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Variations on a theme of Grothendieck: or, I've a feeling we're not in Kansas any more

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  • Johan Martens, Denmark
  • Michael Thaddeus, Columbia University, United States
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 line modulo an action of a split torus; (5) a chain of lines meeting in nodes; 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.
Original languageEnglish
Number of pages27
Publication statusPublished - 1 Nov 2012

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