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Let $X$ be a smooth projective variety and $b:Y\to X$ be the blow up of $X$ along a smooth subvariety $Z\subset X$. I'm trying to compute the complex $Rb_*\mathcal{O}_Y(K_Y+E)$, where $E$ is the exceptional divisor and $K_Y$ is the canonical divisor of $Y$.

I know from Kollar's theorem that $R^ib_*\mathcal{O}_Y(K_Y)=0$ for every $i>0$, and $R^0b_*\mathcal{O}_Y(K_Y)$ is torsion free. Is there any similar result holds for $Rb_*\mathcal{O}_Y(K_Y+E)$?

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By Grothendieck duality $$ Rb_*\mathcal{O}_Y(K_Y + E) \cong Rb_*R\mathcal{H}\mathit{om}(\mathcal{O}_Y(-E),\omega_Y) \cong Rb_*R\mathcal{H}\mathit{om}(\mathcal{O}_Y(-E),b^!\omega_X) \cong R\mathcal{H}\mathit{om}(Rb_*(\mathcal{O}_Y(-E)),\omega_X) \cong I_Z^\vee \otimes \omega_X, $$ where $(-)^\vee$ stands for the derived dual.

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