[Paper Review] Weak normality of families of meromorphic mappings and bubbling in higher dimensions

S. Ivashkovich, Fethi Neji|arXiv (Cornell University)|Apr 20, 2011

Meromorphic and Entire Functions35 references3 citations

TL;DR

This paper investigates weak and gamma convergence of families of meromorphic mappings between complex manifolds, proving that their sets of normality are pseudoconvex under certain conditions. It identifies that exceptional components in weak/gamma limits are rationally connected, enabling detection of failure of strong convergence; applications to Fatou sets of meromorphic self-maps on compact complex surfaces are provided.

ABSTRACT

Minor corrections, to appear in Annali SNS.

Motivation & Objective

• To determine whether sets of normality for families of meromorphic mappings are pseudoconvex under different convergence types.
• To analyze the structure of exceptional components arising in weak and gamma convergent sequences that fail to converge strongly.
• To characterize the failure of strong convergence via geometric properties of limit components.
• To apply the results to the dynamics of meromorphic self-maps on compact complex surfaces, particularly in relation to Fatou sets.

Proposed method

• Introduces and compares three convergence types: strong, weak (w-convergence), and gamma (Γ-convergence), with emphasis on their topological and geometric properties.
• Uses the graph convergence in the Hausdorff metric and cycle topology to analyze convergence behavior.
• Applies projective representation via homogeneous coordinates to describe convergence in terms of uniform convergence of local holomorphic representatives.
• Employs volume bounds on preimages of divisors and integration of pullbacks of Kähler forms to detect bubbling and non-degeneracy in limits.
• Analyzes blow-ups at indeterminacy points to study the dynamical behavior over exceptional curves, particularly in the context of Nori strings.
• Uses explicit examples with rational maps of high degree to demonstrate bubbling phenomena and non-trivial limit structures.

Experimental results

Research questions

• RQ1 Are the sets of normality for weakly or gamma-convergent families of meromorphic mappings pseudoconvex?
• RQ2 What geometric structure do the exceptional components of weak/gamma limits possess?
• RQ3 Under what conditions does a weakly or gamma-convergent sequence fail to converge strongly?
• RQ4 How does bubbling manifest in the limit of meromorphic mappings, and what does it imply about the dynamics?
• RQ5 To what extent do the Fatou sets of meromorphic self-maps coincide with the sets of strong or weak normality?

Key findings

• The sets of normality for weak and gamma convergence are pseudoconvex for a large class of target manifolds.
• Exceptional components in weak and gamma limits are rationally connected, which characterizes the failure of strong convergence.
• Bubbling occurs when the pullback of a Kähler form has non-degenerate total mass, even if the pullback of the form itself degenerates.
• In the example with degree-d maps, the integral of the pullback of the Kähler form remains bounded away from zero, indicating persistent bubbling over a disk.
• Over the indeterminacy point, repeated blow-ups reveal a Nori string of rational curves, with the map sending the final exceptional curve bijectively to a line with multiplicity.
• For algebraically stable maps, the Fatou set coincides with the set of strong normality, as shown by Maegawa's result.