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 $$T_f(r) \leqslant 5 \sum_{i=1}^3 N_f^{[1]}(r, \mathcal{C}_i) + o\big(T_f(r)\big)\quad\parallel,$$ 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 $$H^0\bigl(\mathbb{P}^2,\, E_{2,m}T_{\mathbb{P}^2}^*(\log \mathcal{C}) \otimes \mathcal{O}_{\mathbb{P}^2}(-t)\bigr) = 0$$ 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.