Entire holomorphic curves into projective plane intersecting few generic algebraic curves

Abstract

For $q\le 3$ 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 $f:\mathbb{C}\to {\mathbb{P}}^2(\mathbb{C})$, 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 $\ge 15$ in ${\mathbb{P}}^3(\mathbb{C})$.

There is a gap in the proof of Theorem 2.1. We withdraw the article, at least for the time being