  This paper mainly concerns the biholomorphic invariance of $p$-essential
normality of Hilbert modules on bounded symmetric domains. By establishing new
integral formulas concerning rational function kernels for the Taylor
functional calculus, we prove a biholomorphic invariance result related to the
$p$-essential normality. Furthermore, for quotient analytic Hilbert submodules
determined by analytic varieties, we develop an algebraic approach to proving
that the $p$-essential normality is preserved invariant if the coordinate
multipliers are replaced by arbitrary automorphism multipliers. Moreover, the
Taylor spectrum of the compression tuple is calculated under a mild condition,
which gives a solvability result of the corona problem for quotient submodules.
As applications, we extend the recent results on the equivalence between
$\infty$-essential normality and hyperrigidity.
