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We prove an integral inequality, first proven by Bray-Kazaras-Khuri-Stern onthree-dimensional asymptotically Euclidean (AE) manifolds, for the ADM mass onAE K\"ahler manifolds $(X,g,J)$ of all complex dimensions. This inequalitybounds the ADM mass from below by an integral of the scalar curvature and theHessian of any holomorphic coordinate function on $X$ arising in Hein-LeBrun'swork on the ADM mass on K\"ahler manifolds. Using this, we transfer the proofof the stability of the Positive Mass Theorem from Dong-Song andKazaras-Khuri-Lee from three-dimensional manifolds to AE K\"ahler manifolds. Weconclude that, for any such sequence of AE K\"ahler manifolds, we can findsubsets of each with vanishing boundaries in the limit such that thecomplements converge to Euclidean space in the pointed Gromov-Hausdorff sense.If we impose a uniform lower bound on the Ricci curvature, the result holdswithout taking the complement of a sequence of vanishing sets. Finally, we findthree new families of AE K\"ahler manifolds with vanishing mass in the limitfor which the stability results apply.

Mass Inequality and Stability of the Positive Mass Theorem For K\"ahler Manifolds