Note on quasi-polarized canonical Calabi-Yau threefolds

Let $(X,L)$ be a quasi-polarized canonical Calabi-Yau threefold. In this note, we show that $\mathrm{\mid }mL\mathrm{\mid }$ is basepoint free for $m\ge 4$. Moreover, if the morphism ${\mathrm{\Phi }}_{\mathrm{\mid }4L\mathrm{\mid }}$ is not birational onto its image and $h^0(X,L)\ge 2$, then $L^3=1$. As an application, if $Y$ is a $n$-dimensional Fano manifold such that $-K_Y=(n-3)H$ for some ample divisor $H$, then $\vert mH\vert$isbasepointfreefor$m\geq 4$andifthemorphism$\Phi_{\vert 4H\vert}$isnotbirationalontoitsimage,then$Y$ is either a weighted hypersurface of degree $10$ in the weighted projective space $\mathbb{P}(1,\cdots ,1,2,5)$ or $h^0(Y,H)=n-2$.