Coisotropic actions on compact homogeneous Kähler manifolds
Let \(M\) be a symplectic manifold, and let \(G\) be a compact, connected Lie group acting on \(M\) by symplectomorphisms. The \(G\)-action is called co-isotropic if the principal orbits are co-isotropic; that is, their normal bundles with respect to the symplectic form are contained in their tangent bundles. The main result of this article is the following theorem: Let \(M\) be a compact, homogeneous Kähler manifold, and let \(G\) be a compact, connected group of isometries of \(M\) that acts co-isotropically on \(M\) and fixes a point of \(M\). Then \(M\) is a Hermitian symmetric space without flat factors. This result is deduced from the following proposition: Let \(N = L / K\) be a compact, homogeneous Kähler manifold, where \(L\) is a compact simple Lie group. Then the action of \(K\) on \(N\) is co-isotropic if and only if the pair \((L,K)\) is up to local isomorphy either a Hermitian symmetric space or one of the following : (i) (Sp\((n), T^1 \times \)Sp\((n-1)\)) , \(n \geq 2\) or (ii) (SO\((2n+1), U(n)\)), \(n \geq 3\). From the main theorem the authors obtain the following two corollaries: Corollary 1. Let \(M\) be a compact, homogeneous Kähler manifold admitting a polar action of a compact, connected Lie group \(G\) with a fixed point. Then \(M\) is biholomorphically isometric to a product \(M_1 \times M_2\), where \(M_1\) is a homogeneous Kähler manifold on which \(G\) acts trivially and \(M_2\) is a Hermitian symmetric space. Corollary 2. Let \(M\) be a simply connected, compact, homogeneous Kähler manifold of complex dimension \(n\) that admits an effective and isometric action of the real \(n\)-torus \(T^n\). Then \(M\) is isometric to a product \(M_1 \times \dots \times M_k\), where each \(M_i\) is isometric to some complex projective space of dimension \(n_i \geq 1\) equipped with the Fubini-Study metric.
scientific article; zbMATH DE number 1922417