Characterization of the projective space (Q706154)

Let \(X\) be a smooth projective variety of dimension \(n\geq 2\) over an algebraically closed field of characteristic zero. Let \((\eta,L)\) be a foliation on \(X\) with singular locus \(Z\) (i.e. \(L\) is a line bundle on \(X\), \(\eta :\Omega^1_X\to L^{-1}\) is a non zero map and \(Z\) is the zero locus of the image of the induced map \(\Omega^1_X\otimes L \to O_X\)). In this paper the author proves that, if \(L\) is nef, then either \(Z =\emptyset\) and \(X\) has a fibration whose fibres are projective lines, or \(Z\neq 0\) and \(X\) is isomorphic to the projective space \(\mathbb P^n\). In both cases he gives an explicit geometric description of the leaves of the foliation.