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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).

14C20 Divisors, linear systems, invertible sheaves
14J35 \(4\)-folds
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