Essential normality of quotient submodules over strongly pseudoconvex finite manifolds

TL;DR

This paper proves $p$-essential normality of certain quotient submodules on strongly pseudoconvex domains, confirming parts of the Arveson-Douglas Conjecture and addressing related open problems with applications in $K$-homology.

Contribution

It establishes $p$-essential normality for Bergman-Sobolev quotient submodules over strongly pseudoconvex manifolds with specific geometric conditions, advancing the understanding of operator theory in complex geometry.

Findings

Proves $p$-essential normality for quotient submodules with $p$ exceeding the dimension of the noncompact part.

Partially confirms the geometric Arveson-Douglas Conjecture.

Addresses the trace-class antisymmetric sum problem for truncated Toeplitz operators.

Abstract

We investigate the $p$-essential normality of Hilbert quotient submodules on a relatively compact smooth strongly pseudoconvex domain in a complex manifold satisfying Property (S). For analytic subvarieties that have compact singularities and transversely intersect the strongly pseudoconvex boundary, we prove that the corresponding Bergman-Sobolev quotient submodules are $p$-essentially normal whenever $p$ exceeds the dimension of the noncompact part of the analytic subvarieties. As a consequence, we partially confirm the geometric Arveson-Douglas Conjecture and resolve an open problem regarding the trace-class antisymmetric sum of truncated Toeplitz operators within a broader context. Moreover, we provide applications in $K$-homology and geometric invariant theory.