Smooth 4-folds which contain a \(P^1\)-bundle as an ample divisor (Q1974139)

Let \(X\) be a smooth projective manifold of dimension \(n>2\) and \(A\) an ample divisor on it. Assume that \(A\) is effective and a \(P^4\)-bundle over a smooth projective \((n-2)\)-fold \(S\). \textit{M. L. Fania} and \textit{A. J. Sommese} [Ann. Sci. Norm. Super. Pisa, Cl. Sci., IV. Ser. 15, No. 2, 193-218 (1988; Zbl 0691.14004)] conjectured that if \(A\neq \mathbb{P}^{n-2}\times \mathbb{P}^1\) then the \(P^1\)-bundle map extends to a morphism exhibiting \(X\) as a \(P^1\)-bundle on \(S\). Many authors contributed to proving the conjecture in several cases, e.g. when \(n=4\) and \(S\) fibres onto a curve. This paper completes that result showing that Sommese's conjecture holds for \(n=4\).