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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

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does anyone have an idea ? –  dominik Sep 25 '12 at 5:35
anyone with an idea ??? –  dominik Sep 26 '12 at 5:15
Try looking at this paper of Stenzel: math.osu.edu/~stenzel.3/research/publications/ricci-flat.pdf –  Spiro Karigiannis Oct 10 '12 at 12:19
and also this one of Calabi: archive.numdam.org/ARCHIVE/ASENS/ASENS_1979_4_12_2/… –  Spiro Karigiannis Oct 10 '12 at 12:20
The answer to your question could be probably found in these papers. I didn't take the time to look. My guess is that it almost certainly does not always work. –  Spiro Karigiannis Oct 10 '12 at 12:20

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