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Algebraic index theorem for families. (English) Zbl 0837.58029

Let \(X\to B\) be a smooth bundle with fiber \(F\), \(X\) compact; \(D= \{D_b\}\) – a family of elliptic differential operators on fibers \(F_b\), \(b\in b\). Such a family gives rise to the element \(\text{ind}(D)\) of \(K^0(B)\). The Atiyah-Singer index theorem says that \[ ch(\text{ind}(D))= \int ch[\sigma(D)]\cdot Td(T_{X/B}\otimes \mathbb{C}), \] where \([\sigma(D)]\) is the class in \(K^0(X)\) canonically associated to the principal symbol of \(D\).
The aim of this paper is to generalize the Atiyah-Singer index theorem and prove it as explicitly as possible in purely algebraic context. Generalized Chern classes, flat Fedosov connection, Gelfand-Fuks cohomology, and special hyperconnections are used in proving this generalized theorem.

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
19E20 Relations of \(K\)-theory with cohomology theories
53C05 Connections (general theory)
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