Extending Nirenberg–Spencer’s question on holomorphic embeddings to families of holomorphic embeddings

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 applications, we obtain a stronger version of the Cartan-Fubini type extension theorem for Fano manifolds of Picard number 1 and also prove that two linearly normal projective K3 surfaces in ${\bf P}^g$ are projectively isomorphic if and only if the families of their general hyperplane sections trace the same locus in the moduli space of curves of genus $g >2$.