For a representation $\rho:\pi\_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E\_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes
$$\hat{c}\_k(E\_\rho)\in H^{2k-1}(M,R/Z)$$ 
as defined in <http://www.jstor.org/stable/1971013>


It is claimed at several places in the literature that $\hat{c}\_k(E\_\rho)$ is constant on components of the representation variety. (For example in Reznikov's proof of the Bloch conjecture <http://arxiv.org/pdf/dg-ga/9407007.pdf> where this fact is called rigidity of secondary characteristic classes and is attributed to the above-linked paper of Chern and Simons. However I was not able to find such a statement in that paper.)


Question: is there are a reference for this fact, or is there some well-known fact from which this local constantness follows?



Rigidity of secondary characteristic classes
To answer my own question: the wanted rigidity follows from Theorem 3.4 in <http://arxiv.org/pdf/math/9904131.pdf> together with the main result from P. Ntolo, Homologie de Leibniz d'algébres de Lie semi-simples, Comptes Rendus Acad. Sci. 318 (1994) 


UPDATE: the details of Reznikov‘s original argument have been worked out in a recent paper of Pitsch and Porti: <https://arxiv.org/pdf/1704.01321.pdf>


