Higher-dimensional extension of Abate and López-Hernanz–Rosas theorems

Determine whether Abate’s theorem—that every tangent-to-the-identity biholomorphism in complex dimension two has either a curve of fixed points or parabolic curves whose orbits converge tangentially to a characteristic direction—and the López-Hernanz–Rosas theorem—that the same alternative holds for each characteristic direction—remain valid for tangent-to-the-identity biholomorphisms in complex dimensions three and higher.

Background

The paper establishes two results in complex dimension two. Abate’s theorem states that every tangent-to-the-identity biholomorphism either has a curve of fixed points or admits parabolic curves whose orbits converge tangentially to a characteristic direction. The López-Hernanz–Rosas theorem strengthens the directional statement: for every characteristic direction, either a curve of fixed points tangent to that direction exists or there are parabolic manifolds with orbits converging tangentially to it.

The authors explain that several higher-dimensional partial mechanisms are known: non-degenerate characteristic directions yield parabolic curves, and formal invariant curves yield either curves of fixed points or parabolic manifolds. However, examples show that formal invariant curves or non-degenerate characteristic directions need not exist after blow-ups in higher dimension, leaving the full higher-dimensional validity of both theorems unresolved.