Let $\mathcal{D}^b(X)$ be the bounded derived category of constructible $\mathbb{C}$-sheaves on $X$.


(1) For any closed embedding $i:Z\rightarrow X$ and $A\in \mathcal{D}^b(X)$, there is a natural morphism $i^!(A)\rightarrow i^\*(A)$ which induces a distinguished triangle in $\mathcal{D}^b(Z)$,$$i^!(A)\rightarrow i^\*(A)\rightarrow B\xrightarrow{+1}.$$
(2) For any vector bundle $f:X\rightarrow Y$ and $A\in \mathcal{D}^b(X)$, there is a natural morphism $f\_!(A)\rightarrow f\_\*(A)$ which induces a distinguished triangle in $\mathcal{D}^b(Y)$,$$f\_!(A)\rightarrow f\_\*(A)\rightarrow C\xrightarrow{+1}.$$
Is it possible to describe the cones $B$ and $C$ explicitly? Even in the case that $A=\underline{\mathbb{C}}$ is the constant sheaf on $X$? Any comments will help! Thanks!



Cones of natural morphisms $i^!\rightarrow i^*$ and $f_!\rightarrow f_*$
For (2), every vector bundle can be compactified to a projective space bundle (the Proj of the graded ring of symmetric powers of the dual bundle plus a rank one trivial bundle). Let $j$ be the open immersion into the projective space bundle and $\pi$ the projection from the projective space bundle to $Y$. Then $f= \pi \circ j$ so $f\_! A = \pi\_! j\_! A = \pi\_\* j\_! A$ since $\pi$ is proper while $f\_\* A = \pi\_\* j\_\* A$ so $C$ is $\pi\_\*$ of the mapping cone of $j\_! A \to j\_\* A$, which is just the restriction of $j\_\* A$ to the divisor at $\infty$.


Depending on $A$, this might be easy or difficult to compute.


However, for $A$ the constant sheaf, this is not the best way to describe the mapping cone. This is because, in this case, $f\_\* A$ is simply a constant sheaf and $f\_!A $ is the constant sheaf shifted by twice the dimension $d$ of the vector bundle, so the mapping cone is simply an extension of two constant sheaves, associated to a cohomology class in $H^{2d}(Y,\mathbb Q)$.


The relevant cohomology class is the Euler class or top Chern class of the bundle $X$. The generator for the cohomology of $f\_! \mathbb Q$ can be represented by a section of the bundle $X \to Y$, and the isomorphism between $f\_\* \mathbb Q$ and $\mathbb Q$ can be calculated by pulling back to the zero section, so the cohomology class obtained by sending a generator on the map $f\_\* \mathbb Q \to f\_\* \mathbb Q$ is the pullback of a section to the zero section, i.e. the intersection of an arbitrary section with the zero section.


