Ample vector bundles and del Pezzo manifolds
Let \(X\) be a smooth \(n\)-dimensional complex projective manifold and let a smooth subvariety \(Z\) be a Fano manifold of index \(\dim Z-1\) and Picard number \(\rho(Z)\geq 2\). For an ample vector bundle \(E\) of rank \(r\) on \(X\) \((n\geq r+3)\) is assumed that it has a regular section vanishing along \(Z\). Additionally, let \(Z\) be one of the following del Pezzo manifolds: \[ \mathbb{P}^2\times \mathbb{P}^2,\quad \mathbb{P}(T_{\mathbb{P}^2}),\quad B_q(\mathbb{P}^3),\quad \mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1, \] where \(B_q(\mathbb{P}^3)\) stands for \(\mathbb{P}^3\) blown-up at a point \(q\). The main problem studied in the paper is to find restrictions for the triplet \((X,E, Z)\) under the given conditions and the result is: For \(2\leq r\leq n-3\), \(X\) is determined as \(X= \mathbb{P}_{S}(F)\) where \(F\) is an ample vector bundle of rank \(n-1\) over a surface \(S\) which is \(\mathbb{P}^1\times \mathbb{P}^1\) or \(\mathbb{P}^2\). Moreover \(E\) is determined by the tautological line bundle of \(F\) on \(X\) and by the bundle projection \(f:X\to S\). Also, some applications are obtained.
scientific article; zbMATH DE number 1333301
scientific article