Z. Reichstein, B. Youssin, A non-split torsor with trivial fixed point obstruction, Journal of Algebra, 263 (2003), 255--261.

Abstract: Let G be an algebraic group and X be an irreducible algebraic variety with a generically free G-action, all defined over an algebraically closed field of characteristic zero. It is well-known that X can be viewed as a G-torsor, representing a class [X] in H^1(K, G), where K is the field of G-invariant rational functions on X. We have previously shown that if X has a smooth H-fixed point for some non-toral diagonalizable subgroup of G then [X] is non-trivial. It is natural to ask if the converse is true, assuming G is connected and X is projective and smooth. In this note we show that the answer is ``no".

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