The universal de Rham / Spencer double complex on a supermanifold
Sergio Luigi CacciatoriUniversità degli Studi dell'Insubria, Via Valleggio 11, 22100, Como, Italy and INFN - Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
Simone NojaUniversität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
Riccardo ReUniversità degli Studi dell'Insubria, Via Valleggio 11, 22100, Como, Italy
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 -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.
Cite this article
Sergio Luigi Cacciatori, Simone Noja, Riccardo Re, The universal de Rham / Spencer double complex on a supermanifold. Doc. Math. 27 (2022), pp. 489–518DOI 10.4171/DM/875