2021/03/01 by Mikołaj Rotkiewicz, Rotkiewicz, Mikolaj, Elizaveta Vishnyakova +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2103.00665
openalex publication_date 2021/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We generalize the Donagi and Witten construction of a first obstruction class for splitting of a supermanifold via differential operators using the theory of n-fold vector bundles and graded manifolds. Applying the generalized Donagi--Witten construction we obtain a family of embeddings of the category of supermanifolds into the category of n-fold vector bundles and into the category of graded manifolds. This leads to a realization of any non-split supermanifold in terms of a collection of vector bundles and some morphism between them. Further we study the images of these embeddings into the category of graded manifolds in the case of a Lie supergroup and a Lie superalgebra. We show that these images satisfy universal property of a graded covering or a graded semicovering.