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Stably trivial equivariant algebraic vector bundles. (English) Zbl 0862.14009
In this article, the authors describe a method for constructing non-trivial equivariant algebraic vector bundles over representation spaces for reductive groups (the equivariant Serre problem). The idea is to define invariants, which distinguish bundles, which are explicitly given as subbundles of trivial vector bundles. H. Kraft and G. Schwarz, using another method, also produced many such non-trivial bundles. They considered exclusively the case where the base is a representation with a one-dimensional quotient. The authors do not restrict to this case. They use their method to find the first non-trivial \(G\)-vector bundles over a \(G\)-module, for \(G\) a finite group. For example, they show that the non-trivial \({\mathcal O} (2, \mathbb{C})\)-vector bundles originally given by Schwarz remain non-trivial for large enough dihedral groups. For a certain case, one can choose \(G\) to be the dihedral group of order 14.
The equivariant Serre problem is strongly related to the linearity problem, which asks if there are algebraic actions of a reductive group on affine space which are not linearisable. An action is called linearisable if it is conjugate to a linear action. Schwarz’s first examples of non-trivial \(G\)-vector bundles over \(G\)-modules provided the first examples of non-linearisable actions on affine space. In this article, the authors give a new result, which allows them to use their non-trivial \(G\)-vector bundles for finite groups to obtain non-linearisable actions of finite groups on affine space. They prove, for example, that there exist non-linearisable actions of \(D_{10}\), the dihedral group of order 20, on \(\mathbb{C}^4\). Also, there exist infinite families of inequivalent actions of \(D_{18}\) on \(\mathbb{C}^4\).

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14L30 Group actions on varieties or schemes (quotients)
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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