#### 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".

PDF file

DVI file