Variations on a theme of Grothendieck
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 October 2012
 arXiv:
 arXiv:1210.8161
 Bibcode:
 2012arXiv1210.8161M
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematical Physics;
 Mathematics  Symplectic Geometry;
 14H60;
 14D20;
 14D23;
 20G15
 EPrint:
 28 pages