5
$\begingroup$

Riemann Hilbert Correspondence states for complex manifold $X$, the bounded derived category of $D$ modules on $X$ with cohomology being regular holonomic is equivalent to the bounded derived category of sheaves on $X$ with constructible cohomology.

My question is if we fix a particular stractification $\mathcal{S}$ for $X$ and require the cohomology sheaves is constructible with $\mathcal{S}$, what would be the corresponding subcategory on the $D$ modules side?

$\endgroup$
5
$\begingroup$

One should phrase the constructibility in terms of six-functors and then, since the RH correspondence respects those, one will see what is the corresponding notion.

First, to be a local system for a constructible sheaf translates under RH to the D-module being smooth (i.e., free of finite rank as an $\mathcal{O}$-module).

So, if constructibility of a sheaf $F$ means that for any stratum $i_{\alpha} : S_{\alpha} \to X$ one has that $i_{\alpha}^* F$ is a local system, the corresponding condition for a $D$-module $M$ will be that $i_{\alpha}^* M$ is smooth for all $\alpha$.

(Slightly orthogonally to the question itself; One also needs to be careful that there is a second notion - a sheaf $F$ is $!$-constructible if $i_{\alpha}^! F$ is a local system for all $\alpha$. In some situations (nice equivariant ones usually), $!$-constructibility is equivalent to $*$-constructibility.)

$\endgroup$
  • $\begingroup$ Could you provide a reference for the fact about local system and smooth D-module? $\endgroup$ – userabc Jan 26 at 19:44
  • $\begingroup$ @userabc See Hotta tanasaki and Takeuchi’s book. Specifically the part about integrable connections (which is what Sasha calls smooth) $\endgroup$ – Avi Steiner Jan 27 at 5:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.