Dynamical Aspects of Foliations with Ample Normal Bundle

Masanori Adachi, Judith Brinkschulte

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The uploaded paper proves Brunella’s conjecture in dimension ≥3 with a three-step strategy: (i) strong pseudoconvexity of X\M (Brunella), (ii) vanishing of the Baum–Bott class to produce a holomorphic connection on NF over X\A via Grauert–Riemenschneider, Serre duality, and Kodaira vanishing, and (iii) L2-Hodge/∂∂̄ on the compact Kähler manifold X to localize a1(NF), obtain a flat metric on X\W, and reach a maximum-principle contradiction; see the main theorem and proof outline and details in §3 . The candidate’s proof mirrors this blueprint but makes two material errors: it (a) misapplies the ∂∂̄-lemma on the non-compact open set X\M (the paper instead works on the compact X via zero-extension), and (b) claims global flatness of NF on X\M rather than on X\W containing M. These gaps prevent the model’s argument from being complete/correct.

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript resolves a conjecture of Brunella in dimension ≥3 via a robust blend of convexity, Baum–Bott theory, and L2-Hodge methods on complete Kähler manifolds. The argument is carefully structured: it isolates a holomorphic connection after vanishing of the Baum–Bott class, localizes the first Atiyah class, and uses the ∂∂̄-lemma on the compact ambient manifold to produce a flat metric on the complement of a strongly pseudoconvex neighborhood of the maximal compact analytic set. The final contradiction via a strictly plurisubharmonic exhaustion is clean and convincing.