Essential normality of quotient submodules over strongly pseudoconvex finite manifolds

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.