Special varieties in adjunction theory and ample vector bundles (Q2709854)

Let \(X\) be a smooth projective variety defined over the complex number field. If an ample divisor \(Z\) on \(X\) is a special variety, then the structure of \(X\) is restricted and we can classify \((X,Z)\) in detail. It is natural to consider a generalization of this to the case of ample vector bundles. The authors have studied these under the following assumption: \(\mathcal{E}\) is an ample vector bundle of rank \(r\geq 2\) on a smooth projective variety \(X\) of dimension \(n\) such that there exists a global section \(s\in H^{0}(\mathcal{E})\) whose zero locus \(Z=(s)_{0}\) is a smooth subvariety of \(X\) of dimension \(n-r\geq 1\). NEWLINENEWLINENEWLINEIn Int. J. Math. 6, No.4, 587-600 (1995; Zbl 0876.14027), Forum Math. 9, No.1, 1-15 (1997; Zbl 0876.14026), and in: Higher dimensional complex varieties, Proc. int. conf., Trento 1994, 247-259 (1996; Zbl 0891.14011), \textit{A. Lanteri} and \textit{H. Maeda} have classified polarized pairs \((X,\mathcal{E})\) if \(Z\) is a special variety. (For example, \(Z\) is a projective space, a hyperquadric surface, or a geometrically ruled surface over a smooth curve.) NEWLINENEWLINENEWLINEIn particular, in the last mentioned article they have studied polarized pairs \((X,\mathcal{E},H)\) if \((Z,H_{Z})\) is a scroll over a smooth curve or a hyperquadric fibration over a smooth curve, where \(H\) is an ample divisor on \(X\). The method there is based on the non-nefness of \(K_{Z}+(\dim Z-1)H_{Z}\) or \(K_{Z}+(\dim Z-2)H_{Z}\). NEWLINENEWLINENEWLINEIn this paper, the authors follow this idea, study the non-nefness of \(K_{Z}+tH_{Z}\) for \(\dim_ Z-2\leq t\leq \dim Z+1\) and classify \((X,\mathcal{E}, H)\) under the above assumption. Furthermore they also give a classification of \((X,\mathcal{E}, H)\) if \((Z,H_{Z})\) is a Del Pezzo manifold.