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Ample vector bundles on singular varieties. II. (English) Zbl 0870.14008
[For part I see Math. Z. 220, No. 1, 59-64 (1995; Zbl 0842.14010).]
In this paper, we study the adjunctions of ample vector bundles on projective varieties with at most log-terminal singularities. We prove the following result:
Theorem. Let \(X\) be an \(n\)-dimensional projective variety having at most log-terminal singularities and let \(E\) be an ample vector bundle of rank \(r\geq n\), then \(K_X+c_1(E)\) is nef unless \((X,E)\cong (\mathbb{P}^n,\bigoplus^n{\mathcal O}_{\mathbb{P}^n}(1))\).
The above theorem implies the following corollary which solves part of the conjecture proposed by Lanteri and Sommese:
Corollary. Let \(X\) be a Gorenstein variety of dimension \(n\) with only rational singularities and \(E\) be an ample and spanned vector bundle of rank \(n\). Then \(K_X+c_1(E)\) is nef, unless
\((X,E)\cong (\mathbb{P}^n,\oplus^n{\mathcal P}_{\mathbb{P}^n}(1))\).
Our theorem is also a generalization of a result due to H. Maeda:
Theorem. Let \(X\) be an \(n\)-dimensional projective variety having at most log-terminal singularities and \(L\) be an ample line bundle. Then \(K_X+nL\) is nef, unless \((X,L)\cong (\mathbb{P}^n,{\mathcal O}_{\mathbb{P}^n}(1))\).
Reviewer: Q.Zhang (Columbia)

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J17 Singularities of surfaces or higher-dimensional varieties
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14C20 Divisors, linear systems, invertible sheaves
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