greetings,

Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. $(X,\omega)$ is a Kähler manifold. My question is: are there any examples known where such a construction does not admit a Ricci-flat Kähler metric i.e. where one cannot find a different Kähler metric which is Ricci-flat (maybe in the same cohomology or not)? I would be very thankfull for answers.

best regards dominik

notalways work. – Spiro Karigiannis Oct 10 '12 at 12:20