Certain circle actions on Kähler manifolds
The author considers the following assumption on a circle action on compact Kähler (more generally symplectic) manifolds.NEWLINENEWLINE{ Assumption A}: Let \((M,\omega,J)\) be a \(2n\)-dimensional compact Kähler manifold. Suppose that \((M,\omega,J)\) admits a holomorphic Hamiltonian circle action for which the critical set of the moment map consists of \(3\)-connected components and the extrema is the set of isolated points.NEWLINENEWLINEThen the author proves the following theorem under Assumption A: \(M\) is \(S^1\)-equivariantly biholomorphic to the complex projective space \({\mathbb C}{\mathbb P}^n\) \((n\geq 2)\) or the Grassmannian \(\tilde G_2({\mathbb R}^n)\) consisting of oriented \(2\)-planes in \({\mathbb R}^{n+2}\) \((n\geq 3)\) with standard \(S^1\)-actions.NEWLINENEWLINE{Theorem}: \(M\) is \(S^1\)-equivariantly symplectomorphic to \({\mathbb C}{\mathbb P}^n\) \((n\geq 2)\) or \(\tilde G_2({\mathbb R}^n)\) \((n\geq 3)\) with standard Kähler structures.NEWLINENEWLINEThe classification of Hamiltonian circle actions are determined in low dimensions (See [\textit{L. Godinho} and \textit{S. Sabatini}, ``New tools for classifying Hamiltonian circle actions with isolated fixed points'' (2012), \url{arXiv:1206.3195v1}], [\textit{H. Li, M. Olbermann} and \textit{D. Stanley}, ``One connectivity and finiteness of Hamiltonian \(S^1\)-manifolds with minimal fixed sets'', (2010), \url{arXiv:1010.2505}], [\textit{H. Li} and \textit{S. Tolman}, Int. J. Math. 23, No. 8, Article ID 1250071, 36 p. (2012; Zbl 1319.53090)], [\textit{D. McDuff}, J. Topol. 2, No. 3, 589--623 (2009; Zbl 1189.53073)], [\textit{S. Tolman}, Trans. Am. Math. Soc. 362, No. 8, 3963--3996 (2010; Zbl 1216.53074)]). In [arXiv:1010.2505, Zbl 1319.53090] the authors considered the case that the fixed point set of \(S^1\) consists of \(2\)-components. A characterization of Assumption A is the following formula concerning the fixed point set of \(S^1\)-actions: NEWLINE\[NEWLINE \sum_{F\subset M^{S^1}}^{}(\text{dim}\, (F) +2)=\text{dim}\, (M)+2. NEWLINE\]NEWLINE This may be useful to determine the Borel cohomology \(\displaystyle H^*(ES^1\mathop{\times}_{S^1}{}M)\): Then the author shows that a Hamiltonian \(S^1\)-action with Assumption A satisfies this formula. Assumption A using the equivariant cohomology characterizes the relation of the Kähler class and the first Chern class, that is \(c_1(M)=(n+1)[\omega]\) and \(c_1(M)=n[\omega]\) respectively. This relation implies the classification of Theorem A. Theorem B is a consequence of Moser's classical theorem.
scientific article; zbMATH DE number 6365889