Local H-Principles for Partial Holomorphic Relations

In this paper we introduce the notion of the realifications of an arbitrary \emph{partial holomorphic relation}. Our main result states that if any realification of an open partial holomorphic 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 neighbourhood 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 F. Forstneri\v{c} and M. Slapar on holomorphic immersions, submersions and complex contact structures for instance to holomorphic local $h$--principles for complex even contact, complex Engel or complex conformal symplectic structures.