  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.
