Abstract: I will discuss ``asymptotically complex hyperbolic'' metrics (ACH metrics) satisfying the Einstein equation. Standard examples are the complete K\"ahler-Einstein metrics on bounded strictly pseudoconvex domains of $\mathbb{C}^n$ constructed by Cheng and Yau. If we want to obtain further such metrics, the natural idea is to deform the Cheng--Yau metrics, via analysis on the linearized Einstein operator acting on symmetric $2$-tensors. The foundation of such linear analysis on ACH manifolds was given independently by Roth and Biquard (while an analogous theory for asymptotically \emph{real} hyperbolic manifolds dates back to a previous work of Graham and Lee), and with this linear theory they established that it is possible to deform the complex hyperbolic metric on the ball as desired. Our purpose here is to generalize this result to arbitrary Cheng--Yau metrics, under the technical assumption that the dimension satisfies $n\ge 3$. The crucial idea in the proof is recasting the vanishing of the $L^2$-kernel of the (gauged) linearized Einstein operator to, using an observation made by Koiso in the 1980s, that of the $L^2$ Dolbeault cohomology with values in the holomorphic tangent bundle. Then techniques from several complex variables can be applied.