On the dimension of the global sections of adjoint bundles for quasi-polarized manifold whose anti-canonical bundle is effective, nef and big

Abstract

Let X denote a smooth projective variety of dimension n defined over the field of complex numbers such that the anti-canonical line bundle -K_X of X is nef and big with h⁰(-K_X) > 0, and let L be a nef and big line bundle on X. In this paper, we consider the dimension of the global sections of K_X + mL with m ≥ n - 1 for this case. In particular, under the assumption that K_X + (n - 1)L is nef, we prove that h⁰(K_X + (n - 1)L) > 0 if 6 ≤ n ≤ 9 and L is ample.