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Postcritically finite endomorphisms I: Kobayashi hyperbolicity and tautness
Abstract
Let f:ℙN→ℙN be a postcritically finite endomorphism of degree at least two. Combining Yamanoi's hyperbolicity results with the dynamical property of f, we prove that the complement of its postcritical divisor is taut unless f is a monomial power map or admits a nontrivial equivariant rational fibration with a polarized divisorially postcritically finite base map. We establish analogous alternatives for polarized PCF endomorphisms of rationally connected projective manifolds and, for Kobayashi hyperbolicity, on normal rationally connected projective varieties. In dimension two, the non-taut case can be described, up to iteration and rational semiconjugacy, by monomial power maps and skew products with monomial fiber maps.
Related
- Postcritically finite endomorphisms II: Fatou sets
- Kobayashi Hyperbolicity of General Surfaces via the Poincaré Problem
- Optimal Linear Dependence on Boundary Type for Local Gromov Hyperbolicity of the Kobayashi Metric
- Kobayashi completeness of pseudoconvex domains: finite type and a smooth counterexample
- Gromov hyperbolicity, finite type, and subellipticity on ℂ-convex domains
BibTeX
@misc{sun2026postcritically,
title = {{Postcritically finite endomorphisms I: Kobayashi hyperbolicity and tautness}},
author = {Ruiran Sun and Shengyuan Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11817},
eprint = {2610.11817},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11817},
}
RIS
TY - UNPB
TI - Postcritically finite endomorphisms I: Kobayashi hyperbolicity and tautness
AU - Ruiran Sun
AU - Shengyuan Zhao
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11817
UR - https://arxiv.org/abs/2610.11817
AB - Let f:ℙ^N→ℙ^N be a postcritically finite endomorphism of degree at least two. Combining Yamanoi's hyperbolicity results with the dynamical property of f, we prove that the complement of its postcritical divisor is taut unless f is a monomial power map or admits a nontrivial equivariant rational fibration with a polarized divisorially postcritically finite base map. We establish analogous alternatives for polarized PCF endomorphisms of rationally connected projective manifolds and, for Kobayashi hyperbolicity, on normal rationally connected projective varieties. In dimension two, the non-taut case can be described, up to iteration and rational semiconjugacy, by monomial power maps and skew products with monomial fiber maps.
ER -