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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:

  1. 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.

  2. 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!

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why down votes? Can anybody who did this tell me ? –  Shizhuo Zhang Jun 4 '10 at 14:26
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4 Answers

up vote 5 down vote accepted

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 http://arxiv.org/abs/math.AG/9811155 and http://arxiv.org/abs/math.RT/0104114 . The second one contains some discussion of the general categorical formalism.

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The Kazhdan-Laumon construction is indeed fascinating, as are the other papers cited above. However they deal with a more subtle gluing. If one is simply trying to glue perverse sheaves from an open covering this is just the assertion that perverse sheaves are local/satisfy descent/form a stack (sheaf of categories) (for D modules this is in Beilinson-Bernstein). One can say this as part of a general descent formalism (eg comonads as in the second paper cited), but this is not the "subtle" part of gluing perverse sheaves. The same holds on a derived level, and even for microlocal gluings.. –  David Ben-Zvi Jun 3 '10 at 2:05
    
I am not sure that I quite understand your comment (what is the "subtle" part?); but certainly the problem of gluing sheaves from an open covering can be posed in two ways: assuming that the categories of sheaves on the intersections of the open sets of the covering are given, or when such categories are not defined. The former situation is the situation of a sheaf of categories, and the latter one is the situation of Kazhdan-Laumon gluing. –  Leonid Positselski Jun 3 '10 at 9:38
    
One classically describes descent as a limit of categories (sheaf condition) or in terms of descent data via a coalgebra or comonad. Both are easy and standard for perverse sheaves. The Kazhdan-Laumon construction, as I learned in particular from your beautiful treatment of it, is a nonstandard descent where you glue using Fourier transforms, with important applications in rep theory and NC algebra. Formally it's similar (again it's a categorical limit or comonadic) but it's not necessary for the classical question of describing perverse sheaves on a space from an open cover. –  David Ben-Zvi Jun 3 '10 at 16:08
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As others have said, to glue perverse sheaves given on two open copies of $\mathbb{A}^1$ into a sheaf on $\mathbb{P}^1$, it is necessary (and sufficient) simply to give an isomorphism of these sheaves on the intersection. This is the same procedure as for gluing anything usually called a "sheaf". Beilinson's paper concerns a different kind of "gluing" where the covering is not open, but in the form of a two-piece stratification.

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Ryan Reich has a paper called Notes on Beilinson's "How to glue perverse sheaves" -- it's also available off of Dennis Gaitsgory's Geometric Representation Theory page which is an amazing resource in the area. The paper by MacPherson and Vilonen, Elementary construction of perverse sheaves. Invent. Math. 84 (1986), no. 2, 403--435 provides another perspective on the same construction of perverse sheaves by gluing.

I don't know of a way to construct perverse sheaves in any noncommutative generality, and it seems like an unlikely proposition to me. You can certainly define modules over the de Rham complex and the like in great generality, but the condition of constructibility (not to mention the perverse t-structure) seems very commutative, involving restricting supports and dimensions etc. You can define holonomicity homologically, so once you have a notion of D-modules you can almost reverse engineer perverse sheaves (not sure how you'd describe regularity truly noncommutatively), but I don't see how you'd find a non-definitional version of the Riemann-Hilbert correspondence unless you're secretly in an almost commutative setting. (For that matter D-modules themselves are an almost-commutative notion..)

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Moving this comment here now that I can write them. One should read the version of that paper which is on the arXiv and not the one on Gaitsgory's page (unless you find an arXiv link there), since the physical copy which Google claims is there is quite out of date. –  Ryan Reich Jun 4 '10 at 23:56
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1 Enter your question into google and you get Beilinson's paper with the same name as well as some informal notes on it. It is nearby cycles construction and I don't really think anyone will explain it here better than written in the paper.

2 In short, I do not know. The quantum flag variety is a formal thing. It only has a category of $O$-modules and this has no relation to perverse sheaves, which are topological constructs. Besides the quantum flag has a category of $D$-modules. This category is already a fudge but it has certain useful features. You may be to back-engineer perverse sheaves by defining holonomicity but I dont think there are any sensible perverse sheaves on the category

And remember, 'mud' spelled backwards is 'dum'.

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