The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator on complex manifolds.

中文翻译：

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庞加莱引理、反精确形式和费米子量子谐波振荡器

本文重点介绍了在庞加莱引理中出现的同伦算子的各种性质和应用。第一部分构造了抽象算子演算，其中外导数为抽象导数，同伦算子起到抽象积分的作用。该算子演算可用于制定抽象的微分方程。给出了一个类似于费米子量子谐振子的特征值问题的例子。第二部分介绍了由复流形上的同伦算子生成的 Dolbeault 双复形的对偶复形。