Let $Y$ be an almost Fano variety. That is, $Y$ is an irreducible normal variety such that for some $m$, the pluri-anticanonical bundle $K_{Y_{\text{reg}}}^{-m}$ extends to an ample line bundle over $Y$, where $Y_{\text{reg}}$ denotes the regular part of $Y$.

Let $L \longrightarrow Y$ be an ample line bundle over $Y$, which extends $K_{Y_{\text{reg}}}^{-m}$ for some $m$. If $\pi : \widetilde{Y} \longrightarrow Y$ is any smooth resolution, it is claimed in G. Tian's 1997 paper (reference below) that $\det(\pi)^m$ induces a section of $\pi^{\ast}(L) \otimes K_{\widetilde{Y}}^{-m}$, which does not vanish on $\pi^{-1}(\text{Reg}(Y))$. It therefore follows that \begin{eqnarray} K_{\widetilde{Y}}^{-m} = \pi^{\ast} L + E, \hspace{1cm} (\star) \end{eqnarray}

where $E$ supports in exceptional divisors of $\widetilde{Y}$.

I can't see why $\det(\pi)^m$ induces the section described above and how, as a consequence of this, the anti-canonical bundle admits the splitting $(\star)$. I appreciate any help with this.

Reference: https://link.springer.com/article/10.1007/s002220050176


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