Paper
Improved Degree Bounds for Hyperbolicity of Surfaces and Curve Complements
Authors
Lei Hou, Dinh Tuan Huynh, Joël Merker, Song-Yan Xie
Abstract
This paper establishes new degree bounds for Kobayashi hyperbolicity in dimension two. Our main results are: - A very generic surface in $\mathbb{P}^3$ of degree at least $17$ is Kobayashi hyperbolic. - The complement of a generic curve in $\mathbb{P}^2$ of degree at least $12$ is Kobayashi hyperbolic. These bounds improve the long-standing records in the field. For surfaces, the threshold is lowered from Păun's degree $18$ to degree $17$; for complements, it is lowered from Rousseau's degree $14$ to degree $12$. For complements, we prove a stronger, quantitative version of hyperbolicity via a Second Main Theorem in Nevanlinna theory. Specifically, for every generic smooth curve $\mathcal{C} \subset \mathbb{P}^2$ of degree $d \geqslant 12$ and any nonconstant entire holomorphic curve $f \colon \mathbb{C} \to \mathbb{P}^2$, we establish the following inequality: \[ T_f(r) \leqslant C_d \, N_f^{[1]}(r, \mathcal{C}) + o\big(T_f(r)\big) \quad \parallel, \] where $T_f(r)$ is the Nevanlinna characteristic function, $N_f^{[1]}(r, \mathcal{C})$ denotes the $1$-truncated counting function, and $C_d$ is an explicit constant depending only on $d$. The notation ``$\parallel$'' indicates that the estimate holds for all $r>1$ outside a set of finite Lebesgue measure.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.14881v1</id>\n <title>Improved Degree Bounds for Hyperbolicity of Surfaces and Curve Complements</title>\n <updated>2026-03-16T06:28:56Z</updated>\n <link href='https://arxiv.org/abs/2603.14881v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.14881v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>This paper establishes new degree bounds for Kobayashi hyperbolicity in dimension two. Our main results are:\n - A very generic surface in $\\mathbb{P}^3$ of degree at least $17$ is Kobayashi hyperbolic.\n - The complement of a generic curve in $\\mathbb{P}^2$ of degree at least $12$ is Kobayashi hyperbolic.\n These bounds improve the long-standing records in the field. For surfaces, the threshold is lowered from Păun's degree $18$ to degree $17$; for complements, it is lowered from Rousseau's degree $14$ to degree $12$.\n For complements, we prove a stronger, quantitative version of hyperbolicity via a Second Main Theorem in Nevanlinna theory. Specifically, for every generic smooth curve $\\mathcal{C} \\subset \\mathbb{P}^2$ of degree $d \\geqslant 12$ and any nonconstant entire holomorphic curve $f \\colon \\mathbb{C} \\to \\mathbb{P}^2$, we establish the following inequality: \\[ T_f(r) \\leqslant C_d \\, N_f^{[1]}(r, \\mathcal{C}) + o\\big(T_f(r)\\big) \\quad \\parallel, \\] where $T_f(r)$ is the Nevanlinna characteristic function, $N_f^{[1]}(r, \\mathcal{C})$ denotes the $1$-truncated counting function, and $C_d$ is an explicit constant depending only on $d$. The notation ``$\\parallel$'' indicates that the estimate holds for all $r>1$ outside a set of finite Lebesgue measure.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.CV'/>\n <published>2026-03-16T06:28:56Z</published>\n <arxiv:comment>52 pages (main text) + Maple code verification (vanishing lemmas)</arxiv:comment>\n <arxiv:primary_category term='math.CV'/>\n <author>\n <name>Lei Hou</name>\n </author>\n <author>\n <name>Dinh Tuan Huynh</name>\n </author>\n <author>\n <name>Joël Merker</name>\n </author>\n <author>\n <name>Song-Yan Xie</name>\n </author>\n </entry>"
}