# Differential Geometry – Hypothetical uniqueness of embedding a Riemann manifold in a compact kähler

Inspired by this question (isometric embedding of a real analytic Riemann manifold into a compact Kähler manifold), I ask the following:

Accept $$X$$ is a true analytic Riemannian manifold with a really real embedding $$X ^ mathbb$$ This is Kähler and the Kähler metric is limited to the given Riemann metric $$X$$, in addition, $$X ^ mathbb$$ is equipped with an antiholomorphic involution whose fixed point set is $$X$$, Determine these properties? $$X ^ mathbb$$ unique as a germ of manifolds? I think that's true, but I did not find the exact statement in Lempert, Szöke or Guillemin, Stenzel.

Now let it go $$X$$ be compact. In response to the quoted question, D. Panov has given some necessary words on how to prove this $$X ^ mathbb$$ can be chosen compact. But can $$X ^ mathbb$$ Did you choose a canonical style or is it even unique?

In fact, the case interests me even more $$X$$ Kähler is manifold with a really real embedding in a hypercounter $$X ^ mathbb$$ so that the restriction on the associated Kähler structure is up $$X$$ is the given Kähler structure. We also get one $$S ^ 1$$ Action that rotates the complex structures and sets their fixed point set $$X$$, Feix and Kaledin have proven that these properties are determinative $$X ^ mathbb$$ unique as germ ob $$X$$ is completely can $$X ^ mathbb$$ be chosen to be complete? Canonically unique? As far as I know, the last questions are far from solved.