The Kobayashi pseudodistance on homogeneous manifolds
The author studies homogeneous complex manifolds for which the Kobayashi pseudodistance is not a genuine distance. By a result of \textit{K. Nakajima} [J. Math. Kyoto Univ. 25, 269-291 (1985; Zbl 0583.32066)] every homogeneous hyperbolic complex manifold is biholomorphic to a bounded homogeneous domain. The author shows that if X is an arbitrary homogeneous complex manifold then there exist a bounded homogeneous domain \(\Omega\) and a surjective equivariant holomorphic map \(f:X\to \Omega\) such that any holomorphic map from X to any hyperbolic manifold fibres through \(\Omega\). He also proves that there exist a homogeneous smooth manifold Y with a Riemannian metric h, smooth fibre bundle \(\pi: X\to Y\) and constants \(C_ 1>C_ 2>0\) such that \(C_ 2 d_ h(\pi (x_ 1),\pi (x_ 2))\leq d_ X(x_ 1,x_ 2)\leq C_ 1 d_ h(\pi (x_ 1),\pi (x_ 2)),\) \(x_ 1,x_ 2\in X\), where \(d_ h\) denotes the distance on Y induced by h and \(d_ X\) is the Kobayashi pseudodistance on X. It is well-known that a complex manifold X is hyperbolic iff every holomorphic map from \({\mathbb{C}}\) to X is constant provided X is compact [\textit{R. Brody}, Trans. Am. Math. Soc. 235, 213-219 (1978; Zbl 0416.32013)] or X is homogeneous and admits an invariant Hermitian metric [\textit{A. Kodama}, Tôhoku Math. J., II. Ser. 35, 181-186 (1983; Zbl 0515.32012)]. The author proves that the same result is valid provided \(X=G/H\) is a homogeneous complex manifold for which G is a solvable Lie group. He also discusses the fibres of the fibrations mentioned above.
scientific article; zbMATH DE number 4179767
scientific article