Given a morphism $f:X\rightarrow Y$ between smooth complex varieties, one can define functors from the bounded derived category with holonomic cohomology on $Y$ to the same category on $X$. The easiest one is $Lf^{\*}$ which can be obtained by putting a $D$-module structure on the inverse image of $\mathcal O$-modules and deriving it. From this one can get two more functors:


* $f^!:=D \circ Lf^{\*} \circ D$, where $D$ is the duality functor and
* $f^{\dagger}:=Lf^{\*}[dim X-dim Y] $


Now my question is, under what conditions are these two isomorphic?


Edit: These notations are bad/confusing/wrong, since they are not compatible with the formalism of six functors. See answers below for better notations.



Confusion about D-modules and functors
Just to clear up some notational confusion (there doesn't seem to be any completely standard notation):


**In Bernstein's notes**:


The "easy" pullback (i.e. the one that coincides with the pullback of the underlying $\mathcal O$-module) is denoted $Lf^\Delta$.


$f^! = Lf^\Delta [dim X - dim Y]$ (right adjoint to $f\_!$)


$f^\ast = \mathbb D f^! \mathbb D$ (left adjoint to $f\_\ast$)


Note that $f^\ast$ and $f^!$ agree with the corresponding functors for constructable sheaves (*not* for the underlying $\mathcal O$-modules).


However... **In Hotta, Takeuchi, Tanisaki**


The easy pullback is denoted $Lf^\ast$ to agree with the $\mathcal O$-module functor.


Berstein's $f^!$ is now called $f^\dagger$.


Bernstein's $f^\ast$ is now called $f^\star$


Ok, now in **David Ben-Zvi's answer above (and in many other places)**


The easy pullback is $f^\dagger$, and the rest agrees with Berstein's notation.


In my opinion, the most important notational feature to be preserved is that $(f^\ast , f\_\ast)$ and $(f\_! , f^!)$ form adjoint pairs.


**To answer your question**


When $f$ is smooth, $f^\ast = \mathbb D f^! \mathbb D = f^! [2(dim Y - dim X)]$, and the easy inverse image functor is self dual (and preserves the t-structure).


One way to think about these different pullbacks is that the easy inverse image preserves the structure sheaf $\mathcal O\_Y$. This corresponds to the constant sheaf *shifted in perverse degree* under the RH correspondence. On the other hand $f^\ast$ preserves the "constant sheaf" whereas $f^!$ preserves the dualizing sheaf (for a smooth complex variety, these correspond to the D-modules $\mathcal O[-n]$ and $\mathcal O[n]$).


