On polarized manifolds whose adjoint bundles are not semipositive. (English) Zbl 0659.14002

Fujita, Takao

Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 167-178 (1987).

[For the entire collection see Zbl 0628.00007.]
Let M be a projective variety of dimension n having only rational normal Gorenstein singularities and let L be an ample line bundle. With the line bundle K such that \({\mathcal O}_ M(K)\) is the dualizing sheaf on M, it is shown that:
(0) \(K+(n+1)L\) is nef ( \(= numerically\) effective), i.e., \((K+(n+1)L)\cdot C\geq 0\) for any curve C in M;
(1) \(K+nL\) is not nef if and only if \((M,L)\simeq ({\mathbb{P}}^ n,{\mathcal O}(1));\)
(2) \(K+nL\) is nef but \(K+(n-1)L\) is not nef if and only if (M,L) \(= (a\) hyperquadratic in \({\mathbb{P}}^{n+1},{\mathcal O}_ M(1))\), \(({\mathbb{P}}^ 2,{\mathcal O}(2))\) or M is a scroll over a smooth curve;
(4) The structure of M together with possible types of L is investigated when \(K+(n-1)L\) is nef but \(K+(n-2)L\) is not nef and when \(K+(n-2)L\) is nef but \(K+(n-3)L\) is not nef.
These results are on the line of the author’s works on the structure of polarized manifolds, and the ideas behind the proof are based on Mori’s theory of extremal rays, Mori-Kawamata’s cone theorem and Kawamata-Viehweg’s vanishing theorem.

Reviewer: M.Miyanishi

MSC:

14C20	Divisors, linear systems, invertible sheaves
14N05	Projective techniques in algebraic geometry
14M05	Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)

Keywords:

ample line bundle; nef; numerically effective; polarized manifolds; extremal rays; cone theorem; vanishing theorem