  Nirenberg and Spencer posed the question whether the germ of a compact
complex submanifold in a complex manifold is determined by its infinitesimal
neighborhood of finite order when the normal bundle is sufficiently positive.
To study the problem for a larger class of submanifolds, including free
rational curves, we reformulate the question in the setting of families of
submanifolds and their infinitesimal neighborhoods. When the submanifolds have
no nonzero vector fields, we prove that it is sufficient to consider only
first-order neighborhoods to have an affirmative answer to the reformulated
question. When the submanifolds do have nonzero vector fields, we obtain an
affirmative answer to the question under the additional assumption that
submanifolds have certain nice deformation properties, which is applicable to
free rational curves. As an application, we obtain a stronger version of the
Cartan-Fubini type extension theorem for Fano manifolds of Picard number 1. We
also propose a potential application on hyperplane sections of projective K3
surfaces.
