Riemann-Roch theorems via deformation quantization. II (Q1604334)

A deformation quantization of a complex manifold \(M\) is a formal one-parameter deformation of the structure sheaf \({\mathcal O}_M\), i.e., a sheaf of algebras \(A^t_M\) flat over \(\mathbb{C}[[t]]\) together with an isomorphism of algebras \(\psi: A^t_M\otimes_{\mathbb{C}[[t]]}\mathbb{C}\to {\mathcal O}_M\). The formula \(\{f,g\}={1\over t}[\widetilde f,\widetilde g]+ tA^t_M\), where \(f\), \(g\) are local sections of \({\mathcal O}_M\) and \(\widetilde f\), \(\widetilde g\) are their lifts to \(A^t_M\), defines a Poisson structures on \(M\) called the Poisson structure associated to the deformation quantization \(A^t_M\). A deformation quantization \(A^t_M\) is called symplectic if the associated Poisson structure is nondegenerate. The aim of this second part of the paper is to give a proof of a certain Riemann-Roch type theorem for symplectic deformations. For Part I, cf. ibid. 167, 1-25 (2002; Zbl 1021.53064).