Geodesic
On Fixed Points of Algebraic Actions on $\mathbb{C}^n$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 1, pp. 41-50 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Elliot E. B., An introduction to the algebra of quantics, 2nd ed., Oxford Univ Press, Oxford, 1913

[2] Fankhauser M., “Fixed points for reductive group actions on acyclic varieties”, Ann. Inst. Fourier, 45:5 (1995), 1249–1281 | DOI | MR | Zbl

[3] Fossum R., Iversen B., “On Picard groups of algebraic fibre spaces”, J. Pure Appl. Algebra, 3 (1973), 269–280 | DOI | MR | Zbl

[4] Gelfand I. M., Kapranov M., Zelevinsky A., Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston–Basel–Berlin, 1994 | MR

[5] Kraft H., “Challenging problems on affine $n$-space”, Séminaire Bourbaki, 47ème année, 802, 1994–95, 1–19 | MR

[6] Oliver R., “Weight systems for $SO(3)$-actions”, Ann. of Math., 103 (1979), 637–644 | DOI | MR

[7] Popov V. L., “Gruppy Pikara odnorodnykh prostranstv lineinykh algebraicheskikh grupp i odnomernye odnorodnye vektornye rassloeniya”, Izvestiya AN SSSR, ser. mat., 38 (1974), 294–322 | MR | Zbl

[8] Rosenlicht M., “Toroidal algebraic groups”, Proc. Amer. Math. Soc., 12 (1961), 984–988 | DOI | MR | Zbl