LOG CALABI-YAU STRUCTURE OF PROJECTIVE THREEFOLDS ADMITTING POLARIZED ENDOMORPHISMS

Sheng Meng

correcthigh confidence

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

Audit review

The paper proves that every smooth projective threefold admitting a polarized endomorphism is of Calabi–Yau type (Theorem 1.2), via an f-equivariant MMP to a Q-abelian base, canonical bundle formula, surface analysis, and a detailed treatment of the residual P^2-bundle over an elliptic curve case, including an anti-canonical/Iitaka analysis and a final elimination-by-contradiction step . The candidate solution follows the same overall strategy and reaches the same conclusion. The only substantive nit is a minor overstatement in the pseudo-effective K case (it asserts K_X ~_Q 0 on the original X without adding a boundary), but this is easily fixed by standard birational pullback of a log Calabi–Yau boundary. Overall, both are correct and the proofs are essentially the same in structure.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This work confirms the Broustet–Gongyo conjecture for smooth threefolds, combining an f-equivariant MMP with the lc canonical bundle formula and refined analysis of the residual P\^2-bundle case. The method is conceptually robust and likely extensible. The exposition is clear overall, though a few reductions would benefit from brief justifications and explicit references (e.g., the dim Y=3 birational conclusion, and the pseudo-effective branch).