Superconnection currents and complex immersions (Q910774)

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.