Logarithmic vector fields and hyperbolicity

Abstract

Using vector fields on logarithmic jet spaces we obtain some new positive results for the logarithmic Kobayashi conjecture about the hyperbolicity of complements of curves in the complex projective plane. We are interested here in the cases where logarithmic irregularity is strictly smaller than the dimension. In this setting, we study the case of a very generic curve with two components of degrees $d_{1} \leq d_{2}$ and prove the hyperbolicity of the complement if the degrees satisfy either $d_{1} \geq 4$, or $d_{1} = 3$ and $d_{2} \geq 5$, or $d_{1} = 2$ and $d_{2} \geq 8$, or $d_{1} = 1$ and $d_{2} \geq 11$. We also prove that the complement of a very generic curve of degree $d$ at least equal to 14 in the complex projective plane is hyperbolic, improving slightly, with a new proof, the former bound obtained by El Goul.