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?