Essential normality and the decomposability of algebraic varieties

TL;DR

This paper investigates the essential normality of algebraic subvarieties in the unit ball, showing its invariance under certain maps and linking it to the decomposability of varieties, thereby confirming the Arveson-Douglas conjecture for specific cases.

Contribution

It proves that essential normality is preserved under variety isomorphisms and decompositions, advancing understanding of the Arveson-Douglas conjecture for decomposable varieties.

Findings

Essential normality is invariant under isomorphisms between varieties.

Essential normality is preserved under certain non-invertible maps.

The conjecture holds for varieties decomposing into linear subspaces or disjoint components.

Abstract

We consider the Arveson-Douglas conjecture on the essential normality of homogeneous submodules corresponding to algebraic subvarieties of the unit ball. We prove that the property of essential normality is preserved by isomorphisms between varieties, and we establish a similar result for maps between varieties that are not necessarily invertible. We also relate the decomposability of an algebraic variety to the problem of establishing the essential normality of the corresponding submodule. These results are applied to prove that the Arveson-Douglas conjecture holds for submodules corresponding to varieties that decompose into linear subspaces, and varieties that decompose into components with mutually disjoint linear spans.