Obstructions to Deformation Quantization of Bundles

Let $\left(M, \mathcal{O}_M \right)$ be a smooth algebraic variety over field $\kappa$ of characteristic $0$ with an algebraic symplectic form $\omega$, or a complex manifold with a holomorphic form $\omega$. Furthermore, let $E$ be a vector bundle over $\left(M, \mathcal{O}_M \right)$ and $\mathcal{O}_{\hbar}$ a deformation quantization of $\mathcal{O}_M$ compatible with $\omega$. Assuming that $E$ possesses a deformation quantization to order $\hbar^k$ we consider the problem of extending it to order $\hbar^\ell$ for $\ell > k$, and establish triviality of an obstruction class as a necessary condition for this extension to exist. Furthermore, in the case $\ell \le 2k+1$, we prove that this condition is also sufficient.