It should not be hard to produce a Frechet manifold structure on the space of symplectic submanifolds. A symplectic submanifold of a symplectic manifold has a neighbourhood which is symplectomorphic to the total space of its normal bundle, with a natural (split) symplectic structure. This is a version of Darboux theorem, found, for example, in Dusa McDuff's Park City lectures. Now, the $C^\infty$ symplectic deformations of a zero section in a symplectic bundle are sections of this symplectic bundle. This is a Frechet vector space. We have constructed a Frechet atlas on the space of embeddings.

It should not be hard to produce a Frechet manifold structure on the space of symplectic submanifolds. A symplectic submanifold of a symplectic manifold has a neighbourhood which is symplectomorphic to the total space of its normal bundle, with a natural (split) symplectic structure. This is a version of Darboux theorem, found, for example, in Dusa McDuff's Park City lectures. Now, the $C^\infty$ symplectic deformations of a zero section in a symplectic bundle are sections of this symplectic bundle. This is a Frechet vector space. We have constructed a Frechet atlas on the space of symplectic submanifolds.