CHOW’S THEOREM FOR REAL ANALYTIC LEVI-FLAT HYPERSURFACES

The paper’s Theorem 1 states exactly the claim to be proved and establishes it by (i) proving in P^2 that an infinite family of same‑degree algebraic leaves lies in a pencil (Proposition 3.1), (ii) slicing by generic planes Π ≅ P^2 to force a global pencil in P^n, and (iii) using a tangency‑set argument to identify the extended holomorphic foliation with the Levi foliation; it then notes as a consequence that M is semialgebraic. These steps are visible in the Introduction and the proof of Theorem 1 and Proposition 3.1, together with the slicing and tangency arguments . The candidate solution mirrors this blueprint almost verbatim: local Cartan normal form, identification of a fixed degree, the Bézout–pencil lemma in P^2, slicing to obtain a global pencil, and a tangency argument that the extended foliation coincides with the Levi foliation. Minor differences include (a) an overly strong claim about overlap changes of local first integrals being “real affine” (the paper only needs that they preserve real values along Mreg), and (b) a nonstandard route to semialgebraicity via a single Möbius normalization of the rational first integral; the paper states the semialgebraic consequence and elsewhere leverages known results in that direction. Overall, both are correct, with essentially the same proof skeleton and only minor points requiring clarification .