Rationality of secondary classes

Abstract

We prove the Bloch conjecture : C2(E) £ H^(X,Z(2)) is torsion for holomorphic rank-two vector bundles E with an integrable connection over a complex projective variety X.We prove also the rationality of the Chern-Simons invariant of compact arithmetic hyperbolic three-manifolds.We give a sharp higher-dimensional Milnor inequality for the volume regulator of all representations to PSO(l,n) of fundamental groups of compact n-dimensional hyperbolic manifolds, announced in our earlier paper. The theorem 1.1.Let X be a smooth complex projective variety.Consider a representation p : π λ (X) -> SX(2,C).Let E p be the corresponding rank-two vector bundle over X. Viewing E p as an algebraic vector bundle, denote by c 2 (E p ) the second Chern class in Deligne cohomology group iJ£(X,Z(2)) ([15], [20]).Recall that there is an exact sequence 0 -> J 2 {X) -> i?|>(X,Z(2)) -> # 4 (X,Z(2)), and by the Chern-Weil theory, the image of c 2 (E p ) in H 4 (X, Z(2)) is torsion.Therefore c 2 (E p ) lies in the image of H 3 (X,C/Z) under the natural map H 3 (XX/Z) -> H 3 (X,C/Z(2)) -> iϊ|,(X,Z(2)).It was proved by Bloch [3], Gillet-Soule [24] and Soule [50] that in fact, c 2 (E p ) is an image of the secondary characteristic class Ch(p) of a flat bundle E p (equivalently, of a representation p), lying in iϊ 3 (X,C/Z).The R/Zpart of this class was introduced and studied by Chern-Simons [9] and Cheeger-Simons [8],and will be called Cheeger-Chern-Simons class and denoted ChS(p).The R-part lying in H 3 (X,R) will be called Borel hyperbolic volume class (regulator) and denoted Vol(p).Remark that if p is unitary, then Vol(p) = 0. Next, for a field F denote B(F) the Bloch group of F. Recall that for F algebraically closed there is an exact sequence 0 -> μf 2 -* H 3 (SL(2,F),Z) -> B(F) -> 0 of Bloch-Wigner-Dupont-Sah [19].The dilogarithm function of Bloch-Wigner defines a homomorphism D : B(C) -» C/Q = E/Q Θ iE which splits to

Locations

• Journal of Differential Geometry - View - PDF
• arXiv (Cornell University) - PDF