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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 (\(n>4\))
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