All Regulators of Flat Bundles are Torsion

Cohomological Invariants of a Variation of Flat Connections

“…We note that when X is a smooth complex projective variety, Question 1 (i.e., when r = 0) was answered positively by A. Reznikov [12], and partial results in the quasiprojective case was obtained by Simpson and the author [8].…”

Characteristic classes of Higgs bundles and Reznikov’s theorem

“…It turns out that the classes v 2j+1 necessarily vanish in degrees three and higher over compact Kähler manifolds. This was a conjecture of Bloch [Blo78] and was proved by Reznikov [Rez95].…”

“…The classes have another geometric description, used by Reznikov [Rez95] (who calls these volume regulators), and which fits into the framework of Kamber and Tondeur's characteristic classes for foliated bundles [KT75]. LetX denote the universal cover of X and let ρ : π 1 (X) → GL(n, C) denote the holonomy representation of the flat bundle E. Consider the fiber bundle F =X × ρ GL(n, C)/U(n), whose smooth sections correspond to the space of hermitian metrics on E. For each k, we define…”

The Chern Character of a Parabolic Bundle, and a Parabolic Corollary of Reznikov’s Theorem

“…A parabolic bundle F on a variety X is a collection of vector bundles F α , indexed by a set of weights, i.e., α runs over a multi-indexing set This work is a sequel to [Iy-Si], which in turn was motivated by Reznikov's work on characteristic classes of flat bundles [Re], [Re2]. As a long-range goal we would like to approach the Esnault conjecture [Es2] that the Chern classes of Deligne canonical extensions of motivic flat bundles vanish in the rational Chow groups.…”

“…. , Dm n , and Reznikov's theorem [Re2] applies directly (or alternatively, over a finite Kawamata covering). The only knowledge which we can add is that our formula of Theorem 1.1 gives parabolic Chern classes in terms of the parabolic structure on X deduced from the flat bundle, and Reznikov's theorem can be stated as vanishing of these classes.…”