Narrow Results

Let (X, L) be a polarized manifold of dimension 3. In this paper, we consider a lower bound for h⁰(K_X + 2L). We prove that h⁰(K_X + 2L) > 0 if K_X + 2L is nef, which is a conjecture of Beltrametti–Sommese for polarized 3-folds. Moreover we classify polarized 3-folds (X, L) with h⁰(K_X + 2L) = 1 under the assumption that K_X + 2L is nef.

We find natural numbers m such that the dimensions of global sections of multiple adjoint bundles h⁰(m(K_X + L)) are strictly greater than zero for any quasi-polarized n-folds (X, L) for which X is a complex normal Gorenstein projective variety of dimension n with only rational singularities and K_X + L is nef.

Let m^NEF(n) be the smallest positive integer p such that the dimensions of global sections of multiple adjoint bundles h⁰(m(K_X + L)) are strictly greater than zero for any integer m with m ≥ p and any quasi-polarized n-folds (X, L) for which X is a complex normal Gorenstein projective variety with only rational singularities and K_X + L is nef. In this paper, we prove that m^NEF(n) ≤ 2n - 4 holds for n ≥ 4.