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.
Best Answer
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 $\mathcal O[n]$).