Strong　variational　sufficiency　is　a　newly　proposed　property,　which　turns　out　to　be　of　great　use　in　the　convergence　analysis　of　multiplier　methods.　However,　what　this　property　implies　for　nonpolyhedral　problems　remains　a　puzzle.　In　this　paper,　we　prove　the　equivalence　between　the　strong　variational　sufficiency　and　the　strong　second-order　sufficient　condition　(SOSC)　for　nonlinear　semidefinite　programming　(NLSDP)　without　requiring　the　uniqueness　of　the　multiplier　or　any　other　constraint　qualifications.　Based　on　this　characterization,　the　local　convergence　property　of　the　augmented　Lagrangian　method　(ALM)　for　NLSDP　can　be　established　under　the　strong　SOSC　in　the　absence　of　constraint　qualifications.　Moreover,　under　the　strong　SOSC,　we　can　apply　the　semismooth　Newton　method　to　solve　the　ALM　subproblems　of　NLSDP　because　the　positive　definiteness　of　the　generalized　Hessian　of　augmented　Lagrangian　function　is　satisfied.