Local H-Principles for Partial Holomorphic Relations

Luis Giraldo, Guillermo Sánchez Arellano

Published 2023-04-15Version 1

In this paper we introduce the notion of the realifications of an arbitrary partial holomorphic relation. Our main result is that if the realifications of an open partial holomorphic relation over an Stein manifold satisfy 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 CW-complex with the same homotopy type of the Stein manifold. If the Stein manifold is an open Riemann surface or it has finite type, then that CW-complex is independent of the formal solution. This yields to the existence of local h-principles over that CW-complex. These results broadens F. Forstneri\v{c} and M. Slapar results 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.