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Let $X$, $Y$ be varieties, ${\cal E}\in D^b_{coh}(X\times Y)$. Define the Fourier-Mukai transform with kernel $\cal E$ to be the functor $\Phi({\cal E},-)\colon D^b_{coh}(X)\to D^b_{coh}(Y)$ to be the composition $$ D^b_{coh}(X)\xrightarrow{{\pi_X}^*} D^b_{coh}(X\times Y)\xrightarrow{-\otimes \cal E} D^b_{coh}(X\times Y) \xrightarrow{{\pi_Y}_*} D^b_{coh}(Y) $$ ($\pi$ are the obvious projections) all involved functors are exact, hence one gets an exact functor (in fact one easily sees that $\Phi\colon (\mathcal E,\mathcal F)\mapsto \Phi(\cal E,F)$ is a bifunctor, exact in both components).

Now, I'm stuck with an application of the socalled projection formula $$ \mathbf{R}f_*\big(\mathcal E\stackrel{\mathbf L}{\otimes} \mathbf L f^*\mathcal F\big)\cong \mathbf Rf_*\mathcal E\stackrel{\mathbf L}{\otimes} \mathcal F. $$ Suppose $X=Y=\mathbb P^n(\mathbb C)$, and $\mathcal E = \mathcal O(-k)\boxtimes \Omega^k(k)$, external product of twisted structure sheaves and differential forms sheaf: in particular I would like to prove the isomorphism $$ \Phi\big(\mathcal O(-k)\boxtimes \Omega^k(k),-\big)\cong \mathcal O(-k)\otimes \mathbf R\Gamma(\mathbb P^n(\mathbb C), -\otimes \Omega^k(k)) $$ How can I do? My only trivial achievement was to blindly write down $$ \begin{align*} \Phi\big(\mathcal O(-k)\boxtimes \Omega^k(k),\mathcal F\big) &\cong \mathbf{R}\pi_{X,*}\big(\mathbf{L}\pi_X^*\mathcal F\stackrel{\mathbf L}{\otimes}\mathcal E\big) \\ &\cong \mathcal F\otimes \mathbf{R}\pi_{X,*}\pi_X^*\mathcal O(-k)\otimes \mathbf{R}\pi_{X,*}\pi^*_X\Omega^k(k)\\ \heartsuit &\cong \mathcal F\otimes\mathcal O(-k)\otimes \mathbf{R}\pi_{X,*}\pi^*_X\Omega^k(k) \end{align*} $$ I find the isomorphism in $\heartsuit$ quite intuitive but at the moment I'm not able to formalize it totally. Finding even a faint relation between $\mathbf{R}\pi_{X,*}\pi^*_X\Omega^k(k)\otimes \mathcal F$ and $\mathbf{R}\Gamma(X,\Omega^k(k)\mathcal F)$ seems to me totally obscure.

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