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Geometric integration theory on supermanifolds. (English) Zbl 0839.58014
Sov. Sci. Rev., Sect. C 9, No. 1, i-iv, 1-138 (1992); corrigendum No. 2, 105-106 (1992).
Summary: This is the first detailed and original account of the author’s theory of forms on supermanifolds – a correct and nontrivial analogue of Cartan-de Rham theory based on new concepts. It constructs graded forms of degree \(r |s\) integrable over singular manifolds for any ‘super’ dimension \(r |s\). Complexes of \(\bullet |s\) forms lead to nontrivial ‘de Rham supercohomologies’ which are homotopic invariants of the supermanifolds. The Stokes formula proved by the author allows the cohomologies to be paired with the bordisms of the supermanifolds, which are completely computed in classical terms. The paper develops the apparatus of supermanifold differential topology that is necessary for the integration theory. One of the key features is the identification of a class of proper morphisms (inducing fibrewise monomorphisms of the normal bundles to the carrier), intimately connected with Berezin integration and of fundamental importance in various problems. Examples of applications include the apparatus of supermathematics for classical objects (forms, multivector fields and spinors), a new approach to questions of integral geometry, and an analytic proof of the Atiyah-Singer index theorem. The work also contains a compressed introduction to superanalysis and supermanifolds, free from algebraic formalism and setting out afresh such challenging difficulties as the Berezin integral on a bounded domain and the like.
Reviewer: Reviewer (Berlin)

58C50 Analysis on supermanifolds or graded manifolds