On the supports in the Humilière completion and γ -coisotropic sets

The symplectic spectral metric on the set of Lagrangian submanifolds or Hamiltonian maps can be used to define a completion of these spaces. For an element of such a completion, we define its gamma γ γ -support. We also define the notion of a gamma γ γ -coisotropic set and prove that every gamma γ γ -support is gamma γ γ -coisotropic. We then establish numerous properties of gamma γ γ -supports and gamma γ γ -coisotropic sets. We construct examples of Lagrangians in the completion having large gamma γ γ -support and we study those (called ‘regular Lagrangians’) having small gamma γ γ -support. Finally, we try to understand which singular Hamiltonians (i.e. Hamiltonians continuous on the complement of a set) have a well-defined flow in the Humilière completion.