Local H $H$ -principles for holomorphic partial differential relations

We introduce the notion of the realifications of an arbitrary holomorphic partial differential relation $R$, that are partial differential relations associated with the restrictions of $R$ to totally real submanifolds of maximal dimension. Our main result states that if any realification of an open holomorphic partial differential relation over a Stein manifold satisfies a relative to domain $h$-principle, then it is possible to deform any formal solution into one that is holonomic in a neighborhood of a Lagrangian skeleton of the Stein manifold. If the Stein manifold is an open Riemann surface or it has finite type, then that skeleton is independent of the formal solution. This yields the existence of local $h$-principles over that skeleton. These results broaden those obtained by Forstnerič and Slapar on holomorphic immersions, submersions, and complex contact structures for instance to holomorphic local $h$-principles for the corresponding version in the complex category of some other classical examples of distributions and structures in the smooth category such as complex even contact, complex Engel, and complex twisted locally conformal symplectic structures.