Rational Endomorphisms of Codimension One Holomorphic Foliations

The paper’s Theorem A is established via the LPT16 structure theorem and a hyperbolic/monodromy route: it reduces to the transversely hyperbolic case with Zariski‑dense monodromy and then proves that any rational symmetry must preserve the foliation, forcing finiteness of the transverse action unless the foliation is virtually additive or a surface pullback, which also leads to virtual additivity (Theorem A statement and proof sketch appear explicitly; see , , , ). The candidate solution hinges on two unsupported steps: (i) that the equivariance ρ∘f_* = gρg^{-1} with g of infinite order rules out Zariski‑dense monodromy and forces virtual reducibility; (ii) that f_* has finite‑index image in π1(U), enabling a dynamical argument to make the linear part finite. Neither claim is justified in the paper’s framework; in fact, the paper works precisely with Zariski‑dense monodromy in the hyperbolic case and uses uniqueness of hyperbolic structures to control symmetries (Theorem 5.10 and Theorem 5.13; see , ). The model also ignores regular vs. irregular singularity issues needed when passing from transversely affine to additive. Hence the model’s proof is incomplete/incorrect, while the paper’s argument is coherent and complete.