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Triviality of certain equivariant vector bundles for finite cyclic groups. (English. Abridged French version) Zbl 0813.14033

Consider, the following open question. Is any algebraic action of a finite cyclic group on complex affine space linearizable, that is, is it conjugate to a linear action?
We show that if \(G\) is a finite cyclic group and \(V\) is a complex \(G\)- module consisting of the direct sum of a 1-dimensional nontrivial representation and a module with trivial action, then all algebraic \(G\)- vector bundles with base \(V\) are trivial. As a consequence, the action of \(G\) on the total space of such a bundle is linearizable. This shows, for example, that certain involutions (actions of \(\mathbb{Z}/2 \mathbb{Z})\) on \(\mathbb{C}^ 4\), which until now were not known to be linearizable, are in fact linearizable. The proof is not constructive, in that it does not give an algorithm to linearize such an action.

MSC:

14L30 Group actions on varieties or schemes (quotients)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
20C15 Ordinary representations and characters
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