gamma.