Let $M$ be a symplectic manifold (not Kaehler). Does there exists in a neighbourhood of the zero section in the cotangent bundle $T^{*}M$ a Hyperkaehler structure? I know that by the paper by Feix on can construct such structures if $M$ is Kaehler. Does this also hold or is there a similar construction in the waker case the symplectic case?

Greetings mirta

– Francois Ziegler Sep 23 '13 at 12:02"If a hyperkähler metric on the cotangent bundle $T^*M$ of a complex manifold $M$ exists, $M$ must be real-analytic Kähler; this can be seen from the twistor interpretation."