超流形上的泛de Rham/Spencer二重复形
The Universal de Rham/Spencer Double Complex on a Supermanifold
摘要：
光滑流形或解析流形上层的泛Spencer复形和de Rham复形在$\mathcal{D}$ -模理论中起着重要的作用。在这篇文章中，我们考虑一个二重复形的层推广了这两个复杂的任意的超流形，我们用它来统一的概念，微分和积分形式的实，复和代数超流形。相关的谱序列在第一页给出了微分形式的德拉姆复形和积分形式的复形。对于实和复的超流形，两个谱序列都在第二页收敛到局部常数层。我们利用这一事实表明，微分形式的上同调是同构的积分形式的上同调，他们都计算de Rham上同调的约化流形。此外，我们还证明了，与普通的复流形相比，具有Kähler约化流形的超流形的Hodge-to-de Rham（或Frölicher）谱序列一般不收敛于第一页.
英文摘要：
The universal Spencer and de Rham complexes of sheaves over a smooth or analytical manifold are well known to play a basic role in the theory of $\mathcal{D}$-modules. In this article we consider a double complex of sheaves generalizing both complexes for an arbitrary supermanifold, and we use it to unify the notions of differential and integral forms on real, complex and algebraic supermanifolds. The associated spectral sequences give the de Rham complex of differential forms and the complex of integral forms at page one. For real and complex supermanifolds both spectral sequences converge at page two to the locally constant sheaf. We use this fact to show that the cohomology of differential forms is isomorphic to the cohomology of integral forms, and they both compute the de Rham cohomology of the reduced manifold. Furthermore, we show that, in contrast with the case of ordinary complex manifolds, the Hodge-to-de Rham (or Frölicher) spectral sequence of supermanifolds with Kähler reduced manifold does not converge in general at page one.
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