From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves replaced by vector bundles. Unfortunately I don't know enough algebraic geometry to carryout the translation (for example I'm not sure what the pushforward of quasi-coherent sheafs corresponds to).
Does GHRR just amount to the Atiyah-Singer families index theorem for the fiberwise Dolbeault-Dirac operator?