We study modules over stacks of deformation quantization algebroids on
complex Poisson manifolds. We prove finiteness and duality theorems in the
relative case and construct the Hochschild class of coherent modules. We prove
that this class commutes with composition of kernels, a kind of Riemann-Roch
theorem in the non-commutative setting. Finally we study holonomic modules on
complex symplectic manifolds and we prove in particular a constructibility
theorem.Comment: This paper develops the results of Deformation quantization modules I
(arXiv:0802.1245) and II (arXiv:0809.4309), and also contains new results and
new remarks. It will appear in Asterisque, Soc. Math. France (2012
