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A variety is called $\mathcal D$- affine if the global section functor induces an equivalence between quasi-coherent $\mathcal D$-modules and modules over $\Gamma(X,\mathcal D)$. It is easy to see that affine varieties are $\cal D$-affine. More surprisingly, by an important theorem of Beilinson-Bernstein, (partial) flag varieties are $\mathcal D$-affine as well!

So my question is, how rare are $\mathcal D$-affine varieties? Are there other natural examples of $\mathcal D$-affine varieties?

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This is definitely not an answer, but I don't think there are known examples except partial flag varieties and affine varieties. –  Alexander Braverman Nov 2 '11 at 21:23
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up vote 15 down vote accepted

As Alexander mentions above, it is a conjecture (and one that has stood for a while) that the OP has given a complete list of D-affine projective varieties.

My perspective on this is that $T^*G/P$ (which differential operators quantize) are just not a good model for other cotangent bundles. For example:

  • $T^*G/P$ is a resolution of singularities of its affinization. It's also conjectured that they are the only cotangent bundles of projective varieties with this property.
  • $T^*G/P$ has a complete hyperkähler metric. It's also conjectured that they are the only cotangent bundles of projective varieties with this property.
  • $T^*G/P$ deforms flatly to an affine variety (a generic nilpotent orbit intersected with the appropriate Slodowy slice). I don't know of any cotangent bundles of projective things that satisfy this, though I don't know if anyone else has thought much about it.

Interestingly, if you look for symplectic varieties with the properties above, then you will find some which think they are the cotangent bundle of something D-affine (i.e. have a deformation quantization with a $\mathbb{C}^*$-action which they are affine with respect to). There's a way of "fixing" the cotangent bundles of toric varieties to get D-affinity back, called hypertoric varieties. The affinity theorem in this case was proved by Bellamy and Kuwabara quite recently. An upcoming paper of myself, Braden, Licata and Proudfoot is going to contain some weak versions of these theorems (basically, the proof that some quantizations are affine, without actually proving that any particular one is) in a much more general situation that includes things like quiver varieties and does a recent paper of McGerty and Nevins.

Basically, you should think of the these theorems as coming from the flat deformation to something affine. by going non-commutative, you take some peek out in the direction of this deformation, and see that there is an affine variety right over there.

On some level, one should think of the question of finding out which quantizations of a symplectic variety are affine; Beilinson and Bernstein solve this very combinatorially for the flag variety (it's when for the indexing weight, no inner product with a coroot is a negative integer) and the hypertoric case as a very concrete answer (it's when certain polytopes contain lattice points, though Bellamy and Kuwabara don't state it quite like this). I'm not sure there are any other examples not closely related to these where we know the precise formulation of the localization theorem.

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Great answer! It's worth also mentioning the work of McGerty-Nevins (arxiv.org/abs/1108.6267) which proves D-affinity in the refined sense you mention in a very general setting (derived version here, abelian version to appear). –  David Ben-Zvi Nov 3 '11 at 0:34
    
Concerning flat deformation to affine variety. I thought it is not an exception but general pattern for many things: cotangent to module space of vector bunles on curve - and represenations of fundamental group to GL_n(C) - one is T^*projective another is affine, the complex structres differes by hyper-kaehler rotations as far as I heard. Hilbert scheme of points and C^2 and Calogero-Moser - again same thing. There is some result by Kaledin-Verbitsky where they prove that under certain conditions symplectic resultions always are deformed to affine... –  Alexander Chervov Nov 3 '11 at 7:58
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What about same D-affiniteness for flags for affine Lie algebras ? They are infitnite-dimensional, but moduli space of vector bundles on curves in many respects has similar properties, so may be they are also D-affine ? If we take curves with "marked points" we get some quiver varietes - is it known are they D-affine ? –  Alexander Chervov Nov 3 '11 at 7:59
    
Ben, you're missing a hypothesis in the (second) hyperk\"ahler conjecture, no? cotangent bundles of abelian varieties, for example. –  Thomas Nevins Dec 21 '11 at 4:37
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Warning: the theorem of Beilinson-Bernstein fails to hold in positive characteristic.

Let me elaborate more about $D$-affinity in characteristic $p>0$ (a good reference is the introduction to A. Langer, "$D$-affinity and Frobenius morphism on quadrics"). In general, to prove that a variety is $D$-affine, we have to check two things: (1) that every quasi-coherent (over $\mathcal{O}_X$) $D$-module is globally generated over $D_X$, (2) that $H^i(X, D_X) = 0$ for $i>0$. In the characteristic $p$ case, $D_X$ has the so-called $p$-filtration (found by Haastert) by $\mathrm{End}(F^i_{\star} \mathcal{O}_X)$ by endomorphisms of the Frobenius push-forwards of the structure sheaf. It follows that if $H^i(X, \mathrm{End}(F^i_{\star} \mathcal{O}_X)) = 0$ then (2) holds (and conversely if $X$ is Frobenius split). For example, (2) holds for the projective space, and indeed projective spaces are always $D$-affine. It has been shown by Langer in the aforementioned article that if $n>1$ is odd and $p\geq n$ then the $n$-dimensional smooth quadric is $D$-affine. Since quadrics are Frobenius split, the same calculation of $F^s_\star \mathcal{O}_X$ shows that the even-dimensional quadrics are not $D$-affine in positive characteristic.

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Toric varieties are $\mathcal D$-affine.

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This is false. The only D-affine toric varieties are products of projective spaces: ams.org/mathscinet-getitem?mr=1435566 –  Ben Webster Nov 2 '11 at 22:00
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Hah. That's what I get from posting from memory alone: that was the paper I had in mind, exactly :) –  Mariano Suárez-Alvarez Nov 2 '11 at 22:09
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An understandable mistake. It just happens that one of the ways I try to explain my research is interesting is to think of it as fixing the lack of D-affinity for toric varieties. –  Ben Webster Nov 2 '11 at 22:16
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