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\(\mathbb C^*\)-actions on \(\mathbb C^ 3\) are linearizable. (English) Zbl 0890.14026

The authors give an outline of the proof that every algebraic \(\mathbb{C}^*\)-action on \(\mathbb{C}^3\) is linear in suitably chosen coordinates. Such an action is known to have a fixed point. The proof breaks into cases depending on the dimension of the quotient, the dimension of the fixed point set, and the signs of the weights of the (diagonalized) action on the tangent space at the fixed point. Several cases had been previously settled, and the present article finishes the remaining cases. Combining this result with results of Bialynicki-Birula, Kraft and Popov, it can now be asserted that any action of a connected reductive group on \(\mathbb{C}^3\) is linearizable.

MSC:

14L30 Group actions on varieties or schemes (quotients)
32M05 Complex Lie groups, group actions on complex spaces
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References:

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