Geodesic
The universal de Rham / Spencer double complex on a supermanifold
Documenta mathematica, Tome 27 (2022), pp. 489-518.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 14F10, 14F40, 58A50
Keywords: D-modules, universal de Rham complex, supergeometry 
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     author = {Cacciatori, Sergio Luigi and Noja, Simone and Re, Riccardo},
     title = {The universal de {Rham} / {Spencer} double complex on a supermanifold},
     journal = {Documenta mathematica},
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     volume = {27},
     year = {2022},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a56/}
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Cacciatori, Sergio Luigi; Noja, Simone; Re, Riccardo. The universal de Rham / Spencer double complex on a supermanifold. Documenta mathematica, Tome 27 (2022), pp. 489-518. http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a56/