Holomorphic maps of real submanifolds in complex spaces of different dimensions

We de ne a new local invariant( called degeneracy) associated to a triple ( M, M', H), where M subset of C-N and M' subset of C-N' are real submanifolds of C-N and C-N', respectively, and H : M -->M' is either a holomorphic map, a formal holomorphic map, or a smooth CR-map. We use this invariant to nd sufficient conditions under which finite jet dependence, convergence and algebraicity results hold.