On explicit birational geometry for polarised varieties

In this paper, we investigate the explicit birational geometry for projective $\epsilon$-lc varieties polarised by nef and big Weil divisors. We show that if $X$ is a projective $\epsilon$-lc variety, $H$ is a nef and big Weil divisor with $\dim \overline{{\phi }_H(X)}\ge n-1$ and $L$ is an effective Weil divisor such that $\mathrm{\mid }L-K_X\mathrm{\mid }\mathrm{\ne }\mathrm{\varnothing }$ or $L-K_X$ is nef, then we can find an explicit lower bound of $\text{vol}(H)$ and prove that $\mathrm{\mid }L+m^{\mathrm{\prime }}H\mathrm{\mid }$ is birational for $m^{\mathrm{\prime }}\ge m$, where $m$ is an explicit number which depends only on $n$ and $\epsilon$. This result can be applied to polarised Calabi-Yau varieties, Fano varieties and varieties of general type, generalizing the results in [CEW22] and [Zhu23].