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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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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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