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Moduli spaces of decomposable morphisms of sheaves and quotients by non-reductive groups. (English) Zbl 1034.14023
If \(G\) is a reductive group acting on an algebraic variety, then geometric invariant theory tells you what kind of quotient to expect under which conditions. For non-reductive groups there is no similarly nice theory, and, as emphasized by the examples of Nagata and others, there cannot exist an as nice theory for arbitrary algebraic groups. Thus one is led to the question of which reasonable additional conditions one has to impose in order to achieve good results for actions of non-reductive groups. One such condition is to assume that the algebraic group \(G\) embeds into a larger reductive group \(H\) in such a way that the \(G\)-action on \(M\) extends to an \(H\)-action on some larger space.
In this article this strategy is applied in order to study certain actions on coherent sheaves. More precisely, let \(X\) be a projective variety and \({\mathcal E}\), \({\mathcal F}\) coherent sheaves on \(X\). Then the action of \(G=\operatorname{Aut}({\mathcal E})\times \text{Aut}({\mathcal F})\) on \(\text{Hom} ({\mathcal E},{\mathcal F})\) is considered. In general these sheaves will be decomposable and therefore \(G\) will not be reductive. Polarizations are introduced in order to define an appropriate notion of (semi-)stability. The main result of the paper states that there are bounds for the polarization which imply the existence of a “good quotient” for the semi-stable locus and a “geometric quotient” for the stable locus.

MSC:
14M17 Homogeneous spaces and generalizations
14D20 Algebraic moduli problems, moduli of vector bundles
14L30 Group actions on varieties or schemes (quotients)
Software:
GiANT
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