On parabolic geometry. II. (Q2654727)

The author considers a holomorphic principal \(P\)-bundle \(E_p\) over a connected complex manifold \(M\), equipped with a Cartan connection \(\omega\), the so-called holomorphic parabolic geometry of type \(G/P\), \(G\) is a linear algebraic group defined over \(\mathbb{C}\) and \(P\subset G\) a parabolic subgroup. By extending approach given previously by the author in case of \(G= \text{PGL}(d,C)\), \(G/P= \mathbb{P}_{\mathbb{C}}^{d-1}\) [J. Math. Kyoto Univ. 48, No. 4, Article ID 3, 747--755 (2008; Zbl 1175.53039)] it is proved that if \(M\) has Picard number one and contains a (possibly singular) rational curve, then \(M\) is holomorphically isomorphic to the standard parabolic geometry of type \(G/P\), that is \(M= G/P\), \(E_p= G\) and \(\omega\) is the Maurer-Cartan form.