Two-orbit Kähler manifolds and Morse theory (Q706200)

Let \(M\) be a compact Kähler manifold and let \(K\) be a compact connected Lie group of isometries acting on \(M\) in a Hamiltonian fashion with moment map \(\mu\). Assume that the complexified action of \(K^{\mathbb C}\) on \(M\) has precisely two orbits. The main result of the paper under review asserts that in this situation a) \(K\) is semisimple; b) \(M\) is simply connected projective algebraic; c) the Hodge numbers \(h^{p,q}(M)\) vanish for \(p\neq q\); and d) \(\| \mu\| ^2:M\to\mathbb R\) is a Morse-Bott function with two critical submanifolds. One of the critical components is realized by the closed \(K^{\mathbb C}\)-orbit, the other one by a \(K\)-orbit \(S\). The open \(K^{\mathbb C}\)-orbit retracts onto \(S\). This has strong implications for the Betti numbers of \(M\).