Effective nonvanishing of pluri-adjoint line bundles.

*(English)*Zbl 1344.14006Let \((X,L)\) 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).

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).

Reviewer: Roberto Munoz (Madrid)