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Membrane structures. (Structures membranaires.) (French) Zbl 1134.14016
Ginzburg, Victor (ed.), Algebraic geometry and number theory. In Honor of Vladimir Drinfeld’s 50th birthday. Basel: Birkhäuser (ISBN 978-0-8176-4471-0/hbk). Progress in Mathematics 253, 599-643 (2006).
Let $$X$$ be smooth scheme over a commutative ring $$k$$ containing the field of rational numbers. To any vector bundle $$E$$ over $$X$$ one associates its Chern classes $$c_i^{DR}(E) \in H^{2i}(X, \Omega_X^*)$$. By the work of Beilinson and Drinfeld, it is known that these classes admit a refinement in the form of the Chern-Simons classes $$c_i^{CS}(E) \in H^{i}(X, \Omega_X^{[i,2i-1>})$$ where is essentially obtained by suitable truncations from the de Rham complex $$\Omega_X^*$$ of $$X$$ over $$k$$. The 1-cocycle giving $$c_1^{CS}(E)$$ is not difficult to describe explicitly.
In this paper the author considers the case when $$E=T_X$$ is the tangent bundle of $$X$$ and gives explicit cocycles representing the associated Chern-Simons characters $$ch_i^{CS}(T_X)$$ for $$i=2$$ and $$i=3$$. The background and the main results are clearly described, but the proofs are often just sketched, as they require, in the author’s words, ‘plus laboris quam artis’.
For the entire collection see [Zbl 1113.00007].
##### MSC:
 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14F40 de Rham cohomology and algebraic geometry 57R20 Characteristic classes and numbers in differential topology
##### Keywords:
Chern-Simons classes; de Rham cohomology