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Given a natural number $l$ and a weak Fano $n$-fold $X$ with$\operatorname{dim}\overline{\varphi_{-lK_X}(X)}\geq n-1$, we study the lowerbound of the anti-canonical volume and the upper bound of the anti-canonicalstability index. The method can also be used to give similar bounds forpolarised Calabi-Yau varieties.

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