Beilinson-Deligne's regulators for differentiable or analytic manifolds, by Daniel Delabre

In this paper, we construct and study Beilinson-Deligne Chern classes for differentiable manifolds, whose target are the smooth Deligne's cohomology groups

c_{i, j} : K_i C^\infty (X) \to H_D^{2j-i} (X, Z(j)) .

Their properties are the same as those of the corresponding classes in algebraic geometry: respect of the Adams operations, behavior with respect to products in K-theory and Deligne cohomology, and c_{1, 1} is the determinant map. Thus c_{2, 2} is known on Steinberg's symbols as pull-backs of the Heisenberg bundle and its connection. In the same manner, analytic Beilinson-Deligne Chern classes can be constructed (and the same properties hold).

In a second part, we specialize our study to c_{2, 2} (and smooth classes). We show that, for a large class of manifolds (including Hausdorff, finite-dimensional ones), it makes sense to take infinitesimal deformation of c_{2, 2}. Such deformation yields a central extension of Lie algebras, a cocycle of which is given by:

Y, Z \mapsto Tr (YdZ) .

Then, this extension is a quotient of the Bloch's universal central extension of the C-Lie algebra sl C^\infty (X).

When X = S^1, this is the Segal's cocycle: the extension is the universal central extension of Fréchet-Lie algebra sl C^\infty (X). This shows that the infinitesimal deformation of c_{2, 2} identifies with the infinitesimal deformation of the second Connes-Karoubi's regulator

c_{2}^{CK} : K_2 C^\infty (S^1) \to C^* .

This work is part of my thesis. The original (french) text (and other papers) are accessible by ftp or on my web page.

The dvi file uses the Xy-pic fonts, as well as the jknappen/dc fonts.


Daniel Delabre <delabre@picard.ups-tlse.fr>