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