Invariant Measures for Large Automorphism Groups of Projective Surfaces

The paper (Cantat–Dujardin, arXiv:2110.04213) states and proves a full classification (Theorem A) of ergodic Γ-invariant probability measures on compact Kähler surfaces when Γ is non‑elementary and contains a parabolic, into exactly one of four mutually exclusive cases (a)–(d), including the real‑analytic regularity of densities in (c) and (d). This appears explicitly in the Introduction and Theorem A, with detailed proofs across Sections 3–5 and Appendix A (tori) . The paper further proves uniqueness of the absolutely continuous measure (Lemma 7.1) and a finiteness alternative (Theorem C) . By contrast, the candidate solution asserts that the global exhaustiveness of (a)–(d) is likely open as of 2021‑10‑08, which contradicts the paper’s complete proof and dated result. The model’s partial reasoning (e.g., fibration reduction, disintegration, torus case) aligns with ingredients used in the paper, but its main conclusion is incorrect. The paper also clarifies key technical points (e.g., parabolic implies an invariant genus‑1 fibration, and the tangency analysis and gluing that ensure global real‑analytic structure), see Theorem 3.1 and surrounding development .