The study of p-essentially normal Hilbert modules originated from Arveson's seminal works circa 2000. Recently, there has been an increasing focus on the p-essential normality of Hilbert modules determined by subvarieties. In this talk，we shall discuss the p-essential normality of holomorphic Sobolev quotient submodules over strongly pseudoconvex finite manifolds satisfying Property (S). We prove that holomorphic Sobolev quotient submodules are p-essentially normal whenever p exceeds the dimension of the noncompact part of the corresponding analytic subvarieties. The result differs from the Euclidean case and reveals some distinctive phenomena on non-Stein manifolds. As a consequence, we confirm the geometric Arveson-Douglas Conjecture for subvarieties with isolated singularities, which covers several previously known results. We also resolve an open problem recently proposed by Wang and Xia concerning the trace-class antisymmetric sum of truncated Toeplitz operators, within a broader context.
