Effective non-vanishing of global sections of multiple adjoint bundles for quasi-polarized \(n\)-folds (Q2874710)

Let \(X\) be a complex projective manifold, and let \(L\) be a nef and big line bundle on \(X\) such that the adjoint bundle \(K_X+L\) is nef. By the base point-free theorem we know that some multiple of \(K_X+L\) is globally generated, and by a theorem of \textit{J. Kollár} [Math. Ann. 296, No. 4, 595--605 (1993; Zbl 0818.14002)] we even know that there exists a (very high) effective bound such that \(m(K_X+L)\) is globally generated. When one considers the weaker property of having just one non-zero global section a conjecture of Ionescu and Kawamata claims that \(H^0(X, K_X+L) \neq 0\). This conjecture has been proven by the reviewer for threefolds [J. Algebr. Geom. 21, No. 4, 721--751 (2012; Zbl 1253.14007)], but in higher dimension this problem is very difficult. In the paper under review the author establishes an effective bound, depending only on the dimension \(n\), such that \(m(K_X+L)\) has non-zero global sections. The main theorem is as follows: let \(X\) be a normal Gorenstein projective variety of dimension \(n \geq 4\) with at most rational singularities, and let \(L\) be a nef and big line bundle on \(X\) such that \(K_X+L\) is nef. Then we have \(H^0(X, m(K_X+L)) \neq 0\) for every integer \(m \geq \lceil (n+2)^2/8 \rceil\). The proof is based on adjunction-theoretic computations combined with the explicit resolution of certain systems of linear equations.