zbMATH — the first resource for mathematics

Hyperbolicity of the complement of plane curves. (English) Zbl 0581.32031
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 J. A. Carlson and M. Green in Duke. Math. J. 43, 1-9 (1976; Zbl 0333.32022) has a gap.
Reviewer: G.A.Soifer

MSC:
 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 32Sxx Complex singularities 53A35 Non-Euclidean differential geometry 14H45 Special algebraic curves and curves of low genus 30F30 Differentials on Riemann surfaces
Full Text:
References:
 [1] Brieskorn, E., Knörrer, H.: Ebene algebraische Kurven. Birkhäuser {$$\cdot$$} Basel {$$\cdot$$} Boston {$$\cdot$$} Stuttgart. 1981 [2] Carlson, J.A., Green, M.: Holomorphic Curves in the Plane. Duke Math. J. 43. 1976 · Zbl 0333.32022 [3] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. John Wiley. 1978 · Zbl 0408.14001 [4] Grauert, H., Reckziegel, H.: Hermitesche Metriken und normale Familien holomorpher Abbildungen. Math. Z. 89. 1965 · Zbl 0135.12503 [5] Kobayashi, S.: Intrinsic Distances, Measures and Geometric Function Theorey. B.U.M.S. 82. 1976 · Zbl 0346.32031 [6] Royden, H.L.: Remarks on the Kobayashi Metric. Several Complex Variables II. Lecture Notes in Math. 185. Springer-Verlag, Berlin. 1971 · Zbl 0218.32012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.