Image of an isotropic manifold under lagrangian correspondence is isotropic?

Is the following statement well known?

Let $M,N$ be symplectic (algebraic) manifolds. Let $L \subset M \times N$ be a (smooth) Lagrangian correspondence. For a subset $X \subset M$ we denote $L(X):=(y\in N|(x,y)\in L)$. Let $X \subset M$ be an isotropic subvariety (i.e. its smooth locus is isotropic). Then $L(X)$ is isotropic.

Is it written somewhere? I think I can prove it and it is quite simple, but I'll rather use a reference instead.

Thank you

• You should clarify your definition of $L(X)$. ie. does it consist of those $y\in N$ for which $X\times \{y\} \subset L$ or just those $y$ for which there exists an $x\in X$ such that $(x,y) \in L$ ? And does ''correspondence'' just mean $L$ is a subset of $M \times N$ or something more?
– JHM
Jul 21, 2012 at 16:34
• Plus, i think good advice might be to just forego looking for any reference and providing the one line proof which it requires.
– JHM
Jul 21, 2012 at 17:05
• to J. Martel. Is it better now? Thank you.
– Rami
Jul 22, 2012 at 11:03