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

Abstract

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 $(F,\nabla)$ of a flat algebraic vector bundle on $X^*:=X\setminus D$ with unipotent monodromy around every component of $D$. We define and compare the various constructions of the extended Chern-Simons classes $$ \mathrm{CS}_p(\nabla^{\mathrm{Del}})\;\in\; H^{2p-1}(X,\mathbb{C}/\mathbb{Z}),\, p\geq 1, $$ attached to $(F,\nabla)$. Our main theorem states that $\mathrm{CS}_p(\nabla^{\mathrm{Del}})$ is torsion in $H^{2p-1}(X,\mathbb{C}/\mathbb{Z})$, for every $p\geq 2$, 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.