an:05232514
Zbl 1138.14026
Ohno, Masahiro
Classification of generalized polarized manifolds by their nef values
EN
Adv. Geom. 6, No. 4, 543-599 (2006).
00187600
2006
j
14J60 14N30 14J40 14E30
Nef value; ample vector bundles; extremal contractions
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)].
Gianluca Occhetta (Trento)
Zbl 0954.14006; Zbl 1141.14312