# $D$-modules and quasi-projective varieties?

Until recently I was under the impression, that for any morphism $f:X\rightarrow Y$ of smooth complex varieties there exist functors six functors $f^*,f_{*},...$ between the derived categories of $\cal D$-modules with bounded holonomic cohomology.

However the experts assume, that all varieties are quasi-projective (See for example Bernsteins notes 1.7 or Hotta, Takeuchi Tanisaki 1.4.19)! It seems that this assumption is manily used to guarantee that certain (finite) resolutions exist, but also on a couple of other occasions.

My questions are:

1) Is there a "philosophical" reason why one should restrict to quasi-projective case?

2) Is this assumption crucial or does the whole machinery also work for arbitrary smooth complex varieties?

3) Without the assumption is there still a six functors formalism between unbounded complexes with holonomic cohomology?

4) Does the Riemann-Hilbert correspondence fail for arbitrary smooth complex varieties?

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The restriction to the quasi-projective case is unnecessary. E.g., modern descent techniques for derived categories allow you to pass from the affine case to the general case. I would be very concerned about all six functors existing for unbounded complexes with holonomic cohomologies, but bounded below seems okay (more generally, homotopy colimits of bounded holonomic complexes should be okay, and that's probably as much as holonomicity buys you for free). RH is fine for arbitrary complex varieties (since it's Zariski/etale local). –  Moosbrugger Jul 24 '12 at 12:44
Thanks for your useful comment! Do you know a reference which develops the formalism without the quasi-projective assumption? –  Jan Weidner Jul 24 '12 at 13:20
Yes: math.harvard.edu/~gaitsgde/GL/Crystalstext.pdf or Saito's "D-modules on analytic spaces" must do this in some form. –  Moosbrugger Jul 24 '12 at 13:49
Mhhh, I find both of these quite intimidating. –  Jan Weidner Jul 24 '12 at 14:00
Then an alternative: my memory is that e.g. Borel's book uses quasi-projectivity but very sparsely. You could try to remove the hypothesis for yourself by modifying the proofs. I think he probably uses it in ways that modern science would avoid via Thomason-Trobaugh. –  Moosbrugger Jul 24 '12 at 14:37