Entire holomorphic curves into projective plane intersecting few generic algebraic curves

TL;DR

This paper establishes a Second Main Theorem type estimate for entire holomorphic curves intersecting few algebraic curves in the projective plane, using jet differential bundles, and explores algebraic degeneracy and hyperbolicity of complex surfaces.

Contribution

It introduces new conditions involving logarithmic jet differentials to derive value distribution estimates and applies these to hyperbolicity of generic surfaces.

Findings

Second Main Theorem estimate for entire curves intersecting up to 3 curves

Explicit constant c(q,d_i) in the estimate

New results on algebraic degeneracy and hyperbolicity of surfaces

Abstract

For $q\leq 3$ smooth plane algebraic curves $\mathcal{C}_i$ having simple normal crossings, if the invariant logarithmic $2$-jet differential bundle associated to $(\mathbb{P}^2(\mathbb{C}), \sum_{i=1}^q \mathcal{C}_i)$ has a nonzero section vanishing on some ample divisor, then, for every algebraically nondegenerate entire holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^2(\mathbb{C})$, we have a Second Main Theorem type estimate: \[ T_f(r) \leq c\sum_{i=1}^q\,N_f^{[1]}(r,\mathcal{C}_i) + o\big(T_f(r) \big)\parallel, \] where $T_f(r)$ and $N_f^{[1]}(r,C_i)$ stand for the order function and the $1$--truncated counting functions in the Nevanlinna theory, and where the constant $c=c(q,d_i)>0$ can be computed explicitly. In particular, our result includes the case of $3$ conics in $\mathbb{P}^2(\mathbb{C})$. Moreover, we provide some new results concerning the algebraic degeneracy of certain complex surfaces, e.g., the complex hyperbolicity of a very generic surface of degree $\geq 15$ in $\mathbb{P}^3(\mathbb{C})$.