On the positivity of the dimension of the global sections of adjoint bundle for quasi-polarized manifold with numerically trivial canonical bundle

Abstract

Let (X, L) denote a quasi-polarized manifold of dimension n ≥ 5 defined over the field of complex numbers such that the canonical line bundle KX of X is numerically equivalent to zero. In this paper, we consider the dimension of the global sections of KX + mL in this case, and we prove that h(KX + mL) > 0 for every positive integer m with m ≥ n − 3. In particular, a Beltrametti-Sommese conjecture is true for quasi-polarized manifolds with numerically trivial canonical divisors.