Inspired by this question (Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold) I ask the following:

Suppose $X$ is a real analytic Riemannian manifold with a totally real embedding to $X^\mathbb C$ which is K?hler and the K?hler metric restricts to the given Riemannian metric on $X$. Moreover, $X^\mathbb C$ is equipped with an antiholomorphic involution whose fixed point set is $X$. Does these properties determine $X^\mathbb C$ uniquely as a germ of manifolds? I believe that's true but I haven't found the precise statement of this fact in Lempert, Sz?ke or Guillemin, Stenzel.

Now let $X$ be compact. In the answer to the cited question D.Panov said some necessary words about how to prove that $X^\mathbb C$ can be chosen to be compact. But can $X^\mathbb C$ be chosen is some canonical way or perhaps it is even unique?

In fact I'm even more interested in the case then $X$ is K?hler manifold with a totally real embedding to a hyperk?hler $X^\mathbb C$ such that the restriction of the associated K?hler structure to $X$ is the given K?hler structure. Moreover we are given a $S^1$ action which rotates the complex structures and whose fixed point set is $X$. Feix and Kaledin proved that these properties determine $X^\mathbb C$ uniquely as a germ. If $X$ is complete can $X^\mathbb C$ be chosen to be complete? Canonically? Uniquely? As I understand the last questions are far from being solved.


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