发表机构
Institute for Mathematical Sciences (HBNI); Max-Planck Institute for Mathematics; CNRS, Laboratoire J. A. Dieudonné, UMR 6621 Université Cöte d’Azur（数学科学研究所（HBNI）; 马克斯·普朗克数学研究所; 法国国家科学研究中心，J.A.迪永实验室，UMR 6621蔚蓝海岸大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对光滑复射影簇上平丛的Deligne典范扩张，证明其推广陈-西蒙斯类在p≥2时为挠元，处理拟幂单单值性情形，推广Deligne-Sullivan定理并给出挠界。
AI 中文摘要
设X为光滑复射影簇，D = D₁+…+Dₖ⊂X为具有简单正常交叉的除子。考虑X*:=X\backslash D上具有幂单单值性（围绕D的每个分支）的平坦代数向量丛的Deligne典范扩张(F,∇)，我们定义并比较与(F,∇)相关的推广陈-西蒙斯类CSₚ(∇^Del)∈H^(2p-1)(X,ℂ/ℤ)（p≥1）的各种构造。我们的主要定理指出，对所有p≥2，CSₚ(∇^Del)在H^(2p-1)(X,ℂ/ℤ)中是挠元，这推广了Reznikov、Reznikov2、IS-arXiv、IS-2div的结果。我们通过局部阿贝尔抛物丛处理拟幂单局部单值性的情形，推导陈-西蒙斯类的挠性。我们还将关于光滑流形有限覆盖上平丛平凡性的Deligne-Sullivan定理推广到典范扩张的情形，并给出推广特征类的挠界。
英文摘要
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.
Comments118 pages: v2 has a normalised volume regulator (section 9) with parity convention, as in the literature. A reference is added in Appendix G
