Distinguished varieties and geometric Arveson-Douglas problem in polydiscs

In the present paper, we first revisit the theory of distinguished varieties and develop some new results. These results are used to characterize the essential spectrum and the essential normality of distinguished quotient modules of A α 2 ( D d ) , where α ≥ − 1 is an integer. It is shown that the essential spectrum of a distinguished quotient module is identified with the topological boundary of the variety associated with the corresponding submodule. The essential normality of radical distinguished quotient modules is proven under the assumption on the regularity of the associated variety along ∂ D d . The non-triviality of K -homology elements of these distinguished quotient modules are verified.