On some foliations on ruled surfaces
type: Item
instance of: scholarly article
title: On some foliations on ruled surfaces
zbMATH Open document ID: 0809.53029
author: Akihiro Saeki
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-08-08T00:00:00Z
review text: Let \(X\) be a two-dimensional complex manifold, \(\theta_ X\) the sheaf of germs of holomorphic functions on \(X\) and \(\Theta_ X\) the sheaf of germs of holomorphic vector fields on \(X\). \(C\) a closed Riemann surface, \(\mathcal F\) a one-dimensional foliation on \(X\) (an invertible subsheaf of \(\Theta_ X\) with singular locus of codimension strictly greater than one), and \(N\) a submanifold of \(X\), invariant to respect to \(\mathcal F\) (then defined by a coherent ideal sheaf \({\mathcal J} \subset \Theta_ X\) such that every \(p \in N\), \({\mathcal F}_ p {\mathcal J}_ p \subset {\mathcal J}_ p\)). Let \(X @>\pi>> C\) be a ruled surface, then \(\pi\) a \(\mathbb{P}^ 1\)-bundle and let be \(f\) their fibre. Denote \(C_ 0\) a section from \(\pi\). Let be \(C^ 2_ 0 = \) the minimum of self-intersection numbers of sections of \(\pi\) and take \(e = -C_ 0^ 2\). If \(\text{Num }X\) denotes the group of numerically equivalent classes of divisors on \(X\), from the commutative diagram of the cohomology long exact sequences associate to the exponential sequences on \(X\) and \(C\), follow the isomorphism \(\text{Num }X = H^ 2(X, \mathbb{Z}) = \mathbb{Z} c (C_ 0) \oplus \mathbb{Z} c(f) = \mathbb{Z} \oplus \mathbb{Z} = \mathbb{Z}^ 2\), where \(c(C_ 0)\) and \(c(f)\) are the images by the Chern map of the holomorphic line bundle defined by the divisors \(C_ 0\) and \(f\). Let \(C_ 1 =_{\text{num}} aC_ 0 + bf\) be an irreducible curve on \(X\), with ``\(=_{\text{num}}\)'' the numerically equivalence of divisors on \(X\). The author states the following Main Theorem. Let \(X @>\pi>> C\), \(C_ 0\) and \(f\) as above. Assume that a foliation \(\mathcal F \subset \Theta_ X\) on \(X\) leaves an irreducible curve \(C_ 1 =_{\text{num}} aC_ 0 + bf\) with \(a > 0\) on \(X\) invariant and has no singularities on \(C_ 1\). Then one of the following is the case. I) \(e = 0\) and \(b = 0\). II) \(e < 0\), \(a = 2\) and \(b = 1/2e\) \(a \in \mathbb{Z}\). The demonstration follows…
reviewed by: J.L.Viviente
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 32S65
zbMATH DE Number: 597322
zbMATH Keywords: ruled surface
zbMATH Keywords: Chern map
zbMATH Keywords: Camacho-Sad index formula
MaRDI profile type: Publication
full work available at URL: pjaa.70.17
OpenAlex ID: W2041743742
cites work: Singularities of holomorphic foliations
cites work: Invariant varieties through singularities of holomorphic vector fields
cites work: Q4722336
cites work: Q3782016
cites work: Holomorphic Foliations in Ruled Surfaces
cites work: Q4143433
cites work: On foliations on complex spaces. II
cites work: On foliation on complex spaces
cites work: Q3719039
cites work: Q4218742
cites work: On ruled surfaces of genus 1
DOI: 10.3792/PJAA.70.17
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description: scientific article; zbMATH DE number 597322