A classical approach to smooth supermanifolds

Abstract

This is an extension of the author's Master's thesis written under the supervision of Dr. Gregor Weingart at the National Autonomous University of Mexico. By extension we mean that some of the results have been rewritten and some others have been added to the original work. The purpose of this study is to rewrite di erential supergeometry in terms of classical di erential geometry. This rewriting from rst principles has two main motivations: 1. avoid using local (and usually not very well-de ned) odd coordinates; 2. use both the language and the tools (both highly developed) of classical di erential geometry to state and prove results in supergeometry. Although there is work in this direction (for instance [SV86]) this work's point of view might be useful to translate from the sheaf-theoretic language to one that is better suited for explicit calculations. We now give a summary of the contents. Chapter 1 is about superalgebras. In particular, we give explicit isomorphisms for the space of superderivations of an exterior algebra. The nal section of this chapter is devoted to the twisted action of the symmetric group on the supervector space of tensors with a xed rank; this allows us to construct the supersymmetric and superexterior algebras of a nite-dimensional supervector space. In chapter 2 we begin our study of smooth nite-dimensional supermanifolds. As the title of this monograph indicates, our approach will be rather classical in the following sense: our de nition of a supermanifold is not given in terms of local charts nor in terms of sheaves of superalgebras; we rather study superalgebra bundles over smooth manifolds. That is, supermanifolds in our sense are vector bundles such that the bre at each point is a free supercommutative superalgebra of nite rank. This approach allows the use of tools from di erential geometry which in many cases (e.g. Batchelor's theorem, which we prove as corollary 2.12) simpli es the proofs; furthermore, we prove (theorem 2.11) that both approaches are equivalent. A noteworthy feature of our approach is that supersmooth maps turn out to be generalizations of linear di erential operators (proposition 2.6). With our approach we prove special splittings of the tangent bundle (theorem 2.25) and study the tangential maps of a supersmooth map between two supermanifolds which give conditions for the supermanifold to be split (proposition 2.36). We also prove that the de Rham cohomology of a supermanifold is isomorphic to the de Rham cohomology of the underlying smooth manifold (theorem 2.47) by writing down the exterior derivative in a di erent way (2.45). We include three appendices. The rst one concerns linear di erential operators; there we develop the necessary tools to understand supersmooth maps with our approach. The two other appendices deal with algebraic results of independent interest. The second appendix is the statement and proof of the Cartan Poincare lemma, a result on the (co)homology of a complex that arises in our study. The third appendix deals with the proof of an algebraic fact (lemma 1.26) we use in order to prove a owbox coordinates theorem for supermanifolds (theorem 2.34) which is the foundation of the main results of chapter 2.