# zbMATH — the first resource for mathematics

On a conjecture of Beltrametti-Sommese for polarized 4-folds. (English) Zbl 1327.14038
Let $$(X,L)$$ be a polarized manifold of dimension $$n$$, i.e., $$X$$ is a smooth complex projective variety and $$L$$ is an ample line bundle on $$X$$. M. C. Beltrametti and A. J. Sommese conjectured in Conjecture 7.2.7 on [The adjunction theory of complex projective varieties. De Gruyter Expositions in Mathematics. 16. Berlin: de Gruyter (1995; Zbl 0845.14003)] that for $$n \geq 3$$ the nefness of the adjoint bundle $$K_X+(n-1)L$$ implies its effectivity, that is $$h^0(X, K_X+(n-1)L)>0$$. The conjecture is proved to be true when $$n=3$$ and when $$L$$ is effective (see the Introduction of the paper and references therein). The main result of the paper is to prove the conjecture when $$n=4$$ (see Thm. 3.1). The proof is a case by case analysis where the maximal rationally connected fibration of $$X$$ plays an important role in one of the cases ($$q(X)=0$$ and $$\Omega_X\langle L \rangle$$ not generically nef).

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14J35 $$4$$-folds
Full Text: