Abstract:
This talk is based on joint work with Lei Hou, Dinh Tuan Huynh, and Joël Merker. We establish 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, lowering the threshold from $18$ to $17$ for surfaces (Păun) and from $14$ to $12$ for complements (Rousseau).
Central to the proofs are new vanishing results for certain negatively twisted invariant $2$-jet differentials, obtained through a novel combination of algebraic reduction and computer algebra. Since Demailly's Santa Cruz lectures in 1995, the thresholds for the existence of such differentials — and hence the limits of what $2$-jet techniques can achieve for the Kobayashi conjecture in dimension two — have been recognized as $d = 15$ in the compact case and $d = 11$ in the logarithmic case. While previous approaches were insufficient to reach these targets, the present work establishes the theoretical foundations and algorithmic framework that make them accessible, already improving the known bounds to $d = 17$ and $d = 12,13$, respectively.
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:
\[
T_f(r) \leqslant C_d N_f^{[1]}(r, \mathcal{C}) + o(T_f(r)) \quad \parallel,
\]
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.
