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Logarithmic vector fields and hyperbolicity. (English) Zbl 1189.32015

The paper under review deals with the logarithmic Kobayashi conjecture on the hyperbolicity of complements of curves in the complex projective plane.
The author uses vector fields on logarithmic jet spaces and obtains some positive results as follows. Let \(C\) be a irreducible curve in complex projective plane \(\mathbb P^2\) of degree \(d\). If \(C\) is very generic and \(d \geq 14\), then \(\mathbb P^2\setminus C\) is hyperbolic and hyperbolically embedded in \(\mathbb P^2\).
This gives an improvement of J. El Goul’s results [Osaka J. Math. 40, No. 2, 469–491 (2003; Zbl 1048.32016)], that is, \(d\geq 15\) in El Goul’s paper. The technique of the proof of this result is different from El Goul’s. For the hyperbolicity of complements of reducible curves in \(\mathbb P^2\), the author also proves the following:
Let \(C=C_1 \cup C_2\) be a very generic curves in \(\mathbb P^2\) having two irreducible components \(C_1\) and \(C_2\) with degree \(d_2\geq d_1\). Then \(\mathbb P^2\setminus C\) is hyperbolic and hyperbolically embedded in \(\mathbb P^2\) if the degrees satisfy either \(d_1 \geq 4\), or \(d_1=3\) and \(d_2 \geq 5\), or \(d_1=2\) and \(d_2 \geq 8\), or \(d_1=1\) and \(d_2 \geq11\).
The proof of this theorem is based on the same techniques as in his previous paper [C. R., Math., Acad. Sci. Paris 336, No. 8, 635–640 (2003; Zbl 1034.32017)].

MSC:

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
14J70 Hypersurfaces and algebraic geometry
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References:

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