Reconstruction from category of D-modules on variety

Arinkin has a theorem which says that an abelian variety can be reconstructed from its derived category of coherent D-modules.

D.Orlov conjectured that this theorem is true for any variety.

My question is:

Is this conjecture proved or disproved? I wonder know the related work, examples and any other related observations, comments.

Thanks

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Looking in MathSciNet for papers which reference the paper where Orlov makes the conjecture might be a good start :) –  Mariano Suárez-Alvarez Aug 3 '10 at 5:13
I am not an expert. I have heard algebraic geometers talk about "Fourier-Mukai" transform in this context. –  Bruce Westbury Aug 3 '10 at 6:55
Ah! You said $D$-modules! –  Bruce Westbury Aug 3 '10 at 6:56
@Mariano Suarez-Alvarez: Could you please clarify the status of your comment? Do you mean that looking at MathSciNet is a good idea in general? I don't even know which, if any, Orlov's paper has the statement (I heard it in conversation); perhaps the OP could provide references? Or does the smile indicate that you know of a paper that is relevant; if so, could you please be more specific? It's not that I fail to enjoy wasting time by randomly tracing MathSciNet, of course. –  t3suji Aug 3 '10 at 14:49
It might also be good to add the "noncommutative geometry" tag. –  Peter Samuelson Aug 3 '10 at 18:39

1 Answer

As far as I understand from your statement of the conjecture, the conjecture is false, although there are similar statements that are true. If I understand correctly, a weaker question (more likely to have the answer yes) would be "can one recover a variety from its category of D-modules."

For a non-example of the weaker question, if $X = Spec(\mathbb{C}[x])$ and $Y = Spec(\mathbb{C}[x^2,x^3])$, then D(X) and D(Y) are Morita equivalent. If X is a smooth curve and Y is another curve, then D(X) is Morita equivalent to D(Y) iff X and Y are homeomorphic (in the example above, the normalization map gives a homeomorphism $X \to Y$). If $X = Spec(\mathbb{C}[x])$, then the natural numbers parameterize isomorphism classes of curves Y with D(X) Morita equivalent to D(Y).

A similar-sounding statement which is true is "If X and Y are smooth curves, they are isomorphic iff D(X) and D(Y) are isomorphic (as algebras)." A paper with these and many more facts can be found here http://arxiv.org/abs/math/0304320

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The conjecture was probably about projective varieties over complex numbers. –  Zoran Skoda Aug 4 '10 at 14:12
That's probably true - I hadn't heard the conjecture before so all I had to go on was the OP's statement. It would be nice if there were a link to the actual conjecture, if it's written somewhere. –  Peter Samuelson Aug 4 '10 at 19:01