Infinite dimensional symplectic space

Question

Is the infinite dimensional space of symplectic compact sub-varieties of a symplectic variety, symplectic?

The symplectic form $\Omega$ at $M$ would be:

$$\Omega (X,Y)= \int_M \omega (X_x, Y_x) dx $$

where $\omega$ is the symplectic form of the variety, and $X,Y$ are vectors at $M$.