Holomorphic group actions with many compact orbits (Q909078)

In Arch. Math. 37, 364-371 (1981; Zbl 0456.57017), the reviewer proved that the set of closed orbits of a complex group G acting on a compact Kähler space X is a closed complex subspace and each irreducible component is isomorphic to \(Q\times E\) where Q is homogeneous projective rational manifold and \(E\to Z\) is a principal Seifer fibration with torus fiber, T. The G-orbits in this component are isomorphic to \(Q\times T\). A counter-example for the non-Kähler case involving the Iwasawa manifold is also given. Results for the algebraic case were discovered independently by \textit{J. Konarski} in Lect. Notes Math. 956, 79-91 (1982; Zbl 0497.14022). In the present paper, the authors reprove a special case of this result under a slightly different hypothesis. In particular, they assume that the compact complex manifold X satisfies the following condition: Let \(\lambda_ X: Aut_{{\mathcal O}}(X)\to Aut_{{\mathcal O}}(Alb(X))\) be the natural homomorphism induced from the map of X to its Albanese variety, Alb(X), and let \(\rho_ X: aut_{{\mathcal O}}(X)\to aut_{{\mathcal O}}(Alb(X))\) be the associated homomorphism of Lie algebras. Define \({\mathcal L}(X)\) to be the kernel of \(\rho_ X\). Assume that for every compact submanifold Y of X the restriction of \({\mathcal L}(X)\cap stab(Y)\) to \(aut_{{\mathcal O}}(Y)\) lies in \({\mathcal L}(Y)\). This condition is always satisfied by compact Kähler spaces (i.e., meromorphic images of Kähler manifolds). They further assume that the set of closed orbits in X has an interior point, and then deduce that X has the above mentioned fiber structure. The Iwasawa manifold counter-example is given in this paper as well. The authors were apparently unaware of the earlier work in the subject.