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Regulators and characteristic classes of flat bundles. (English) Zbl 0976.14005
Gordon, B. Brent (ed.) et al., The arithmetic and geometry of algebraic cycles. Proceedings of the CRM summer school, Banff, Alberta, Canada, June 7-19, 1998. Vol. 2. Providence, RI: American Mathematical Society (AMS). CRM Proc. Lect. Notes. 24, 47-92 (2000).
Let \(X\) be a smooth complex algebraic variety. For a vector bundle \(E\) with a flat connection \(\nabla\), Cheeger and Simons have defined characteristic classes \({\widehat c}_p(E, \nabla) \in H^{2p-1} (X,{\mathbb C}/{\mathbb Z}(p))\), where \({\mathbb Z}(p) = (2\pi i)^p{\mathbb Z}\) (the Tate twist). A flat vector bundle has a unique algebraic structure for which the connection has regular singularities, hence one can associate to it Beilinson Chern classes \(c^B_p(E) \in H^{2p}_D(X,{\mathbb Z}(p))\). The first result of this paper is the following.
Theorem. For all \(p\geq 1\), the natural map \(H^{2p-1}(X, {\mathbb C}/ {\mathbb Z}(p)) \to H^{2p}_D(X, {\mathbb Z}(p))\) maps \({\widehat c}_p(E,\nabla)\) to \(c_p^B(E)\).
The authors generalise this theorem to algebraic vector bundles with a more general class of connections called \(F^1\)-connections. The analogous result for Kähler manifolds is formulated and proved. Let \(B = B GL_n({\mathbb C})^{\delta}\) denote the classifying space for the general linear group of complex \(n\times n\) matrices with discrete topology. Let \({\widehat c}_p \in H^{2p-1}(B, {\mathbb C}/{\mathbb Z}(p))\) be the universal Cheeger-Simons class (i.e. the class of the universal flat bundle).
Theorem. The image of \({\widehat c}_p\) in \(H^{2p-1}(B, {\mathbb C}/{\mathbb R}(p))\) is the Borel regular element, a canonical cohomology class defined by Borel.
The authors conjecture that \({\widehat c}_p\) is the Beilinson-Chern class of the universal flat bundle on \(B\), but they do not succeed in proving the conjecture.
For the entire collection see [Zbl 0935.00034].

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
57R20 Characteristic classes and numbers in differential topology
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