Classification of embedded projective manifolds swept out by rational homogeneous varieties of codimension one (Q657356)

A classical problem in the adjunction theory of complex projective manifolds is to determine those projective manifolds \(X\) admitting a special variety as an ample divisor \(A\). The guiding philosophy is that this forces \(X\) to be even more special, that is, the anticanonical divisor \(-K_X\) should be even more positive than \(-K_A\). In [Math. Ann. 342, No. 3, 557--563 (2008; Zbl 1154.14037)] the author gave a classification of such pairs \((X, A)\) when \(A\) is supposed to be a homogeneous manifold. The recent work by \textit{M. C. Beltrametti} and \textit{P. Ionescu} [Math. Z. 260, No. 1, 229--234 (2008); erratum 235--236 (2008; Zbl 1146.14027)] and \textit{R. Muñoz} and \textit{L. E. Solá Conde} [in: Interactions of classical and numerical algebraic geometry. A conference in honor of Andrew Sommese, Notre Dame, IN, USA, May 22--24, 2008. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 496, 303--315 (2009; Zbl 1203.14053)] suggests that it is possible to obtain classification results under a relaxed positivity hypothesis on the embedding \(A \subset X\), for example one may assume that the normal bundle \(N_{A/X}\) is nef. The paper under review contributes to this more general problem: let \(X \subset \mathbb P^N\) be a projective manifold of dimension \(n \geq 3\) and \(A\) an \((n-1)\)-dimensional rational homogeneous manifold such that the Picard group is generated by \(\mathcal O_A(1)\). If there exists a subvariety \(Z \subset X\) such that \(N_{Z/X}\) is nef and the polarised pairs \((A, \mathcal O_A(1))\) and \((Z, \mathcal O_Z(1))\) are isomorphic, then either \(X\) is a projective space, quadric, Grassmannian of lines or \(E_6\)-variety or \(X\) admits a fibration onto a curve such that the general fibre is projective equivalent to \((A, \mathcal O_A(1))\). The basic strategy of the proof is quite simple: the author shows that the family of lines on \(A\) extends to a covering family of \(X\), by a theorem of \textit{L. Bonavero, C. Casagrande} and \textit{S. Druel} [J. Eur. Math. Soc. (JEMS) 9, No. 1, 45--57 (2007; Zbl 1107.14015)] this family is contracted by an extremal contraction onto a point or a curve. If the contraction maps to a point, the divisor \(A\) is ample and the author's preceding paper applies.