In the 2001's paper 'Hyperk?hler metrics on cotangent bundles' B.Feix gives a construction of a hyperk?hler metric on a neighbourhood of zero section in $T^*X$ where $X$ is a real analytic K?hler manifold.
As for me, this construction is awful. It's not difficult to understand the general steps of it and I'm really inspired by them but if you look at details it's just awful. Almost everything is written in local coordinates without explaining why everything can be glued globally, the article starts with choosing a local antiholomorphic involution on $X$ without mentioning it properly, she calls 'the complexified K?hler form on $X^\mathbb C$' a form which is not the analytic continuation of the K?hler form on $X$ etc.
I've spent several days trying to rewrite the paper in a coordinate-free way changing some steps as it'd be more similar to constructions of a K?hler metric on a neighbourhood of zero section in $T^*X$ where $X$ is a real analytic Riemannian manifold due to Lempert, Sz?ke and Guillemin, Stenzel. I'm not completely successful yet, though. So I ask, perhaps this paper is already written? I know about D.Kaledin's paper but his approach is rather different, I want something which looks like Feix's construction.