A Second Main Theorem for Entire Curves Intersecting Three Conics

Abstract

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 provides a systematic method for proving vanishing results for negatively twisted jet differentials--a key component in complex hyperbolic geometry--thereby filling a missing piece in two-dimensional complex hyperbolicity theory.