Invariant holomorphic foliations on Kobayashi hyperbolic homogeneous manifolds (Q2790269)

Let \(M\) be a Kobayashi hyperbolic homogeneous manifold. The authors consider holomorphic foliations \(\mathcal{F}\) on \(M\), invariant under a transitive group \(G\) of biholomorphisms. Recall that, an homogeneous and Kobayashi hyperbolic manifold is biholomorphic to a homogeneous Siegel domain of type II, due to [\textit{K. Nakajima}, J. Math. Kyoto Univ. 25, 269--291 (1985; Zbl 0583.32066)].NEWLINENEWLINEThe main result is that the leaves of \(\mathcal{F}\) are the fibers of a holomorphic \(G\)-equivariant submersion \(\pi:M\longrightarrow N\) onto a \(G\)-homogeneous complex manifold \(N\). In addition, if \(\mathcal{Q}\) is an automorphism family of a hyperbolic convex (possibly unbounded) domain \(D\) in \(\mathbb{C}^n\), then the fixed point set of \(\mathcal{Q}\) is either empty or a connected complex submanifold of \(D\).NEWLINENEWLINEThe present results are related to [\textit{A. Behague} et al., Monatsh. Math. 153, No. 4, 295--308 (2008; Zbl 1142.57016)].