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Recent developments on characteristic classes of flat bundles on complex algebraic manifolds. (English) Zbl 0884.32022
This is a concise account on refined characteristic classes of flat bundles on complex algebraic manifolds.
A flat bundle \((E_{an},\bigtriangledown_{an})\) on a complex analytic manifold \(X_{an}\), is an analytic bundle \(E_{an}\) endowed with an integrable connection \(\bigtriangledown_{an}\), i.e. a connection with vanishing curvature. \(Ker\bigtriangledown_{an}\) is then a local system.
By a theorem of Deligne, when \(X\) is algebraic, so are \(E\) and \(\bigtriangledown\). This gives characteristic classes \(c_i^{CH}(E)\in CH^i (X)\) in the Chow groups of codimension \(i\) cycles on \(X\), modulo rational equivalence. There is a cycle map: \[ CH^i(X)\rightarrow H^{2i}_D(X,Z(i)),\quad c_i^{CH}(E)\mapsto c_i^D(E) \] to Deligne-Beilinson cohomology groups, which are extensions: \[ \begin{split} \rightarrow H^{a-1}(X_{an},C/Z)/F^b\rightarrow H^a_D(X, Z(b))\rightarrow\\ Ker(F^bH^a(X_{an},C)\rightarrow H^a(X_{an},C/Z)\rightarrow 0\end{split} \] where \(F^\bullet\) is the Hodge filtration.
Refined classes \(c_i^{an}(E,\bigtriangledown)\) and \(\hat c_i(E,\bigtriangledown)\) can be defined in \(H^{2i-1}(X_{an},C/Z)\), which map to \(c_i^D(E)\). By a result of Reznikov, on a smooth projective \(X\), these classes are torsion for \(i\geq 2\), since lying in \[ H^{2i-1}(X_{an},Q/Z)\subset H^{2i}_D(X,Z(i)). \]

The secondary analytic classes \(c_i^{an}(E,\bigtriangledown)\) can be lifted to algebraic classes (algebraic Chern-Simons theory).

32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)