On parabolic geometry of type \(PGL(d,\mathbb C)/P\) (Q1029299)

The author considers a holomorphic principal \(P\)-bundle \(E_p\) over a connected complex manifold \(M\) of complex dimension \(d-1\), equipped with a Cartan connection \(\omega\), the so-called holomorphic parabolic geometry of type \(\text{PGL}(d,C)/P\). It is proved that if there is a nonconstant holomorphic map \(f: \mathbb{C}\mathbb{P}^1\to M\), then \(M\) is biholomorphic to the projective space \(\mathbb{C}\mathbb{P}^{d-1}\). Also, that if \(M\) is a Fano projective variety, then \(M\) is biholomorphic to the projective space \(\mathbb{C}\mathbb{P}^{d-1}\).