Ample vector bundles on open algebraic varieties
type: Item
instance of: scholarly article
title: Ample vector bundles on open algebraic varieties
zbMATH Open document ID: 0796.14010
author: Shigeharu Takayama
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1994-10-05T00:00:00Z
review text: 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 \(D\).
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 32J25
zbMATH DE Number: 562831
zbMATH Keywords: nef
zbMATH Keywords: jet
zbMATH Keywords: current
zbMATH Keywords: \(L^ 2\)-cohomology
zbMATH Keywords: Lelong number
zbMATH Keywords: very ample vector bundle
zbMATH Keywords: divisor
zbMATH Keywords: ample line bundles
reviewed by: Krzysztof Dabrowski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195166422
OpenAlex ID: W1978968294
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DOI: 10.2977/PRIMS/1195166422
description: scientific article; zbMATH DE number 562831