A complex local system on a space S gives rise to "secondary" Chern classes in $H^{2p-1}(S;C/Z(p))$, refining the usual "primary" Chern classes in $H^{2p}(S;Z(p))$. In fact, Esnault in a survey article describes four methods of defining such classes, of which 3 are proved to be equivalent by means of her "modified splitting principle". I will explain how to show that the remaining 1 out of 4 definitions, that of Cheeger-Simons, agrees with the others. Then, changing gears, I will describe some arithmetic analogs of the phenomenon of rigidity of secondary Chern classes. This has bearing on another question from Esnault's article, and leads us to some motivic speculations.
Dustin Clausen (IHES)
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