2012/10/30 by Johan Martens, Martens, Johan, Michael Thaddeus +1 · 3 citations
Mathematics · #14D20 #14D23 #14H60 #20G15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1210.8161
openalex publication_date 2012/10/30 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Grothendieck and Harder proved that every principal bundle over the projective line with split reductive structure group (and trivial over the generic point) can be reduced to a maximal torus. Furthermore, this reduction is unique modulo automorphisms and the Weyl group. In a series of six variations on this theme, we prove corresponding results for principal bundles over the following schemes and stacks: (1) a line modulo the group of nth roots of unity; (2) a football, that is, an orbifold of genus zero with two marked points; (3) a gerbe over a football whose structure group is the nth roots of unity; (4) a chain of lines meeting in nodes; (5) a line modulo an action of a split torus; and (6) a chain modulo an action of a split torus. We also prove that the automorphism groups of such bundles are smooth, affine, and connected.