A non-split torsor with trivial fixed point obstruction

✍ Scribed by Z. Reichstein; B. Youssin

• Book ID: 104140673
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 82 KB
• Volume: 263
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00171-6

✦ Synopsis

Let G be a linear algebraic group and X be an irreducible algebraic variety with a generically free G-action, all defined over an algebraically closed base 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 H of G then [X] = 1. 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."