User joan - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:52:11Z http://mathoverflow.net/feeds/user/27043 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108983/unique-symplectic-form-in-an-adapted-complex-structure Unique symplectic form in an adapted complex structure Joan 2012-10-06T06:07:51Z 2012-10-09T11:00:09Z <p>Hallo,</p> <p>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 :).</p> <p>Greetings Joan</p> http://mathoverflow.net/questions/108983/unique-symplectic-form-in-an-adapted-complex-structure Comment by Joan Joan 2012-10-08T15:22:10Z 2012-10-08T15:22:10Z is there any reference ? http://mathoverflow.net/questions/108983/unique-symplectic-form-in-an-adapted-complex-structure Comment by Joan Joan 2012-10-08T15:21:52Z 2012-10-08T15:21:52Z does this set of metrics have a manifold structure or something like that ? http://mathoverflow.net/questions/108983/unique-symplectic-form-in-an-adapted-complex-structure Comment by Joan Joan 2012-10-08T07:06:53Z 2012-10-08T07:06:53Z But I am asking not only the forms compatible with $J$ but also that fixes $M$ as a Lagrangian manifold! http://mathoverflow.net/questions/108983/unique-symplectic-form-in-an-adapted-complex-structure Comment by Joan Joan 2012-10-07T04:24:08Z 2012-10-07T04:24:08Z But I didn't fix $g$. $g$ comes with the symplectic form.