A Second Main Theorem for Entire Curves Intersecting Three Conics

We establish a Second Main Theorem for entire holomorphic curves $f:\mathbb{C}\to {\mathbb{P}}^2$ intersecting a generic configuration of three conics $\mathcal{C}={\mathcal{C}}_1+{\mathcal{C}}_2+{\mathcal{C}}_3$ in the complex projective plane ${\mathbb{P}}^2$. Using invariant logarithmic $2$-jet differentials with negative twists, we prove the estimate $$T_f(r)⩽5\sum _{i=1}^3{N}_f^{[1]}(r,{\mathcal{C}}_i)+o(T_f(r))\parallel ,$$ where $T_f(r)$ is the Nevanlinna characteristic function, and $N_f^{[1]}(r,{\mathcal{C}}_i)$ is the $1$-truncated counting function. The key innovation of our approach is establishing new vanishing lemmas of the form $$H^0({\mathbb{P}}^2,E_{2,m}{T}_{{\mathbb{P}}^2}^{\ast }(\log \mathcal{C})\otimes {\mathcal{O}}_{{\mathbb{P}}^2}(-t))=0$$ for specific pairs $(m,t)$, achieved by combining algebro-geometric arguments with computer-assisted computations through a mod-$p$ reduction technique. This yields a systematic method for proving vanishing results for negatively twisted jet differentials – a key component in complex hyperbolic geometry.