For any holomorphic family of bounded strongly pseudoconvex domains in a Kähler manifold, the variation of complete Kähler-Einstein metrics is positive definite on the total space when the total domain is strongly pseudoconvex, and extends as a positive current under a scalar-curvature bound.
Choi, Variations of K¨ ahler-Einstein metrics on stronlgy pseudoconvex domains, Math
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Holomorphic family of strongly pseudoconvex domains in a K\"ahler manifold
For any holomorphic family of bounded strongly pseudoconvex domains in a Kähler manifold, the variation of complete Kähler-Einstein metrics is positive definite on the total space when the total domain is strongly pseudoconvex, and extends as a positive current under a scalar-curvature bound.