Formal principle with convergence for rational curves of Goursat type (2404.05941v1)

Abstract: We propose a conjecture that a general member of a bracket-generating family of rational curves in a complex manifold satisfies the formal principle with convergence, namely, any formal equivalence between such curves is convergent. If the normal bundles of the rational curves are positive, the conjecture follows from the results of Commichau-Grauert and Hirschowitz. We prove the conjecture for the opposite case when the normal bundles are furthest from positive vector bundles among bracket-generating families, namely, when the families of rational curves are of Goursat type. The proof uses natural ODEs associated to rational curves of Goursat type and corresponding Cartan connections constructed by Doubrov-Komrakov-Morimoto. As an example, we see that a general line on a smooth cubic fourfold satisfies the formal principle with convergence.