Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion

We consider a class of analytic subsets M˜ of an open neighborhood of the closed unit ball in Cⁿ. Such an M˜ gives rise to a submodule R and a quotient module Q of the Hardy module H²(S) on the unit sphere S⊂Cⁿ. We show that, as predicted by the geometric Arveson-Douglas conjecture, the quotient module Q is p-essentially normal for p>d=dim_CM˜. We further show that, more interestingly, the quotient module Q exhibits a behavior that is only found on the Bergman space and the Fock space: an operator A in the Toeplitz algebra on Q is compact if and only if its Berezin transform vanishes near M˜∩S.