The universal de rham/spencer double complex on a supermanifold

• Verfasst von: Cacciatori, Sergio L. [VerfasserIn]
• Verfasst von: Noja, Simone [VerfasserIn]
• Verfasst von: Re, Riccardo [VerfasserIn]
• Titel: The universal de rham/spencer double complex on a supermanifold
• Verf.angabe: Sergio L. Cacciatori, Simone Noja, and Riccardo Re
• Jahr: 2022
• Umfang: 30 S.
• Fussnoten: Gesehen am 01.09.2022
• Titel Quelle: Enthalten in: Documenta mathematica
• Ort Quelle: Berlin : Deutsche Mathematiker-Vereinigung e.V., 1996
• Jahr Quelle: 2022
• Band/Heft Quelle: 27(2022), Seite 489-518
• ISSN Quelle: 1431-0643
• DOI: doi:10.25537/dm.2022v27.489-518
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: D-modules
• Sach-SW: super riemann surfaces
• Sach-SW: supergeometry
• Sach-SW: universal de Rham complex
• K10plus-PPN: 1815583061

Abstract

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 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 Frolicher) spectral sequence of supermanifolds with Kahler reduced manifold does not converge in general at page one.