Torsion of extended Chern-Simons classes for canonical extensions of flat bundles

Let $X$ be a smooth complex projective variety and $D=D_1+\cdots +D_k\subset X$ a divisor with simple normal crossings. Consider Deligne’s canonical extension $(F,\mathrm{\nabla })$ of a flat algebraic vector bundle on $X^{\ast }:=X\setminus D$ with unipotent monodromy around every component of $D$. We define and compare the various constructions of the extended Chern-Simons classes $${\mathrm{C}\mathrm{S}}_p({\mathrm{\nabla }}^{\mathrm{D}\mathrm{e}\mathrm{l}})\in H^{2p-1}(X,\mathbb{C}\mathrm{/}\mathbb{Z}),p\ge 1,$$ attached to $(F,\mathrm{\nabla })$. Our main theorem states that ${\mathrm{C}\mathrm{S}}_p({\mathrm{\nabla }}^{\mathrm{D}\mathrm{e}\mathrm{l}})$ is torsion in $H^{2p-1}(X,\mathbb{C}\mathrm{/}\mathbb{Z})$, for every $p\ge 2$, extending \cite{Reznikov}, \cite{Reznikov2}, \cite{IS-arXiv},\cite{IS-2div}). We treat the case of quasi-unipotent local monodromies via locally abelian parabolic bundles, and deduce the torsion of Chern-Simons classes. We also extend the Deligne-Sullivan theorem \cite{DeSu} on triviality of flat bundle on a finite covering of a smooth manifold, to that of a canonical extension, and provide torsion-bounds on the extended characteristic classes.