It might be a stupid question.
How to glue perverse sheaves? I am considering the following example. Flag variety of $sl_2$, which is $P^1$. Consider category of perverse sheaves on $P^1$. Denoted by $Perv(P^1)$. The big cells of flag variety of $sl_2$ give the affine cover for it. In this case, they should be two $A^1$. We can also consider category of perverse sheaves on $A^1$. My question is:
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How to glue two pieces of perverse sheaves on 1-dimensional affine spaces to that of projective spaces? More general, is there any gluing machinery which can globalize the perverse sheaves? It seems that Beilinson had a paper talking about this, but what I preferred is some expository notes explaining with some examples.
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Maybe I need to ask this in another question. How can one define perverse sheaves on noncommutative space. I am aware that there is a paper by Amnon Yekutieli, James J. Zhang talking about perverse sheaves on noncommutative space. However, what they considered was not really a noncommutative space from my understanding. (If I made mistake or bullshit, point out please). They consider quasi coherent sheaves of (not necessarily) commutative algebra on commutative scheme. Which does not fit my need. I am considering the following example: Quantized flag variety of $sl_2$ i.e. $Proj(O_q(G/N))$ in the sense of Lunts–Rosenberg (see also Erik Backelin and Kobi Kremnitzer and Tanisaki). It is a noncommutative scheme. I wonder whether James Zhang has also defined dualizing complexes for this case. How to define category of perverse sheaves on quantized flag variety?
The motivation for this question is I think there should be quantum version of Riemann–Hilbert correspondence. Which should describe the categorical equivalence:
$Perv(Proj(O_q(G/N))$ and category of quantum holonomic D-modules on quantized flag variety.
At present, I have more interest to know the answer of question 1. Thank you!
Best Answer
Beilinson's How to glue perverse sheaves explains how one can glue perverse sheaves on a variety from perverse sheaves on a closed subvariety and its open complement (assuming that the closed subvariety is the set of zeros of a regular function on the big variety).
Gluing perverse sheaves from an open covering is a very different problem. There is a paper by Kazhdan and Laumon Gluing of perverse sheaves and discrete series representations, devoted to the second problem. Further developments in this direction include the papers arXiv:math/9811155 and arXiv:math/0104114. The second one contains some discussion of the general categorical formalism.