Quantum moment map and obstructions to the existence of closed Fedosov star products

It is shown that the normalized trace of Fedosov star product for quantum moment map depends only on path component in cohomology class symplectic form and closed formal 2-form required to define connections (Theorem 1.3). As an application we obtain a family obstructions existence products naturally attached manifolds 1.5) K\"ahler 1.6). These are integral invariants depending form. Restricted compact re-discover obstruction found earlier [29].