Vanishing of Invariant 2-Jet Differentials and Improved Hyperbolicity Degree Bounds in Dimension Two

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 {\em 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 (P\u{a}un) 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 consequently the limits of what $2$-jet techniques can accomplish toward 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 unable to reach these targets, the present work provides both the theoretical foundations and the algorithmic framework required to access them, and has already improved the known bounds to $d = 17$ and $d = 12, 13$, respectively. As an unexpected byproduct, our computational method reveals the existence of nonzero negatively twisted invariant $2$-jet differentials with $(m,t) = (3,1)$ for hyperelliptic-type equations of degree at least $11$ in the logarithmic case and degree at least $15$ in the compact case, further illuminating the geometry of these special jet differentials.