Linear Hamiltonian circle actions that generate minimal Hilbert bases
The following theorem is the main result of the paper. Let \(S^1\times {\mathbb C}^k\to {\mathbb C}^k\) be a linear Hamiltonian circle action with relatively prime weights \(n_1,\dots,n_k\). Assume that the algebra \(C^\infty({\mathbb R}^{2k})^{S^1}\) is generated by \(k^2\) polynomials \(f_1,\dots,f_{k^2}\) which is the minimal possible number of such polynomials for a fixed number \(k\). Denote the greatest common divisor of all the weights except \(n_i\) by \(d_i\). Assume furthermore that at least three of the numbers \(d_1,\dots, d_k\) are not equal to one. Then the Poisson structure on the singular orbit space \({\mathbb C}^k/S^1\), embedded into \({\mathbb R}^{k^2}\) by \(F=(f_1,\dots,f_{k^2})\), cannot be extended to \({\mathbb R}^{k^2}\). The paper is a continuation of [Electron. Res. Announc. Am. Math. Soc. 1, No. 2, 48-56 (1995; Zbl 0849.58029)] by the author and answers a question of Cushman and Weinstein in the negative.
scientific article; zbMATH DE number 1409072