LEVI-FLATS IN CPn: A SURVEY FOR NONEXPERTS

Rasul Shafikov

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper (survey) correctly outlines Siu’s proof for n ≥ 3: extend the Levi foliation, use positivity of the induced normal bundle to build a global leafwise strictly plurisubharmonic potential via a ∂-solvability step on Stein complements (which crucially uses n ≥ 3), and then contradict the maximum principle on leaves . The model’s solution mirrors the curvature-positivity core but wrongly asserts the existence of a globally defined potential φ = log||∂ρ||^2 on M; without globalizing this potential (the delicate cohomological step that fails in n = 2), the harmonic-measure contradiction cannot be applied. Thus, the model proof is incomplete at a critical step, whereas the paper’s argument is complete.

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The survey accurately synthesizes multiple proofs of the nonexistence of real-analytic Levi-flats in CP\^n for n>2, with a clear, correct outline of Siu’s method that foregrounds the decisive cohomological step on Stein complements and the dimension restriction. Its explanations are accessible to nonexperts yet technically reliable, with appropriate references. No corrections are necessary, though a few cross-references and clarifications could further aid readers.