Coniveau of classes of flat bundles trivialized on a finite smooth covering of a complex manifold
Let \((E,\bigtriangledown)\) be an algebraic vector bundle with a flat connection on a smooth complex variety, \(X\). There are (analytic) Chern classes \(c_i (E,\bigtriangledown) \in H^{2i - 1} (X_{an}, \mathbb{C}/ \mathbb{Z} (i))\), mapping to the Deligne-Beilinson Chern class, \(c_i^{\mathfrak D} (E) \in H^{2i - 1} \bigl( X_{an}; \mathbb{C}/ \mathbb{Z} (i) \bigr)/F^i \subseteq H^{2i}_{\mathfrak D} \bigl( X; \mathbb{Z} (i) \bigr)\). The author shows that \(c_i (E,\bigtriangledown)\) lies in the bottom stage of the coniveau filtration, \(L^{i - 1} H^{2i - 1} (X_{an}; \mathbb{Q} (i)/ \mathbb{Z} (i))\). From this she deduces that the corresponding classes in étale cohomology also lie in \[ L^i H^{2i - 1}_{\text{ét}} (X; \mathbb{Q}/ \mathbb{Z}), \] which affirmatively answers a question of [\textit{B. Kahn}, \(K\)-theory 5, No. 6, 555-566 (1992; Zbl 0770.12006)] in the smooth case.
scientific article; zbMATH DE number 720849
scientific article