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

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 $h^{0}(-K_{X})>0$, 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 $m\geq n-1$ for this case. In particular, under the assumption that $K_{X}+(n-1)L$ is nef, we prove that $h^{0}(K_{X}+(n-1)L)>0$ if $6\leq n\leq 9$ and $L$ is ample.