Characteristic classes of flat bundles

Abstract

which are finctorial and additive for exact sequences compatible with V. They are uniquely determined by those two properties and the definition of c,(E, V) (due to P. Deligne [2], (1.3), [6]). The group H2(X, Z( 1) + fix+A1) is identified with the group of isomorphism L classes (E, V) of rank one bundles E with an Al-connection V. Therefore V is integrable if and only if (E, V) lies in the subgroup H2(X, Z (1) + A’). This defines c,(E, V). M. Karoubi ([ll] and [12]) constructed with K-theory and cyclic homology classes E (E)E H2p-’ (X, @/Z(p)) when X is a simplicial set and E is a flat bundle. He told the aLthor that his classes are functorial and additive (and that he will write it down in the planned “Homologie cyclique et K-thkorie III”). This would imply c,(E, V) = E,(E) for flat bundles (2.252). The cohomology H2p(X, Z(p) + L?,) maps to the Deligne cohomology H2p(X, h(p),), whereZ(p),=Z(p)+Ox+. . . +i2f;-‘. Our classes c,(E, V) lif the Chern classes c;(E) in the Deligne cohomology (2.25.1) (see [Z] for a definition of c;(E)). If X has an Hodge structure, for example if X is algebraic proper over C, then the projection of c,(E, V) in H2p(X, Z(p)+ R(p)) is identified with c;(E) (2.251). J. Cheeger and J. Simons ( [4] ) constructed in a differential geometric framework classes ep(E)~ H 2p-1(X, R/E) when X is a C” manifold and E is a flat bundle. In general the relationship between c*,(E) and c;(E) is not known. When E is unitary and X has an Hodge structure, S. Bloch [3] and C. Soul& [13] proved that e,(E) lifts c;(E). Therefore in this case our classes lift the Cheeger-Simons classes (2.25.3) via the map C/B(p) -+ WPMP) =$ R/Z. Our method consists of two parts: the definitions of the “r-construction” and of the “Tproduct”. Iff: P + X is the flag bundle of E(2.7), the connection V defines a morphism T: n: +f*A’. The integrability condition implies that (rd)2 =0 and that T extends to the