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Holomorphic gerbes and the Beilinson regulator. (English) Zbl 0821.19001

Kassel, Christian (ed.) et al., \(K\)-theory. Contributions of the international colloquium, Strasbourg, France, June 29-July 3, 1992. Paris: Société Mathématique de France, Astérisque. 226, 145-174 (1994).
Beilinson has defined a regulator map \(c_{m,i}\) from \(K_ i (X)\) to \(H^{2m - i} (X, \mathbb{Z} (m)_ D)\), the Deligne cohomology. For \(m = i = 2\) the last group has an interpretation as a group of isomorphism classes of line bundles with a holomorphic connection. Then the symbol technique gives a possibility to define \(c_{2,2}\) in an explicit way. This paper solves this question for \(m = 2\) and \(i = 1\). First of all the Deligne group \(H^ 3 (X, \mathbb{Z} (2)_ D)\) can be interpreted as a group of isomorphism classes of holomorphic gerbes (these are some kinds of sheaves of categories over \(X)\) having the holomorphic connective structure defined by the author. The holonomy of two-forms (instead of 1- forms in Deligne’s construction) gives the desired map.
For the entire collection see [Zbl 0809.00016].

MSC:

19E20 Relations of \(K\)-theory with cohomology theories
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
18G50 Nonabelian homological algebra (category-theoretic aspects)
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