ON THE STABILITY OF HOLOMORPHIC FAMILIES OF ENDOMORPHISMS OF Pk

François Berteloot, Xavier Buff

correcthigh confidence

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

Audit review

The uploaded 2025 note proves (i) μ-stability implies weak stability for arbitrary holomorphic families on P^k (Proposition 1.2) and (ii) on simply connected parameter spaces, the seven notions listed—pluriharmonicity of L, postcritical mass bound, ramification current convergence, μ-stability, weak stability, asymptotically weak stability, and partially weak stability—are all equivalent (Theorem 4.1). This directly contradicts the model’s claim that μ-stable ⇔ weakly stable was only known in the universal family or for k=1 and was “likely open as of cutoff.” The paper’s proofs are coherent and close the full cycle of implications 1⇒2⇒3⇒4⇒5⇒6⇒7⇒1, using the dd^cL formula and the Lyapunov approximation formula to tie the dynamics to L, and existing results on equilibrium webs/laminations. Hence the paper is correct and the model’s status assessment is outdated.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This note resolves a widely anticipated equivalence among key stability notions for holomorphic endomorphisms of projective space, extending previous partial results to arbitrary families and giving a compact proof architecture. The arguments are clean and rely on well-established tools (dd\^cL, Lyapunov approximants, and the web/lamination correspondence). A few globalization details are concise and could benefit from an explicit reference or lemma, but the mathematical content is sound and important for the field.