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Classification of generalized polarized manifolds by their nef values. (English) Zbl 1138.14026
The paper under review deals with generalized polarized manifold, i.e., pairs \((M,\mathcal E)\) with \(M\) an \(n\)-dimensional complex projective manifold and \(\mathcal E\) an ample vector bundle of rank \(r\) on \(X\).
Problems related to the nefness and the ampleness of the line bundle \(K_M + \det \mathcal E\), in case \(r \geq n-2\) have been considered by many authors; we refer the reader to the paper’s introduction for an accurate overview on previous results.
In the present paper the author studies pairs \((M,\mathcal E)\) as above from a slightly different view point, considering the nef value \(\tau\) of the pair, which is the minimum real number \(t\) such that the line bundle \(K_M + t\det \mathcal E\) is nef, and giving a classification of the pairs such that \(\tau r \geq n-2\).
The case \(\tau r \geq n-1\) was previously considered by the author in [RIMS Kokyuroku 1078, 75–85 (1999; Zbl 0954.14006)]. The proof is based on a case by case analysis of the extremal contraction of the manifold \(M\), using traditional results and techniques of Mori theory as well as recent results as the characterization of the projective space given by Cho, Miyaoka and Shepherd Barron and Kebekus.
Some minor improvements and corrections have been published in an Addendum [Adv. Geom. 7, No. 2, 315–316 (2007; Zbl 1141.14312)].

MSC:
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14N30 Adjunction problems
14J40 \(n\)-folds (\(n>4\))
14E30 Minimal model program (Mori theory, extremal rays)
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