Geodesic
On Fixed Points of Algebraic Actions on $\mathbb{C}^n$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 1, pp. 41-50

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It is shown that the action regarded for a rather long time by experts as a possible example disproving the conjecture on the existence of fixed points for reductive algebraic group actions on affine spaces is not an action on an affine variety, and therefore provides no example of this kind. Moreover, it is shown that the actions naturally related to the original one provide no examples of this kind as well. 
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P. I. Katsylo; V. L. Popov. On Fixed Points of Algebraic Actions on $\mathbb{C}^n$. Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 1, pp. 41-50. http://geodesic.mathdoc.fr/item/FAA_2000_34_1_a3/