Global generation and very ampleness for adjoint linear series (Q2283343)

Fujita's conjecture states that for a smooth projective variety \(X\) of dimension \(n\) and \(L\) an ample line bundle on \(X\), the adjoint bundle \(K_X+(n+1)L\) is globally generated and \(K_X+(n+2)L\) is in fact very ample. There are several partial results on this conjecture (see the Introduction of this paper and references therein). In particular (see Thm. 1.1, by \textit{D. S. Keeler} [Am. J. Math. 130, No. 5, 1327--1336 (2008; Zbl 1159.14003)]) if \(L\) is ample and globally generated and \(A\) an ample line bundle on \(X\), then \(K_X+nL+A\) is globally generated and \(K_X+(n+1)L+A\) is very ample. The first result of the paper under review (see Thm. 1.2) generalizes this result to the weaker hypothesis on \(A\) to be nef and not numerically trivial. Secondly, the paper deals with vector bundles on compact Kähler manifolds. In Thm. 1.4 it is shown that if \(X\) is a compact Kähler manifold, \(L\) is ample and globally generated and \((E,h)\) a Hermitian holomorphic vector bundle with Nakano semi-positive curvature, then \((K_X+nL)\otimes (E\otimes A)\) is globally generated if \(A\) is a nef but not numerically trivial line bundle. In particular, this is applied in Thm. 1.6 to prove the following: for \(f : X \to Y\) a holomorphic submersion between two complex projective varieties, \(L\) an ample and globally generated line bundle on \(Y\), and \(A\) a nef but not numerically trivial line bundle on \(Y\), and \(s \geq 1\) \[ f_*(K_{X/Y})^{\otimes s}\otimes (K_Y+(\dim Y)L+A) \] is globally generated.