Chapter II - Sectional Genus and Adjoint Bundles
Published online by Cambridge University Press: 17 March 2010
Summary
Throughout this chapter we assume that ℝ is the complex number field ℂ, but most of the results are true if char(ℝ) = O. How they can be generalized in positive characteristic cases is an interesting problem.
The contents of this chapter are relatively independent of those in Chapter I. It is not necessary to have read Chapter I, but it could be helpful.
Semipositivity of adjoint bundles
(11.1) Throughout this section let (M, L) be a polarized manifold with dim M = n ≥ 2. Let K be the canonical bundle of M. We will consider whether or not K + tL is nef for t > O.
The theory of adjoint bundles was first developed by Sommese in the case BS│L│ = ø (cf. §18 in Chapter III) using Apollonius method. However, our approach is technically independent of his, since we do not assume BS│L│ = ø.
Our main tool is Mori-Kawamata theory, and the result (0.4.16) is especially important. Thus, if K + tL is not nef, there is an extremal curve R with (K + tL)R < O and the contraction morphism Φ: M → W of R (cf. (0.4.16)).
(11.2) Theorem. K + nL is nef unless (M, L) ≃ (ℙn, O(1)). In particular K + tL is always nef if t > n.
The proof will be given in (11.6).
Remark. The nefness of K + tL for t > n follows from Mori's theory [Mor3].
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- Information
- Classification Theory of Polarized Varieties , pp. 93 - 138Publisher: Cambridge University PressPrint publication year: 1990