Essential normality and the decomposability of algebraic varieties (Q374034)

Let \(d\) be a fixed positive integer and let \(\mathbb{C}[z]=\mathbb{C}[z_1,\dots, z_d]\) denote the algebra of complex polynomials in \(d\) variables. The Drury-Averson space \(H^2_d\) is the reproducing kernel Hilbert space on the unit ball \(\mathbb{B}_d\) of \(\mathbb{C}^d\) generated by the kernel functions \(k_\lambda (z)=\frac{1}{1-\langle z,\lambda \rangle},\,\, \lambda \in\mathbb{B}_d\). For an ideal \(I\) of \(\mathbb{C}[z]\), let \(\mathcal{F}_I=H^2_d \ominus I\) and \(V(I)=\{z\in \mathbb{B}_d\mid p(z)=0\text{ for all }p\in I\}\) and for a subset \(V\) of \(\mathbb{B}_d\), let \(I(V)=\{p \in \mathbb{C}[z] \mid p(z)=0\text{ for all } z\in V\}\). If \(V=V(I)\) and \(I=I(V)\), which is the case whenever \(I\) is a radical ideal, then one writes \(\mathcal{F}_V\) for \(\mathcal{F}_I\) and \(\mathcal{A}_V\) for \(\mathcal{A}_I\).NEWLINENEWLINEIn this paper, the authors study the Arveson-Douglas conjecture which states that, if \(V\) is a homogenous variety in the unit ball \(\mathbb{B}_d\), then the submodule \(\mathcal{F}_V\) is \(p\)-essential normal for every \(p>\dim V\). They prove that, if \(W\) is a homogenous variety in the unit ball \(\mathbb{B}_{d'}\), for some positive integer \(d'\), and if \(\mathcal{A}_V\) is isomorphic to \(\mathcal{A}_W\), then \(\mathcal{F}_V\) is \(p\)-essential normal if and only if \(\mathcal{F}_W\) is \(p\)-essential normal. They also establish a similar result for maps between varieties that are not necessarily isomorphic. They also consider when it is possible to decompose \(V\) as \(V=V_1\cup V_2\cup \cdots \cup V_n,\) where \(V_1,\dots,V_n\) are homogeneous varieties in \(\mathbb{C}^d\) with the property that that the algebraic sum \(\mathcal{F}_{V_1}+ \cdots +\mathcal{F}_{V_n}\) is closed.