On entire curves tangent to foliation
type: Item
instance of: scholarly article
title: On entire curves tangent to foliation
zbMATH Open document ID: 1147.32019
author: Marco Brunella
published in: Journal of Mathematics of Kyoto University
publication date: 2008-06-18T00:00:00Z
review text: Let \(X\) be a complex projective manifold of dimension \(n\), equipped with a holomorphic foliation by curves of dimension one \(\mathcal{F}\), with a possibly nonempty singular set Sing(\(\mathcal{F})\). The paper is devoted to prove that if \(T \subset X^0\) is an embedded \((n-1)\)-disc transverse to \(\mathcal{F}\), and \(S \subset T\) is an embedded \(k\)-disk, \(1 \leq k \leq n-1\), then there exists a (canonical) splitting of \(\varepsilon(S) := S \cap \{ p \in X^0 | \) there exists a non constant \(f: \mathbb{C} \rightarrow X, f(0)=p, \) tangent to \(\mathcal{F}\}\), \(\varepsilon(S) = \mathcal{P}(S) \cup \mathcal{Z}(S)\), where \(\mathcal{Z}(S)\) is thin in \(S\) and \(\mathcal{P}(S)\) is either complete pluripolar in \(S\) or full. Moreover, in the last case (\(\mathcal{P}(S) =S\)) there exists a meromorphic map \(F: S \times \mathbb{C} \rightarrow X\) such that for every \(s \in S\) the restriction \(F(s,\cdot): \mathbb{C} \rightarrow X\) is (after removing indeterminacies) an entire curve tangent to \(\mathcal{F}\), and it sends \(0\) to \(s \in S \subset X\). The author also includes an interesting appendix where the interest of the theorem is explained to prove a weak version of a Lang conjecture (for surfaces) what asserts that if \(Y\) is a smooth complex projective surface of general type, then there exists a pluripolar subset of \(Y\) which contains all the entire curves.
Mathematics Subject Classification ID: 32S65
zbMATH DE Number: 5290072
zbMATH Keywords: complex projective manifold
zbMATH Keywords: pluripolar set
zbMATH Keywords: entire curve
reviewed by: Carlos Galindo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250692286
OpenAlex ID: W1937865346
DOI: 10.1215/KJM/1250692286
description: scientific article; zbMATH DE number 5290072