A vector bundle characterization of \({\mathbb{P}}^ n\) (Q913883)

Let X be a normal Cohen-Macaulay irreducible n-dimensional complex projective variety and let E be an ample rank-n holomorphic vector bundle over X spanned by its global sections and such that \(c_ n(E)=1\). The main result of the paper is that in the above situation we have \((X,E)=({\mathbb{P}}^ n,{\mathcal O}_{{\mathbb{P}}^ n}(1)^{\oplus n})\) if either \(n\leq 2\) or \(n=3\) and X is Gorenstein. The same assertion, stated in the paper as a conjecture for any normal irreducible complex projective variety and any \(n\geq 1\), has been subsequently proved in the smooth case by \textit{J. A. Wisniewski} [Math. Z. 200, No.3, 409-427 (1989; Zbl 0668.14004)]. An application to 3-dimensional scrolls over a smooth surface is given.