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Superconnection currents and complex immersions. (English) Zbl 0696.58006

The present paper contains complete proofs of the results which have been announced in C. R. Acad. Sci., Paris, Sér. I 307, No.10, 523-526 (1988; Zbl 0652.32020). Given an immersion \(M'\to M\) of complex manifolds, a vector bundle \(\eta\) on \(M'\), and a finite complex (\(\xi\),v) of Hermitian vector bundles on M which provides a projective resolution of the sheaf of sections of \(\eta\) the author transfers Quillen’s family \((\omega_ u)_{u\in {\mathbb{R}}_+}\) of superconnection currents from \({\mathbb{Z}}_ 2\)- graded bundles to (\(\xi\),v) and investigates the limit \(\omega_{\infty}\). He proves its existence and expresses \(\omega_{\infty}\) in terms of integrals of Gaussian shaped differential forms on the normal bundle N of \(M'\). Under certain compatibility assumptions for the metrics on \(\xi\), N, and \(\eta\) the limit \(\omega_{\infty}\) is explicitly calculated using Chern-Weil representatives of \(Td^{-1}(N)ch(\eta)\). For later applications to intersection theory (Bismut, Gillet, Soulé; to appear) the speed of convergence \(\omega_ u\to \omega_{\infty}\) and the behaviour of the wave front sets are precisely controlled. This together with complicated algebraic identities make the present paper quite technical.
Reviewer: K.Lamotke

MSC:

58A50 Supermanifolds and graded manifolds
53C05 Connections (general theory)
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
57R42 Immersions in differential topology
57R20 Characteristic classes and numbers in differential topology
53C65 Integral geometry
58A10 Differential forms in global analysis

Citations:

Zbl 0652.32020
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References:

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