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G-vector bundles and the linearization problem. (English) Zbl 0703.14009

Group actions and invariant theory, Proc. Conf., Montreal/Can. 1988, CMS Conf. Proc. 10, 111-123 (1989).
[For the entire collection see Zbl 0679.00007.]
This expository paper reports results about vector bundles on varieties on which a linear algebraic group is acting compatibly on bundle and variety. The motivating question is whether every action of a linearly reductive algebraic group G on an affine space is conjugate to a linear action (the linearization problem). As the author notes, this problem has been solved in the negative by G. Schwarz: if the linearization problem had a positive solution, then every G-vector bundle over every representation space of G would be trivial. However, Schwarz found G- vector bundles over the adjoint representation of G which are not G- trivial for \(G=SL_ 2({\mathbb{C}}).\)
Other results reported on include: G-vector bundles are locally trivial in the Zariski topology when the (reductive) group G acts trivially on the base; this implies that all G-vector bundles are trivial when all vector bundles are trivial. Also, it is shown that G-vector bundles over representation spaces of G are locally trivial, and a characterization is given of G-vector bundles arising from pullbacks of bundles on categorical quotients.
The final section of the paper is devoted to G-line bundles. Here a unified approach to the exact sequences relating the group of G-line bundles to the Picard group is given.
Reviewer: A.R.Magid

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14L30 Group actions on varieties or schemes (quotients)
20G15 Linear algebraic groups over arbitrary fields

Citations:

Zbl 0679.00007