Superconnections and the Chern character. (English) Zbl 0569.58030

Let \(u: E^ 0\to E^ 1\) be a homomorphism of complex vector bundles over a smooth manifold M. If u is an isomorphism over an open subset U, then u determines a class in the relative K-group K(M,U). The author presents a new way to construct the Chern character in de Rham cohomology of such a K-class. The construction involves a ”super” (or \({\mathbb{Z}}_ 2\)-graded) variant on \(E=E^ 0\oplus E^ 1\) of the concept of connection and was motivated by the problem of generalizing the heat kernel proof of the Atiyah-Singer index theorem to prove a local index theorem for families of elliptic operators. One finds a suitable representative for the Chern character of the index of the family as a differential form, whose de Rham class is independent of the choice of superconnection.
Reviewer: M.Craioveanu


58J20 Index theory and related fixed-point theorems on manifolds
53C05 Connections (general theory)
55N15 Topological \(K\)-theory
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