Entire holomorphic curves into projective plane intersecting few generic algebraic curves

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 $f\text{\hspace{0.17em}⁣}:\mathbb{C}\to {\mathbb{P}}^2(\mathbb{C})$, we have a Second Main Theorem type estimate: [ T_f® \leq c\sum_{i=1}^q,N_f^{[1]}(r,$\mathcal{C}$_i)

• o\big(T_f® \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 ${\mathbb{P}}^3(\mathbb{C})$.