论文标题
Universal de Rham/Spencer Double Complex上
The Universal de Rham/Spencer Double Complex on a Supermanifold
论文作者
论文摘要
众所周知，在平滑或分析的歧管上，带束带的普遍的Spencer和De Rham复合物在$ \ Mathcal {d} $ - 模块的理论中起着基本作用。在本文中，我们考虑了将两种复合物概括为任意超级if的双层双带，我们使用它来统一在真实，复杂和代数的超级男性群上的差异和积分形式的概念。相关的频谱序列使DE RHAM复合体的差异形式和第一个集成形式的复合物。对于真实和复杂的超级命中，两个光谱序列在第二页上收敛到局部恒定的捆。我们利用这一事实表明，差异形式的共同体学是对整体形式的共同学的同构，并且它们都计算了减少的歧管的De Rham的共同体。此外，我们表明，与普通复杂歧管的情况相反，超级男性群的hodge-to-de-de rham（或frölicher）光谱序列与Kähler降低的歧管通常不会在第一页上收敛。
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.
