$A_\infty$-形变及其导出范畴
$ A_\infty $-Deformations and their Derived Categories
AI总结：
本文研究带曲率$A_\infty$-形变的导出范畴定义难题，通过迭代优化曲率与同调分裂，建立形变Kadeishvili定理，并用树构造极小模型上的$A_\infty$-结构。
摘要：
处理带曲率的$A_\infty$-形变颇具挑战，而定义它们的导出范畴似乎是不可能的。本文展示了如何接纳曲率，并在这些明显的困难之下构建导出范畴。我们通过一个形变的Kadeishvili定理显式描述它们的$A_\infty$-结构。本文首先面向非专业读者回顾$A_\infty$-形变，解释为何$A_\infty$-形变需要曲率，以及曲率为何被认为会引发问题。我们证明，无穷小带曲率的$A_\infty$-形变实际上并无问题，并给出其扭曲完备化和极小模型的正确定义。经典Kadeishvili定理在带曲率形变中失效；我们展示如何通过迭代优化曲率并无穷小地改变同调分裂来对其加以调整。极小模型上最终的$A_\infty$-结构以树的形式构造。
英文摘要：
Handling curved $ A_\infty $-deformations is challenging and defining their derived categories seems impossible. In this paper, we show how to welcome the curvature and build derived categories despite the apparent difficulties. We explicitly describe their $ A_\infty $-structure by a deformed Kadeishvili theorem. This paper starts with a review of $ A_\infty $-deformations for non-specialists. We explain why $ A_\infty $-deformations require curvature and why curvature supposedly causes problems. We prove that infinitesimally curved $ A_\infty $-deformations are in fact non-problematic and provide the correct definitions for their twisted completion and minimal model. The classical Kadeishvili theorem collapses for curved deformations. We show how to adapt it by iteratively optimizing the curvature and changing the homological splitting infinitesimally. The final $ A_\infty $-structure on the minimal model is constructed in terms of trees.
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