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Equivariant algebraic vector bundles over cones with smooth one-dimensional quotient. (English) Zbl 0928.14013
The paper is concerned with some aspects of the general problem of constructing and distinguishing equivariant algebraic vector bundles over a base space which is an affine variety with an algebraic action of a complex reductive group \(G\), i.e. affine \(G\)-variety. There are two important classes of base spaces for equivariant vector bundles. These are homogeneous spaces and representations. In the case of homogeneous spaces all equivariant vector bundles are known and easy to describe. For representations, far less is known. What the authors do in this paper is to introduce a class of affine \(G\)-varieties called weighted \(G\)-cones which from the point of view of the \(G\)-action are somewhat more complex than homogeneous spaces but far simpler than representations. In this way, one is able to describe some of the equivariant vector bundles over weighted \(G\)-cones in theorem \(A\) of the paper. As an application the authors apply the results to describe families of equivariant vector bundles over representation in theorems \(B\) and \(C\).

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
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