AI 中文总结
研究在特定条件下向量丛\(E\)的变形量子化从\(\hbar^k\)阶扩展到\(\hbar^\ell\)阶（\(\ell>k\)）的问题，建立障碍类平凡性为必要条件，在\(\ell\leq2k + 1\)时此条件也是充分条件。
AI 中文摘要
设\((M,\mathcal{O}_M)\)是特征为\(0\)的域\(\kappa\)上具有代数辛形式\(\omega\)的光滑代数簇，或具有全纯形式\(\omega\)的复流形。设\(E\)是\((M,\mathcal{O}_M)\)上的向量丛，\(\mathcal{O}_{\hbar}\)是与\(\omega\)兼容的\(\mathcal{O}_M\)的变形量子化。假设\(E\)具有到\(\hbar^k\)阶的变形量子化，考虑将其扩展到\(\ell>k\)的\(\hbar^\ell\)阶的问题，并建立障碍类的平凡性作为该扩展存在的必要条件。此外，在\(\ell\leq2k + 1\)的情况下，证明该条件也是充分的。
英文摘要
Let $\left(M, \mathcal{O}_M \right)$ be a smooth algebraic variety over field $κ$ of characteristic $0$ with an algebraic symplectic form $ω$, or a complex manifold with a holomorphic form $ω$. 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 $ω$. 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.
