The modified Cartan conjecture

In 1928 H. Cartan stated a conjecture about holomorphic curves in ${\mathbb{P}}^n$ parametrized by the unit disc, omitting $n+2$ hyperplanes in general position. He proved it for $n=2$. In 1996 the first-named author constructed counterexamples for all $n\ge 3$, and proposed a modified form of Cartan’s conjecture which he proved for $n=3$. In this paper we prove this modified conjecture in all dimensions. As a byproduct we prove a quantitative version of the classical theorem that analytic functions are linearly dependent if and only if their Wronskian determinant is zero. We interpret our results in terms of the Kobayashi–Royden pseudometric on the complement of $n+2$ hyperplanes, and on smooth algebraic varieties in the algebraic torus.