## 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).

### MSC:

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

### Keywords:

flat bundles; characteristic classes