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If I understand correctly for a proper map, the upper shriek functor is the same as the pullback functor. Now, for some complex compact manifold, I read that $f^{!}(\mathbb{C}_{X})$, where $\mathbb{C}_{X}$ is the constant sheaf and f map X to a point, is the orientation sheaf. How can it be ? I am thinking that $f^{!}(\mathbb{C}_{X})$ is the constant sheaf on X. Where am I going wrong ?

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Unless I am misunderstanding something, there are a few things that are not quite correct.

Let $f:X\rightarrow Y$ be any proper map between topological spaces. It is not true that $f^!=f^*$, but only that $f_!=f_*$ (direct image with proper support and direct image).

In fact, $f^!$ does not even exist in general. However, there are derived versions of these functors : $f^*,Rf_*,Rf_!$. We still have $Rf_*=Rf_!$ for proper map, and now $Rf_!$ has a right adjoint $Rf^!$ (at least for reasonable spaces). Note that $f^*$ is the left adjoint of $Rf_*$ so even if $f$ is proper, $f^*$ and $Rf^!$ may be very different.

Now let $f:X\rightarrow pt$ be the projection onto a point where $X$ is a manifold of (real) dimension $n$. Then it is not quite true that $Rf^!\mathbb{C}$ is the orientation sheaf, but only true up to a shift : $Rf^!\mathbb{C}=\mathfrak{or}_X\otimes\mathbb{C}[n]$.

Note however that complex manifold are oriented, so the orientation sheaf is trivial anyway, and we have $f^!\mathbb{C}=\mathbb{C}_X[2n]$ if $X$ is a complex manifold of complex dimension $n$.

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  • $\begingroup$ thank you for your response. Is there any good lectures on the subject ? $\endgroup$ – epsilones Sep 13 '15 at 11:48
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    $\begingroup$ I like Iversen Cohomology of sheaves which is really not hard to read. There is also Kashiwara Schapira Sheaves on manifold, it is very complete, but way more advanced. Of course, there are plenty other references... $\endgroup$ – Roland Sep 13 '15 at 11:55

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