Ample vector bundles on open algebraic varieties
Let \(X\) be a smooth complex projective variety, let \(D\) be an effective, reduced divisor on \(X\) with only simple normal crossings. A vector bundle \(E\) on \(X\) is called ample modulo \(D\) if for every torsion free sheaf \({\mathcal F}\) on \(X\), for large \(m\), \({\mathcal F} \otimes S^ m(E) |_{X \backslash D}\) is generated by \(H^ 0 (X,{\mathcal F} \otimes S^ m(E))\). A line bundle \(L\) is said to be very ample modulo \(D\) if the rational map \(\Phi_{| L |}\) gives an embedding of \(X \backslash D\) into a projective space. The author generalizes a theorem of \textit{J.-D. Demailly} on ample line bundles [J. Differ. Geom. 37, No. 2, 323-374 (1993; Zbl 0783.32013)]. He proves: Theorem 1. There is a function \(C(n)\) in \(n \in \mathbb{N}\) such that, for \(X\) and \(D\) as above and \(L\) a nef line bundle on \(X\) being ample modulo \(D\), the bundle \(2(K_ X + D)+mL\) is very ample modulo \(D\) for \(m \geq C (\dim X)\). A corollary to the proof is theorem 2: Under the notation above, let \(K_ X + D\) (resp. \(-(K_ X+D))\) be nef on \(X\) and ample modulo \(D\). Then there is a function \(C(n)'\) in \(n \in \mathbb{N}\) such that \(m(K_ X + D)\) (resp. \(-m(K_ X + D))\) is very ample modulo \(D\) for \(m>C(\dim X)'\). The methods of the proofs are similar to the analytic methods of Demailly such as Hörmander's \(L^ 2\) estimates for \(\overline \partial\), Lelong number theory and Aubin-Yau's solution of the Calabi conjecture. The author gives his ideas on how to apply theorem 2 to the construction of the moduli space of open algebraic varieties of general type. He also collects some elementary properties of vector bundles which are ample modulo \(D\).
scientific article; zbMATH DE number 562831