Suppose we are given a projective smooth variety $X$ and a locally free sheaf $\mathcal{E}$. Now consider $\pi:\mathbb{P}(\mathcal{E})\rightarrow X$. One can define the relative dualizing sheaf as it is done for example in Fourier--Mukai transfors in algebraic geometry (Huybrechts). It is $\omega_{\pi}=\omega_{\mathbb{P}(\mathcal{E})}\otimes \pi^*\omega_X^{\vee}$. My question is now: If we restrict $\omega_{\pi}$ to a fiber over some point $x\in X$, which is $\mathbb{P}^n$ do we get something that is isomorphic to $\omega_{\mathbb{P}^n}$ ?
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Sure, since the pullback part restricts to constant, and note that the top wedge of differential sheaf base changes to top wedge of differential sheaf on fiber, you get the dualizing sheaf. – Oct 17 '15 at 21:26