A Second Main Theorem for Entire Curves Intersecting Three Conics

TL;DR

This paper proves a Second Main Theorem for entire curves intersecting three generic conics in the complex projective plane, using invariant jet differentials and novel vanishing lemmas to advance understanding in complex hyperbolic geometry.

Contribution

It introduces new vanishing lemmas for negatively twisted jet differentials via a combination of algebraic geometry and computer-assisted methods, advancing techniques in complex hyperbolic geometry.

Findings

Establishes a Second Main Theorem estimate for entire curves and three conics.

Develops new vanishing lemmas for jet differentials using a mod-p reduction approach.

Provides a systematic method for proving vanishing results in complex hyperbolic geometry.

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 yields a systematic method for proving vanishing results for negatively twisted jet differentials -- a key component in complex hyperbolic geometry.