Kähler tubes of constant radial holomorphic sectional curvature (Q1382647)

Let \(\pi: N\to P\) be a \(C^\infty\) complex vector bundle of real rank \(2k\) over a Kähler manifold \(P\) and let \(M^k(4\lambda)\) be a complex space form of constant holomorphic sectional curvature \(4\lambda\) and complex dimension \(k\). Using these data, the authors have introduced in [\textit{A. Lluch} and \textit{V. Miquel}, Geom. Dedicata 61, 51-69 (1996; Zbl 0869.53024)], inspired by earlier work of J. H. Eschenburg, the notion of a model tube of radial holomorphic sectional curvature \(4\lambda\). The work of Eschenburg concerns the construction of tubes about totally geodesic submanifolds \(P\) having constant radial sectional curvature, and it turns out that they are defined from any vector bundle over the center \(P\) of the tube. Using a metric and a special connection on \(N\), these model tubes may be endowed with an almost Hermitian structure and a central problem is to determine whether this structure can be Kähler or not, i.e., whether the tube can be a Kähler model tube. This is the central theme of the paper under review. The authors succeed in determining completely (up to holomorphic isometries) the \(C^\infty\) complex vector bundles giving rise to Kählerian model tubes when \(P\) is simply connected or \(P\) is a complex hyper surface with \(H^1(P,\mathbb{R})= 0\). Several interesting related questions are treated.