# zbMATH — the first resource for mathematics

Effective nonvanishing of pluri-adjoint line bundles. (English) Zbl 1344.14006
Let $$(X,L)$$ be a complex polarized manifold ($$X$$ smooth and $$L$$ ample). When the adjoint bundle $$K_X+L$$ is nef is known that the linear system $$m(K_X+L)$$ is non empty when $$m$$ is big enough, by Shokurov’s nonvanishing theorem. In particular, it is conjectured (by F. Ambro [J. Math. Sci., New York 94, No. 1, 1126–1135 (1999; Zbl 0948.14033)] and Y. Kawamata [Asian J. Math. 4, No. 1, 173–181 (2000; Zbl 1060.14505)], see Conjecture 1.1) that $$m=1$$ would be enough, i.e., from $$K_X+L$$ to be nef it would follow to be effective. A first step towards this fact (as Y. Fukuma [J. Pure Appl. Algebra 215, No. 2, 168–184 (2011; Zbl 1200.14019)] proposed, see Problem 1.2) could be to provide a bound $$m_n$$ on $$m$$ just depending on $$n=\dim(X)$$, that is $$m \geq m_n$$ implies $$m(K_X+L)$$ is effective. The main result of this paper is Theorem 1.3 where it is shown that $$m_n \leq n(n+1)/2+2$$.
The outline of the proof is described in the Introduction: the nefness of $$K_X+L$$ allows to produce a morphism onto a smooth projective $$Y$$ and a big line bundle $$B$$ on $$Y$$ where the question now is the effectivenes of adjoint bundles of the type $$K_Y+B+mN$$ where $$N$$ is nef and big. This relates the problem with that of the Fujita’s freenes conjecture ($$|K_X+mL|$$ is free when $$m \geq n+1$$) and some techniques and results on this problem are applied (by Kawamata, Shokurov, Angehrn, Siu and Tsuji, see references in the Introduction of the paper under review).

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14J40 $$n$$-folds ($$n>4$$)
##### Keywords:
polarized manifolds; adjoint bundles; effective divisors
Full Text: