On the Nonemptiness of the Adjoint Linear System of Polarized Manifolds

Abstract

Let (X, L) be a polarized manifold over the complex number field with dim X = n. In this paper, we consider a conjecture of M. C. Beltrametti and A. J. Sommese and we obtain that this conjecture is true if n = 3 and h⁰(L) ≥ 2, or dim Bs |L| ≤ 0 for any n ≥ 3. Moreover we can generalize the result of Sommese.