Direct connection-theoretic comparison on the tautological Rees object

Develop a direct connection-theoretic realization of the comparison between the Deligne-Beilinson Chern class and the regulator class on the tautological bisimplicial Rees object, despite the absence of a projective completion carrying the logarithmic connection structure required to evaluate the class as a Chern-Simons invariant.

Background

The paper compares the regulator with the Deligne-Beilinson Chern class on a universal tautological object using algebraic K-theory and A-realizations. Connections are introduced only afterward, on the specific Rees bundle associated with the given geometric data, where projective completion and the Dupont–Hain–Zucker and Burgos–Gil theories are available. The universal tautological object itself has no connection, and the authors explicitly identify the inability to perform the connection-theoretic comparison directly there as an unresolved issue.