Hallo,

I am reading the paper "Hyperkaehler structures on total spaces of holomorphic cotangent bundles" by Kaledin where he puts a hyperkähler structure on a neigbourhood of the $0$-section in the cotangent bundle of $M$, where $M$ is a real analytic Kaehler manifold. I have also read the papers of Stenzel, Guillemin, Lempert and Szoke where they put in a neighbourhood of the $0$-section an adapted complex structure of the cotangentbundle. My question is the following: starting with a real analytic Kaehler manifold $(M,I,\omega)$. First, make the hyperkaehler construction i.e. we have in a neigbourhoord of the $0$-section in the cotangent bundle $T^{*}M$ a hyperkaehler structure i.e. $I_{0}, J_{0}, K_{0}$ (three complex structures) and with the hyperkaehler metric $h$ (that restricts on the $0$-section to the original metric on $M$). Second, do the construction with an adapted complex structure i.e. in a neighbourhood of the $0$-section in the cotangent bundle the exists a kaehler structure $J_{1}$, $\omega_{1}$ such that the corresponding kaehler metric restricts to the original metric on the $0$-section (well, tis has also some aditional properties: that the leaves of the Riemannian foliation are complex submanifolds). I am asking if there is any relationship between these two constructions i.e. is there some transformation that brings the Kaehler structure $J_{1}$ in relationship with the hyperkaehler structures $I_{0}, J_{0}, K_{0}$? Or does $J_{1}$ conincide with one of the three ? Or are the constructions totally independent? I hope for a lot of answers. Thanks a lot.

Ben