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Geometry of twisting cochains. (English) Zbl 0641.32021
The results here extend the work of D. Toledo and Y. L. Tong on the Hirzebruch-Riemann-Roch formula for complex manifolds [Topology 15, 273-301 (1976; Zbl 0355.58014)]. Their proof of the formula involves the construction, for any holomorphic vector bundle E over a smooth complex manifold X, of a certain Čech cocycle which for compact X integrates to give the Euler characteristic $\chi (X,E)=\Sigma (-1)^ i\dim_{{\mathbb{C}}}H^ i(X,E).$ The argument is based on Serre- Grothendieck duality for the diagonal embedding $$\Delta$$ : $$X\to X\times X$$ and comparison of local Koszul resolutions for the sheaf of the diagonal by a ‘twisting cochain’ o) in Hilbert space. We succeed in constructing an operator-valued symbol for finite sections of such operators, and we set up necessary and sufficient conditions for the finite section method to be applicable. In the present paper we extend these results to the Banach space case. Our results essentially generalize well-known statements operators, for singular integral operators and for singular integral operators with Carleman shift.
Reviewer: Z.Binderman
##### MSC:
 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 57R20 Characteristic classes and numbers in differential topology 14C40 Riemann-Roch theorems
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##### References:
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