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PreprintCC BY 4.0arXiv

Degree growth of rational and meromorphic self-maps

  • Fei Hu
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Abstract

Let f be a dominant rational self-map of a normal projective variety over an algebraically closed field of arbitrary characteristic, or a dominant meromorphic self-map of a compact Kähler manifold. We prove that in every codimension, the degree sequence of f differs from its dynamical-degree growth by at most a polynomial factor, with an explicit universal bound on the polynomial exponent. In particular, subexponential but superpolynomial degree growth cannot occur. At a strict dynamical-degree break, we prove the sharper asymptotic degi,ω(fn)=Cf,ω,i λi(f)n+O(ηn), for every √(λi−1(f)λi+1(f))<η<λi(f). This extends the degree-asymptotic theorem of Dang--Favre from codimension one in characteristic zero to arbitrary codimension and arbitrary characteristic, and recovers the surface theorem of Boucksom--Favre--Jonsson. Both results rely on a uniform estimate for the temporal discrete logarithmic Laplacian of the degree sequences, obtained by applying the Ptolemy inequality to ordered Hodge--Lorentz packages on varying finite-time orbit spaces.

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Cite

BibTeXRIS
BibTeX
@misc{hu2026degree,
  title = {{Degree growth of rational and meromorphic self-maps}},
  author = {Fei Hu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18094},
  eprint = {2610.18094},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18094},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Degree growth of rational and meromorphic self-maps
AU  - Fei Hu
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18094
UR  - https://arxiv.org/abs/2610.18094
AB  - Let f be a dominant rational self-map of a normal projective variety over an algebraically closed field of arbitrary characteristic, or a dominant meromorphic self-map of a compact Kähler manifold. We prove that in every codimension, the degree sequence of f differs from its dynamical-degree growth by at most a polynomial factor, with an explicit universal bound on the polynomial exponent. In particular, subexponential but superpolynomial degree growth cannot occur. At a strict dynamical-degree break, we prove the sharper asymptotic deg_i,ω(f^n)=C_f,ω,i λ_i(f)^n+O(η^n), for every √(λ_i−1(f)λ_i+1(f))<η<λ_i(f). This extends the degree-asymptotic theorem of Dang--Favre from codimension one in characteristic zero to arbitrary codimension and arbitrary characteristic, and recovers the surface theorem of Boucksom--Favre--Jonsson. Both results rely on a uniform estimate for the temporal discrete logarithmic Laplacian of the degree sequences, obtained by applying the Ptolemy inequality to ordered Hodge--Lorentz packages on varying finite-time orbit spaces.
ER  -