Hallo,

I ave the following question: Due to Stenzel, Lempert, Szöke ect. we know that a Riemannian manifold $(M,g)$ admits a complex structure on an neighbourhood of the cotangent bundle. This complex structure $J$ is unique due to bruhat and whitney complexification method. Well, with this complex structure there comes a symplectic form $\omega$ that is Kähler and morover $M$ is a Lagrangian submanifold in this complexified neighbourhood of the zero section in the cotangent bundle. My question is: is this Kähler form $\omega$ unique? Does there exists a different (or in the same cohomology) Kähler form with the same properties? If its not unique to which extend does uniqueness fail? Or, how can one caracterize the space of Kähler forms on a neighbourhood of the zero section in the cotangent bundle that fixes $M$ as a Lagrangian submanifold? (actually in the last question I am very interested) I hope to get a lot of answers and I apologize if the question is too trivial/hard :).

Greetings Joan

Riemannianmanifold $(M, g_0)$ then $T^*M = TM$ acquires a natural Riemannian metric $g$ from $g_0$; this is what I thought you meant. Since this is not the case, are you asking: "I have a complex manifold $(N, J)$. What are all symplectic forms compatible with $J$." ? If this is the case, I believe there are lots of these forms. See arxiv.org/abs/math/0004122 which applies to your question when $M$ is a torus (so $T^*M = (C^\times)^n$). – Eugene Lerman Oct 7 '12 at 11:47