On explicit birational geometry for weak Fano varieties and polarised Calabi-Yau varieties

Given a natural number $l$ and a weak Fano $n$-fold $X$ with $\dim \overline{{\phi }_{-l{K}_X}(X)}\ge n-1$, we study the lower bound of the anti-canonical volume and the upper bound of the anti-canonical stability index. The method can also be used to give similar bounds for polarised Calabi-Yau varieties.