Torsors on moduli spaces of principal G-bundles on curves.

Let G be a semisimple complex algebraic group with a simple Lie algebra , and let ℳ G 0 denote the moduli stack of topologically trivial stable G -bundles on a smooth projective curve C. Fix a theta characteristic κ on C which is even in case dim is odd. We show that there is a nonempty Zariski open substack κ of ℳ G 0 such that H i (C , ad (E G) ⊗ κ) = 0 , i = 1 , 2 , for all E G ∈ κ . It is shown that any such E G has a canonical connection. It is also shown that the tangent bundle T U κ has a natural splitting, where U κ is the restriction of κ to the semi-stable locus. We also produce an isomorphism between two naturally occurring Ω M G r s 1 -torsors on the moduli space of regularly stable M G r s . [ABSTRACT FROM AUTHOR]