## 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)

### Keywords:

Chern class; connection theory; Atiyah-Singer index theorem
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