Deformations of principal bundles on the projective line

Abstract

The study of bundles on IP 1 apparently has a long history (see [22, Chap. I, Sect. 2.4]). Grothendieck proved that any principal bundle on IP~ with a complex reductive Lie groups as structure group admits a reduction of structure group to a maximal torus, unique up to Weyl group action [9]. Harder gave a simple proof of this result which works for IP 1 over arbitrary fields 1-11]. In this paper we study the deformations of principal bundles over IPL Let G be a split reductive group over the field k. By the result of Grothendieck-Harder and Zariski locally trivial G-bundle on IP ~ is associated to the G,,-bundle k Z 0 ~ I P 1 by a 1-PS 2: G,,--,G. Let us denote this G-bundle by E~. Let E-,S • 1 be a G-bundle with an isomorphism Eso=E[s o x lP~---E~. We then call E a deformation of Ex parametrized by S,s o. We say that the Gbundle E' tends or degenerates to the G-bundle E, and write E' ,~E, if there is a deformation E ~ S x l P 1 of E such that in every neighbourhood of the base point socS, (E~o-~E), there is an s such that E ,~ E ' . We prove (Theorem 7.4) that if 2,/~ are dominant 1-PS then E , , ~ E z if and only if # < 2 , i.e. 2 p is a positive integral combination of simple coroots (or, equivalently (2-/~, ~oi)e2g + for every fundamental weight co i. See Sect. 2.5). Note that the set of dominant # such that # < 2 is the same as the set of dominant weights occuring in the indecomposable (or irreducible, if char k=0) representation of the dual group G ~ (see Sect. 2.6) with highest weight 2 (cf. [16, Sect. 21.3]). The deformation theory of G-bundles on IP ~ seems to be much the same as the representation theory of the dual group G ~ (cf. [9, p. 123]). It would be interesting to find a more intrinsic connection between them. The G-bundles E and E" are said to be algebraically equivalent if there is a G-bundle E ~ S x lP 1, with S connected, such that E ~ E s and E ' ~ E s, for some s, s 'eS. We prove (Theorem 7.7) that the algebraic equivalence classes of Zariski locally trivial G-bundles are classified by the fundamental group of G (i.e. the quotient of the lattice of 1-PS of G by the lattice of coroots). This result