Ample vector bundles on singular varieties. II (Q1359543)

[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 \((X,L)\cong (\mathbb{P}^n,{\mathcal O}_{\mathbb{P}^n}(1))\).