A complex case of Vojta’s general abc conjecture and cases of Campana’s orbifold conjecture

Abstract:

We show a truncated second main theorem of level one with explicit exceptional sets for analytic maps into $\mathbb P^2$ intersecting the coordinate lines with sufficiently high multiplicities. The proof is based on a greatest common divisor theorem for an analytic map $f:\mathbb C\mapsto \mathbb P^n$ and two homogeneous polynomials in $n+1$ variables with coefficients which are meromorphic functions of the same growth as the analytic map $f$. As applications, we study some cases of Campana’s orbifold conjecture for $\mathbb P^2$ and finite ramified covers of $\mathbb P^2$ with three components admitting sufficiently large multiplicities. In addition, we explicitly determine the exceptional sets. Consequently, it implies the strong Green-Griffiths-Lang conjecture for finite ramified covers of $\mathbb G_m^2$.