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

Abstract

Let X be a smooth complex projective variety and D = D₁+⋯+D_k⊂ X a divisor with simple normal crossings. Consider Deligne's canonical extension (F,∇) of a flat algebraic vector bundle on X^*:=X∖ D with unipotent monodromy around every component of D. We define and compare the various constructions of the extended Chern-Simons classes CS_p(∇^Del) ∈ H^2p−1(X,ℂ/ℤ), p≥ 1, attached to (F,∇). Our main theorem states that CS_p(∇^Del) is torsion in H^2p−1(X,ℂ/ℤ), for every p≥ 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.

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BibTeX RIS

BibTeX

@misc{iyer2026torsion,
  title = {{Torsion of extended Chern-Simons classes for canonical extensions of flat bundles}},
  author = {Jaya NN Iyer and Carlos Simpson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.20877},
  eprint = {2608.20877},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.20877},
}

RIS

TY  - UNPB
TI  - Torsion of extended Chern-Simons classes for canonical extensions of flat bundles
AU  - Jaya NN Iyer
AU  - Carlos Simpson
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2608.20877
UR  - https://arxiv.org/abs/2608.20877
AB  - Let X be a smooth complex projective variety and D = D_1+⋯+D_k⊂ X a divisor with simple normal crossings. Consider Deligne's canonical extension (F,∇) of a flat algebraic vector bundle on X^*:=X∖ D with unipotent monodromy around every component of D. We define and compare the various constructions of the extended Chern-Simons classes CS_p(∇^Del) ∈ H^2p−1(X,ℂ/ℤ), p≥ 1, attached to (F,∇). Our main theorem states that CS_p(∇^Del) is torsion in H^2p−1(X,ℂ/ℤ), for every p≥ 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.
ER  -

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