Characteristic classes of holomorphic or algebraic foliated fiber bundles.
type: Item
instance of: scholarly article
title: Characteristic classes of holomorphic or algebraic foliated fiber bundles.
zbMATH Open document ID: 0833.57012
DOI: 10.1007/BF00961455
author: Max Karoubi
published in: \(K\)-Theory
publication date: 1996-03-24T00:00:00Z
review text: In a previous paper [\(K\)-theory 4, No. 1, 55-87 (1990; Zbl 0716.57018)], the author introduced the notions of multiplicative bundles and multiplicative \(K\)-theory and proposed to utilize them to describe various secondary characteristic classes in a unified manner. In the present paper, he gives and studies several examples in which his point of view applies and shows how these notions are used to clarify the nature of secondary classes in different situations. Let \(X\) be a smooth manifold. If \(X\) has additional geometric structures such as foliations or complex structures, the de Rham complex of \(X\), \(\Omega^* (X)\) may have some filtrations. For example, let \(F^1\) be the differential ideal generated by transverse forms of foliations then \(F^r= (F^1 )^r\) gives a filtration on a foliated manifold. Another example is the Hodge filtration on a complex manifold given by \(F^r \Omega (X)= \bigoplus_{p\geq r, p+q =n} \Omega^{p, q} (X)\). The manifolds equipped with such filtrations on its de Rham complex are called filtered manifolds. Let \(G\) be a Lie group and \(P= P_1+ P_2+ \cdots\), \(P_r\in I^r (G)\) an invariant series of \(G\). A multiplicative \(G\)-bundle on a filtered manifold \(X\) is a triple \((E, D, \omega)\), where \(E\) is a \(G\)- bundle, \(D\) is a connection on \(E\) and \(\omega= \omega_1+ \omega_2+ \cdots\), \(\omega_r\in \Omega^{2r-1}\) is a sum of differential forms such that \(P_r (D)\equiv d\omega_r \bmod F^r\). The set of isomorphism classes of multiplicative \(G\)-bundles forms the multiplicative \(K\)-theory associated to \(P\) which is denoted by \(MK^{(P)} (X)= MK(X)\). The most typical example is the one where \(G= GL_n\) and \(P= \text{Ch}= \sum \text{Ch}_r\), (Chern character). The following is a fundamental exact sequence which is used in many places of the paper: \[ [X,G ]\to \bigoplus_r H^{2r -1} (\Omega^* (X)/ F^r)\to MK(X)\to [X,BG ]\to \bigoplus_r H^{2r} (\Omega^* (X)/ F^r). \] To describe the content we list the tit…
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R32
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 32L99
zbMATH DE Number: 590953
zbMATH Keywords: Chern character
zbMATH Keywords: invariants of flat bundles
zbMATH Keywords: regulators
zbMATH Keywords: multiplicative \(K\)-theory
zbMATH Keywords: secondary characteristic classes
zbMATH Keywords: foliations
zbMATH Keywords: complex structures
zbMATH Keywords: transverse forms of foliations
zbMATH Keywords: Hodge filtration on a complex manifold
zbMATH Keywords: multiplicative \(G\)-bundle
zbMATH Keywords: filtered manifold
zbMATH Keywords: foliated vector bundles
zbMATH Keywords: truncated Weil algebras
zbMATH Keywords: holomorphic bundle with a partially holomorphic metric connection
zbMATH Keywords: complex algebraic bundles
zbMATH Keywords: complex algebraic manifolds
zbMATH Keywords: Hodge-Deligne filtrations
zbMATH Keywords: Chern-Cheeger-Simons invariant
zbMATH Keywords: cohomology of Deligne-Belinson
zbMATH Keywords: differential characters
reviewed by: Tadayoshi Mizutani
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 590953