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 P 2 \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 : C ↦ P n f:\mathbb C\mapsto \mathbb P^n and two homogeneous polynomials in n + 1 n+1 variables with coefficients which are meromorphic functions of the same growth as the analytic map f f . As applications, we study some cases of Campana’s orbifold conjecture for P 2 \mathbb P^2 and finite ramified covers of P 2 \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 G m 2 \mathbb G_m^2 .