Corollary 19.4.56C08CC234CAB9

Proof. After replacing $X$ by a modification, we may assume that $E$ is a prime divisor on $X$. Note that since $X$ is projective, we can always find an admissible flag with $E$ being the first component.

Then (19.4.57) follows from theorem 19.4.51 and the fact that $\Delta _{Y_{\bullet }}(\theta ,V_{\theta })=\Delta _{Y_{\bullet }}(L)$ for any admissible flag $Y_{\bullet }$ on $X$. $\square$