Kobayashi Hyperbolicity of General Surfaces via the Poincaré Problem
Abstract
We prove that a general surface in ℙ3 of degree at least $18$ contains no rational or elliptic curves, strengthening the classical result of Clemens by replacing the original very general assumption by a genuine Zariski-open condition. Previously, nonexistence results in the ``general'' setting were known only in much higher degrees. Combining this with established algebraic degeneracy results for entire curves, we deduce the Kobayashi hyperbolicity of a general surface in ℙ3 of degree at least $18$, thereby resolving a question asked by Demailly--El Goul. Our proof uses foliations induced by $2$-jet differentials. Two independent such differentials give rise to a multi-foliation tangent to all rational and elliptic curves. We establish a Poincaré-type bound for its algebraic leaves, yielding a mechanism to upgrade very general statements to general ones. Our method also applies to complements of plane curves. In particular, we prove that the complement of two general cubic curves in ℙ2 is hyperbolically embedded.
Related
- Postcritically finite endomorphisms I: Kobayashi hyperbolicity and tautness
- Optimal Linear Dependence on Boundary Type for Local Gromov Hyperbolicity of the Kobayashi Metric
- Hierarchical hyperbolicity, ball quotients, and the moduli space of smooth cubic surfaces
- Non-Kähler Hyperbolicity: Surfaces and Codimension One
- Kobayashi's problem for power semigroups of inverse semigroups
BibTeX
@misc{xie2026kobayashi,
title = {{Kobayashi Hyperbolicity of General Surfaces via the Poincaré Problem}},
author = {Song-Yan Xie and Shengyuan Zhao},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06644},
eprint = {2609.06644},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05749189v1},
}
RIS
TY - UNPB
TI - Kobayashi Hyperbolicity of General Surfaces via the Poincaré Problem
AU - Song-Yan Xie
AU - Shengyuan Zhao
PY - 2026
DA - 2026/09/06
PB - HAL
DO - 10.48550/arxiv.2609.06644
UR - https://hal.science/hal-05749189v1
AB - We prove that a general surface in ℙ^3 of degree at least $18$ contains no rational or elliptic curves, strengthening the classical result of Clemens by replacing the original very general assumption by a genuine Zariski-open condition. Previously, nonexistence results in the ``general'' setting were known only in much higher degrees. Combining this with established algebraic degeneracy results for entire curves, we deduce the Kobayashi hyperbolicity of a general surface in ℙ^3 of degree at least $18$, thereby resolving a question asked by Demailly--El Goul. Our proof uses foliations induced by $2$-jet differentials. Two independent such differentials give rise to a multi-foliation tangent to all rational and elliptic curves. We establish a Poincaré-type bound for its algebraic leaves, yielding a mechanism to upgrade very general statements to general ones. Our method also applies to complements of plane curves. In particular, we prove that the complement of two general cubic curves in ℙ^2 is hyperbolically embedded.
ER -