On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds
Let \((X,L)\) be a polarized manifold, i.e., \(X\) is a smooth complex projective variety and \(L\) is an ample line bundle on \(X\). Denote by \(n\) the dimension of \(X\). It is of interest to study the properties of the adjoint bundles \(K_X+mL\) (\(K_X\) stands for the canonical bundle of \(X\) and \(m\) is an integer) of \((X,L)\). In particular it is a natural question to know when the adjoint linear systems \(| K_X+mL| \) are non empty. The paper under review gives some answers to this question when \(n=3\) and \(n=4\). In particular, when \(n=3\) an explicit lower bound (\(>0\)) for \(h^0(X,K_X+L)\) is provided when the Kodaira dimension of \(X\) is non-negative. Moreover, under the assumption that the Iitaka dimension of \(K_X+L\) is non-negative, it is shown that the emptiness of \(| K_X+L| \) implies not only negative Kodaira dimension of \(X\) but \(h^1({\mathcal O}_X)=0\). When \(n=4\), for \(X\) of non-negative Kodaira dimension, a lower bound (also \(>0\)) is given for \(h^0(X,K_X+mL)\), \(m \geq 2\). If moreover \(h^1({\mathcal O}_X)>0\) then \(| K_X+L| \) is non-empty. The results of the paper provide partial answers to a famous conjecture of Beltrametti and Sommese which states that for \(n \geq 3\) the nefness of \(K_X+(n-1)L\) implies the non-emptiness of \(| K_X+(n-1)L| \). In particular, for \(n=3,4\) the conjecture is true unless \(h^1({\mathcal O}_X)=0\) and Kodaira dimension of \(X\) equal to \(-\infty\).
scientific article; zbMATH DE number 5186651
scientific article