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Locally approximable CR functions, a sharp maximum modulus principle and holomorphic extension

Mauro Nacinovich, Egmont Porten

Abstract

We introduce a notion of locally approximable continuous CR functions on locally closed subsets of reduced complex spaces, generalizing both holomorphic functions and CR functions on CR submanifolds. Under additional assumptions of set-theoretical weak pseudoconcavity we prove optimal maximum modulus principles for these functions. Restricting to real submanifolds (possibly with CR singularities) of complexmanifolds, we generalize results on holomorphic extension known for CR submanifolds.

Locally approximable CR functions, a sharp maximum modulus principle and holomorphic extension

Abstract

We introduce a notion of locally approximable continuous CR functions on locally closed subsets of reduced complex spaces, generalizing both holomorphic functions and CR functions on CR submanifolds. Under additional assumptions of set-theoretical weak pseudoconcavity we prove optimal maximum modulus principles for these functions. Restricting to real submanifolds (possibly with CR singularities) of complexmanifolds, we generalize results on holomorphic extension known for CR submanifolds.
Paper Structure (13 sections, 33 theorems, 134 equations)

This paper contains 13 sections, 33 theorems, 134 equations.

Table of Contents

  1. Set-theoretical pseudo-convexity and concavity
  2. Complex subvarieties and smooth real submanifolds of complex manifolds
  3. Intersections with complex submanifolds
  4. Locally approximable generalized $CR$ functions
  5. Closure operations on sheaves of continuous functions
  6. Locally approximable $CR$ functions
  7. Maximum modulus principle $\mathrm{I}$
  8. Maximum modulus principle $\mathrm{I{I}}$
  9. Applications to $CR$ manifolds.
  10. Holomorphic extension to ambient neighbourhoods
  11. Isolated $CR$ singularities
  12. Massive sets of $CR$ singularities
  13. Some remarks on $CR$ singularities

Key Result

Lemma 1.1

For a point $\mathrm{p}$ of $\mathsf{A}$ the following are equivalent:

Theorems & Definitions (82)

  • Lemma 1.1
  • proof
  • Remark 1.2
  • Definition 1.1
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Corollary 1.5
  • proof
  • Lemma 1.6
  • ...and 72 more