[Paper Review] Continuity principle and extension properties of meromorphic mappings with values in non K\"ahler manifolds

Sergei Ivashkovich|arXiv (Cornell University)|Apr 17, 1997

Meromorphic and Entire Functions3 citations

TL;DR

This paper establishes a non-Kähler analogue of the Levi Continuity Principle for meromorphic mappings into complex spaces, proving that such mappings extend across analytic sets of codimension ≥2 under bounded cycle geometry conditions. It further identifies geometric obstructions to singularity removal in spaces with pluriclosed Hermitian metrics, generalizing Hartogs-type extension theorems beyond the Kähler setting.

ABSTRACT

. In this paper we are proving an analogue of E. Levi Continuity Principle for meromorphic mappings with values in general complex spaces. We also describe the singularities of meromorphic mappings into complex spaces carrying pluriclosed Hermitian metric forms and geometrical obstructions to removing them. Contents 0. Introduction 2 0.1. Continuity principle 2 0.2. Hartogs-type extension theorem and spherical shells 5 0.3. Kahler case: Griffiths approach revisited 8 0.4. Applications, generalisations, open questions 9 1. Continuity principle. 11 1.1. Cycle space associated to a meromorphic map 11 1.2. Analyticity of C f;C 14 1.3. Proof of the Continuity principle 15 1.4. Spaces from G k have bounded cycle geometry 17 1.5. Construction of Example 1 18 2. Hartogs-type extension and spherical shells 19 2.1. Generalities on pluripotential theory 19 2.2. Metrics on complex spaces and proof of Theorem 2 in dimension two 22 2.3. Meromorphic families of analytic sets 26 2.4. Proof of Theorem...

Motivation & Objective

• To generalize the Levi Continuity Principle to meromorphic mappings with values in non-Kähler complex spaces.
• To investigate the structure and analyticity of cycle spaces associated with meromorphic maps in non-Kähler geometry.
• To characterize singularities of meromorphic mappings into complex spaces equipped with pluriclosed Hermitian metrics.
• To identify geometric obstructions preventing the removal of isolated singularities in such mappings.
• To extend Hartogs-type extension theorems to non-Kähler settings using pluripotential theory and meromorphic families of analytic sets.

Proposed method

• Constructs the cycle space $ C_f $ associated with a meromorphic map $ f $, proving its analyticity in the ambient complex space.
• Applies bounded cycle geometry conditions on spaces from $ G_k $ to control the behavior of meromorphic mappings.
• Employs pluripotential theory to analyze plurisubharmonic functions and their role in extension properties.
• Uses meromorphic families of analytic sets to model the extension of mappings across analytic sets of codimension ≥2.
• Analyzes the interplay between pluriclosed Hermitian metrics and the geometry of singularities in complex spaces.
• Proves extension theorems in dimension two by leveraging metric-induced curvature and analytic capacity conditions.

Experimental results

Research questions

• RQ1 Can the Levi Continuity Principle be extended to meromorphic mappings with values in non-Kähler complex spaces?
• RQ2 What geometric conditions on the target space ensure the analyticity of the associated cycle space?
• RQ3 How do pluriclosed Hermitian metrics influence the removable singularity problem for meromorphic mappings?
• RQ4 What are the obstructions to extending meromorphic mappings across analytic sets in non-Kähler manifolds?
• RQ5 To what extent can Hartogs-type extension theorems be generalized beyond the Kähler case?

Key findings

• The cycle space $ C_f $ associated with a meromorphic map $ f $ is analytic in the ambient complex space, enabling the application of analytic continuation techniques.
• Meromorphic mappings into spaces with bounded cycle geometry extend across analytic sets of codimension ≥2, generalizing classical Hartogs' theorem.
• In the presence of a pluriclosed Hermitian metric, singularities of meromorphic mappings are obstructed by geometric invariants tied to curvature and pluripotential theory.
• The paper constructs an explicit example (Example 1) demonstrating the failure of extension when cycle geometry is unbounded.
• In dimension two, the extension of meromorphic mappings is guaranteed under the existence of a pluriclosed Hermitian metric and suitable analytic capacity bounds.
• The results provide a non-Kähler framework for understanding extension phenomena, extending Griffiths' approach beyond Kähler geometry.