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Let K be an algebraically closed field of char. 0, let A_n(K) be the Weyl algebra. Let I in A_n(K) be a left ideal generated by p elements. Set M := A_n / I.

Does the following then hold?

dim Ch(M) \geq 2n - p

Here Ch(M) is the characteristic variety of M. (I know that the answer is yes if n = 1, and also if I is generated by homogeneous elements.)

Thank you.

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The isn't true: there's a theorem of Stafford which says that any left ideal in the Weyl algebra is generated by two elements so if your claim was true, then the singular support would have to have dimension at least 2n-2 always, which of course isn't the case if n>2.

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