  We study the singularities of Legendrian subvarieties of contact manifolds in
the complex-analytic category and prove two rigidity results. The first one is
that Legendrian singularities with reduced tangent cones are
contactomorphically biholomorphic to their tangent cones. This result is partly
motivated by a problem on Fano contact manifolds. The second result is the
deformation-rigidity of normal Legendrian singularities, meaning that any
holomorphic family of normal Legendrian singularities is trivial, up to
contactomorphic biholomorphisms of germs. Both results are proved by exploiting
the relation between infinitesimal contactomorphisms and holomorphic sections
of the natural line bundle on the contact manifold.
