标题： 非正则变形的全纯函数代数在多圆盘和球体上的$\mathbb C^n$ 显示英文标题

标题： Nonformal deformations of the algebras of holomorphic functions on the polydisk and on the ball in $\mathbb C^n$

摘要： 我们构造了弗雷歇$\mathcal O(\mathbb C^\times)$-代数$\mathcal O_{\mathrm{def}}(\mathbb D^n)$和$\mathcal O_{\mathrm{def}}(\mathbb B^n)$，它们可以解释为代数$\mathcal O(\mathbb D^n)$和$\mathcal O(\mathbb B^n)$的非形式（或更准确地说，全纯）变形，这些代数分别是多圆盘$\mathbb D^n\subset\mathbb C^n$和球体$\mathbb B^n\subset\mathbb C^n$上的全纯函数代数。 我们的代数在$q\in\mathbb C^\times$上的纤维与之前引入的“量子多圆盘”和“量子球”代数$\mathcal O_q(\mathbb D^n)$和$\mathcal O_q(\mathbb B^n)$同构。 我们证明代数$\mathcal O_{\mathrm{def}}(\mathbb D^n)$和$\mathcal O_{\mathrm{def}}(\mathbb B^n)$在$\mathbb C^\times$上给出连续的弗雷歇代数丛，这些是$\mathbb D^n$和$\mathbb B^n$的严格形变量子化（在里费尔的意义下）。 我们还给出了$\mathcal O_{\mathrm{def}}(\mathbb D^n)$的非交换幂级数解释，并将其应用于证明$\mathcal O_{\mathrm{def}}(\mathbb D^n)$在$\mathcal O(\mathbb C^\times)$上不是拓扑投射的（更不用说是拓扑自由的了）。最后，我们考虑$\mathcal O(\mathbb D^n)$和$\mathcal O(\mathbb B^n)$的相应形式变形，并证明它们可以通过标量扩张从全纯变形中得到。 显示英文摘要

摘要： We construct Fr\'echet $\mathcal O(\mathbb C^\times)$-algebras $\mathcal O_{\mathrm{def}}(\mathbb D^n)$ and $\mathcal O_{\mathrm{def}}(\mathbb B^n)$ which may be interpreted as nonformal (or, more exactly, holomorphic) deformations of the algebras $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$ of holomorphic functions on the polydisk $\mathbb D^n\subset\mathbb C^n$ and on the ball $\mathbb B^n\subset\mathbb C^n$, respectively. The fibers of our algebras over $q\in\mathbb C^\times$ are isomorphic to the previously introduced ``quantum polydisk'' and ``quantum ball'' algebras, $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$. We show that the algebras $\mathcal O_{\mathrm{def}}(\mathbb D^n)$ and $\mathcal O_{\mathrm{def}}(\mathbb B^n)$ yield continuous Fr\'echet algebra bundles over $\mathbb C^\times$ which are strict deformation quantizations (in Rieffel's sense) of $\mathbb D^n$ and $\mathbb B^n$. We also give a noncommutative power series interpretation of $\mathcal O_{\mathrm{def}}(\mathbb D^n)$ and apply it to showing that $\mathcal O_{\mathrm{def}}(\mathbb D^n)$ is not topologically projective (and a fortiori is not topologically free) over $\mathcal O(\mathbb C^\times)$. Finally, we consider respective formal deformations of $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$, and we show that they can be obtained from the holomorphic deformations by extension of scalars.