Characteristic classes of holomorphic or algebraic foliated fiber bundles.
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 titles of each section and some aspects of the content. Section 1: Characteristic classes of foliated vector bundles. The author discusses the characteristic classes of foliations from his point of view and shows how to construct cocycles from the ones in truncated Weil algebras. Section 2: Characteristic classes of complex analytic bundles. The base manifold \(X\) is a complex manifold mostly with Hodge filtration. It is proved that if \(E\) is a holomorphic bundle with a partially holomorphic metric connection \(D\) (metric connection with \(D''= d''\)) defines a class determined by \((E, D, 0)\) in \(MK(X)\). This enables the author to define the homomorphism \(\Phi: K(X)\to MK(X)\) from the Grothendieck analytic \(K\)-theory to the multiplicative \(K\)-theory and to show it is an isomorphism when \(X\) is a Riemann surface. Section 3: Characteristic classes of complex algebraic bundles. Section 4: Regulators and the multiplicative \(K\)-theory of complex algebraic manifolds. In these sections, the filtrations are the Hodge- Deligne filtrations and similar calculations as in Section 2 are made. Also, the author recovers the homomorphism studied earlier [Astérisque, 149 (1987; Zbl 0648.18008)]\ concerning the theory of regulators. Section 5: Comparison with the other invariants of the flat bundles. Let \(E\) be a \(G\)-bundle with a connection \(D\) and \(P\) an invariant polynomial such that \(P(R) =0\) (\(R\) is the curvature of \(D\)). In this situation the map \(\gamma: MK^{(P)} (X)\to H^{2r-1} (X; \mathbb{C}/ \mathbb{Z})\) is defined and it is shown that the Chern-Cheeger-Simons invariant of \((E, D)\) is the image of the class of \([(E, D, 0)]\in MK^{(P)} (X)\). Section 6: Functorial properties of the multiplicative \(K\)-theory. Axiomatic characterization of characteristic classes. Let \(X\) be a filtered manifold and \({\mathcal O}\) be the sheaf \(U\to {\mathcal O} (U)= \text{Ker } d: \Omega^0 (U)\to \Omega^1 (U)/ F^1 (\Omega^1 (U))\). As in the case of usual characteristic classes of vector bundles, the existence and uniqueness of the characteristic classes (in \(MK(X)\)) for \(GL_n ({\mathcal O})\) bundles is proved. Section 7: Cohomological properties of the multiplicative \(K\)-theory. Relation with the cohomology of Deligne-Belinson. First the author constructs a space \(MK_X\) whose homotopy group \(\pi_n (MK_X)\) is naturally isomorphic to the multiplicative \(K\)-theory \(MK_n (X)\). Secondly, he introduces `multiplicative cohomology' to define `generalized Chern classes' for \(GL_n ({\mathcal O})\) bundles. It is pointed out that these classes are the same as the ones introduced by \textit{H. Esnault} [Topology 27, No. 3, 323-352 (1988; Zbl 0699.32016)]. The relations with the differential characters of Cheeger-Simons are also mentioned.
holomorphic bundle with a partially holomorphic metric connection
scientific article; zbMATH DE number 590953
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