# Hypothetical uniqueness of an embedding of a Riemannian manifold to a compact Kähler one

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.