Hyperbolicity of the complement of plane curves (Q1068263)

A complex manifold \(X\subseteq {\mathbb{P}}_ 2\) is a hermitian hyperbolic complex manifold if there exists a positive definite continuous hyperbolic hermitian metric on X which is bigger than a positive multiple of Fubini-Study metric. The main result of this paper is the following Theorem. Assume that for a holomorphic curve \(D\subseteq {\mathbb{P}}_ 2\) of genus \(g\geq 2\), \(b^*\) the number of all irreducible singularities of the dual curve \(D^*\) of D, \(\chi\) (D) the Euler number, the inequality \(b^*+\chi (D)<0\) is valid. Moreover assume that every tangent at \(D^*\) intersects \(D^*\) in at least two distinct point. Then \(X={\mathbb{P}}_ 2\setminus D\) is a hermitian hyperbolic complex manifold. The authors note that the proof of a theorem similar to the theorem above given in the paper of \textit{J. A. Carlson} and \textit{M. Green} in Duke. Math. J. 43, 1-9 (1976; Zbl 0333.32022) has a gap.