  We consider deformations of singular Lagrangian varieties in symplectic
spaces. We show the coherence of the direct image sheaves of relative
infinitesimal Lagrangian deformations. Using this result, we prove that, under
some assumptions, a Lagrangian deformation is analytically rigid provided that
this is the case infinitesimally. This gives generalisations of recent results
by Colin de Verdi\`ere concerning Lagrangian curves.
