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Local H-Principles for Partial Holomorphic Relations

Luis Giraldo, Guillermo Sánchez Arellano

Abstract

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. Forstnerič 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.

Local H-Principles for Partial Holomorphic Relations

Abstract

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 --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 --principles over that skeleton. These results broaden those obtained by F. Forstnerič and M. Slapar on holomorphic immersions, submersions and complex contact structures for instance to holomorphic local --principles for complex even contact, complex Engel or complex conformal symplectic structures.
Paper Structure (24 sections, 25 theorems, 40 equations)

This paper contains 24 sections, 25 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Holomorphic partial differential relations and their h--principles
  3. Review of the smooth case.
  4. Holomorphic partial differential relations
  5. Types of h--principles for germs of sections of holomorphic partial differential relations
  6. Statements of the Theorems
  7. H--principles for germs over skeletons of finite type.
  8. Theorems for Stein manifolds of arbitrary type
  9. The case of Riemann surfaces
  10. Holomorphic approximation
  11. Adapted skeletons and properly attached sets
  12. Parametric Mergelyan Theorem for sections of holomorphic vector bundles
  13. Proofs of the Theorems
  14. Extending holonomy and pseudo-holonomy through totally real submanifolds
  15. Extending holonomy and pseudo-holonomy through adapted skeletons
  16. ...and 9 more sections

Key Result

Proposition 2.7

Let $\pi:X\to B$ be a holomorphic vector bundle and $\mathcal{R}\subset X_\mathbb{C}^{(1)}$ be a holomorphic partial differential relation. If $\mathcal{R}$ is a THR then it has ample realifications.

Theorems & Definitions (83)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • Definition 2.9
  • ...and 73 more