I'm sorry for the self-citation. But your question is largely answered in the monograph Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck, or the arxiv version, joint work of Jean-Michel Bismut, Shu Shen, and me.
In the above work we proved the Grothendieck-Riemann-Roch theorem for coherent sheaves on compact complex manifolds. The target theory is the Bott-Chern cohomology.
We use antiholomorphic flat superconnections as a tool to study coherent sheaves on complex manifolds, on which we can build Chern-Weil theory. We then proved the Grothendieck-Riemann-Roch theorem as a version of the family index theorem.
