2020/09/11 by Hanspeter Kraft, Kraft, Hanspeter, Andriy Regeta +3
Mathematics · #14L30 #14R20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2009.05559
openalex publication_date 2020/09/11 · openalex created_date 2023/04/12 · openalex updated_date 2026/07/28
An affine varieties with an action of a semisimple group G is called "small" if every non-trivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group \mathbbK^* commuting with the G-action. We show that X is determined by the \mathbbK^*-variety XU of fixed points under a maximal unipotent subgroups U of G. Moreover, if X is smooth, then X is a G-vector bundle over the quotient X// G. If G is of type An (n>1), Cn, E6, E7 or E8, we show that all affine G-varieties up to a certain dimension are small. As a consequence we have the following result. If n>4, every smooth affine SLn-variety of dimension <2n is an SLn-vector bundle over the smooth quotient X//SLn, with fiber isomorphic to the natural representation or its dual.