I'm looking for references on constructible sheaves and the six operation formalism on analytic stacks (stacks fibered over complex analytic spaces). Does anyone have some suggestions?

Basically I want it to be an analytic version of Laszlo and Olsson's articles "The six operations for sheaves on Artin stacks, I, II", or a stack version of Dimca's book "Sheaves in Topology".


I don't know an answer to your question, but there is already a literature on D-modules on algebraic stacks (over $C$ say) and I'm not sure I would expect a significant difference with your topic, though I might be missing something. The basic reference for this is Chapter 7 (essentially a book in itself) of Beilinson and Drinfeld's "Quantization of Hitchin Hamiltonians and Hecke Eigensheaves", available here. There's also a treatment in my paper with David Nadler on Character Theory of a Complex Lie Group, here, in a more homotopical language, and I would imagine most of the arguments translate to your setting, though perhaps I'm missing something..

Edit: to be slightly more detailed, the argument is explained in Section 4.1 of our paper (I'll try to paraphrase to your setting). You write a smooth stack as a colimit of spaces (say as a geometric realization of a simplicial analytic space) with smooth maps, this defines the constructible derived category (or more precisely its $\infty$-version) as a (homotopy) limit of categories along the simplices with maps given by pullbacks ($f^!$'s in our setting -- those are the natural functors on all $D$-modules). By base change and composition you find that $f_*$ and $f^!$ are then defined for maps of smooth stacks. You also check Verdier duality descends (after normalizing by dimensions) to this limit. That gives you the other two functors, and the adjunction relations descend automatically as well.

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  • $\begingroup$ Dear David: thank you for giving me two excellent references. I read a bit about them (though I know almost nothing about geometric Langlands), and they are still a little different from what I want. I'd like a ref that discusses the foundations, hopefully in the style of Laszlo-Olsson. In particular, we'd better not to make the smoothness assumption (in which case the constant sheaf serves as a dualizing complex, and we don't need BBD glueing lemma, generalized to unbdd complexes in LO, to construct it). Also, it should discuss proper/smooth base change, purity etc. in the style of SGA4. $\endgroup$ – shenghao Feb 11 '11 at 0:52

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