Geometric Arveson-Douglas conjecture for the Drury-Arveson space: the case of one-dimensional variety. (English) Zbl 07811957
Summary: We consider a class of analytic subsets \(\tilde{M}\) of an open neighborhood of the closed unit ball in \(\mathbf{C}^n\). Such an \(\tilde{M}\) gives rise to a submodule \(\mathcal{R}\) and a quotient module \(\mathcal{Q}\) of the Drury-Arveson module \(H_n^2\) in \(n\) variables. The geometric Arveson-Douglas conjecture predicts that the quotient module \(\mathcal{Q}\) is \(p\)-essentially normal for \(p > d = \dim_{\mathbf{C}} \tilde{M}\). We prove this conjecture for the case of dimension \(d = 1\). In fact, we prove that if \(d = 1\), then \(\mathcal{Q}\) is 1-essentially normal, which is a stronger result than the original prediction.
MSC:
| 47Axx | General theory of linear operators |
| 32Axx | Holomorphic functions of several complex variables |
| 47Bxx | Special classes of linear operators |
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